JUST SOLVE IT

COMBINATORICS

permutation - grouped

professor chang's books

(2018 AMC8) Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?

(2018 AMC8) Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture. If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column?

permutation – unique case work without exception

(2017 AIME I) Consider arrangements of the 9 numbers 1,2,3,4,5,6,7,8,9 in a 3x3 array. For each such arrangement, let a, b, and c be the medians of the numbers in rows 1, 2, and 3, respectively, and let m be the median of {a,b,c}. Find the number of arrangements for which m=5.

counting shapes - rectangles between lines

rectangles on a chess board

(1997 AIME) The nine horizontal and nine vertical lines on an 8x8 checkerboard form r rectangles, of which s are squares. Find the ratio of s to r.

counting shapes - triangles with positive area

(2017 AIME I) Fifteen distinct points are designated on triangle ABC: the 3 vertices A, B and C; 3 other points on side AB; 4 other points on side BC; and 5 other points on side CA. Find the number of triangles with positive area whose vertices are among these 15 points.

counting - handshake

twins and triplets

(2011 12A) At a twins and triplets convention, there were 9 sets of twins and 6 sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place?

counting - tabulate schemes

(2007 AIME I) A frog is placed at the origin on the number line, and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of 3, or to the closest point with a greater integer coordinate that is a multiple of 13. A move sequence is a sequence of coordinates which correspond to valid moves, beginning with 0, and ending with 39. For example, 0,3,6,13,15,26,39 is a move sequence. How many move sequences are possible for the frog?

counting - write down all cases

(2020 AIME II) While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.

star/bar – loaded bins

Alice has 24 apples

(2019 AMC8) Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?

(SIM) Alice has 24 apples. In how many ways can she share them with Becky and Chris?

(SIM) Alice has 24 apples. In how many ways can she share them with Becky and Chris so that one of them has at least one apple and the other two of them have at least three apples?

(2011 AIME II) Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles whose right hand neighbor is the other color. An example of such an arrangement is GGRRRGGRG. Let m be the maximum number of red marbles for which such an arrangement is possible, find the number of ways he can arrange the m+5 marbles to satisfy the requirement.

star/bar – unique stars

5 rings on 4 fingers

(2000 AIME II) Given eight distinguishable rings, let n be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find n.

(SIM) Given eight distinguishable rings, let n be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, and it is required that each finger have a ring. Find n.

(SIM) Given eight distinguishable rings, let n be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is not significant, but it is required that each finger have a ring. Find n.

(SIM) Given eight distinguishable rings, let n be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is not significant, and it is not required that each finger have a ring. Find n.

(2020 AMC8) Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed?

star/bar – multinomial expansion

find canceled terms

(2006 12A) The expression (x+y+z)²⁰⁰⁶+(x-y-z)²⁰⁰⁶ is simplified by expanding it and combining like terms. How many terms are in the simplified expression?

(1996 AIME) Find the smallest positive integer n for which the expansion of (xy-3x+7y-21)ⁿ, after like terms have been collected, has at least 1996 terms.

linear combinatorics – fixed patterns

one-faced gold and silver coins

(2005 AIME I) Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins.

blue and green flags on two flagpoles

(2008 AIME II) There are two distinguishable flagpoles, and there are 19 flags, of which 10 are identical blue flags, and 9 are identical green flags. Find the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags on either pole are adjacent.

(SIM) There are two distinguishable flagpoles, and there are 19 flags, of which 10 are identical blue flags, and 9 are identical green flags. Find the number of distinguishable arrangements using all of the flags in which no two green flags on either pole are adjacent.

(SIM) There are two distinguishable flagpoles, and there are 19 flags, of which 10 are identical blue flags, and 9 are distinguishable green flags. Find the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags on either pole are adjacent.

(1990 AIME) A fair coin is to be tossed 10 times. Find the probability that heads never occur on consecutive tosses.

linear combinatorics – fixed patterns with exceptions

(2022 12A) Azar and Carl play a game of tic-tac-toe. Azar places an X in one of the boxes in a 3x3 array of boxes, then Carl places an O in one of the remaining boxes. After that, Azar places an X in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third O. How many ways can the board look after the game is over?

linear combinatorics – restricted bins

matilda's 3 math books

(2013 AIME I) Matilda has three empty boxes and 12 textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Matilda packs her textbooks into these boxes in random order, find the probability that all three mathematics textbooks end up in the same box.

selection with replacement - re-construct without replacement

(2020 AIME I) Let S be the set of positive integer divisors of 20⁹. Three numbers are chosen independently and at random with replacement from the set S and labeled a,b,c in the order they are chosen. Find the probability that both a divides b and b divides c.

committee forming

(2017 AIME I) For nonnegative integers a and b with a+b≤6, let T(a,b)=(6Ca)(6Cb)(6Ca+b). Find the sum of all T(a,b), where a and b are nonnegative integers with a+b≤6.

block walk - no restriction

meet at times square

(2003 12A) Objects A and B move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object A starts at (0,0) and each of its steps is either right or up, both equally likely. Object B starts at (5,7) and each of its steps is either to the left or down, both equally likely. Which of the following is closest to the probability that the objects meet?

block walk - with restrictions

meet at times square by skipping central park

(2010 12A) A 16-step path is to go from (-4,-4) to (4,4) to with each step increasing either the x-coordinate or the y-coordinate by 1. How many such paths stay outside or on the boundary of the square -2≤x≤2, -2≤y≤2, at each step?

round robin - chase the unchanged

everyone got 50% points from bottom 10

(1985 AIME) In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 0.5 point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned in games against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?

round robin - find winning patterns

no one wins the same numbers of games

(1999 AIME) Forty teams play a tournament in which every team plays every other(39 different opponents) team exactly once. No ties occur, and each team has a 50% chance of winning any game it plays. Find the probability that no two teams win the same number of games.

(1996 AIME) In a five-team tournament, each team plays one game with every other team. Each team has a 50% chance of winning any game it plays. (There are no ties.) Find the probability that the tournament will produce neither an undefeated team nor a winless team.

round robin - focus on individual team

beat you first beat you last

(2006 AIME II) Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a 50% chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumulated to decide the ranks of the teams. In the first game of the tournament, team A beats team B. Find the probability that team A finishes with more points than team B.

(2001 AIME II) Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each 1/3. Find the probability that Club Truncator will finish the season with more wins than losses.

circular combinatorics – selecting is back filling

king arthur's 25 knights

(1983 AIME) Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Find the probability that at least two of the three had been sitting next to each other.

2-d combinatorics – fix lines or columns

black and white board

(2007 AIME I) In a 6x4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Find the number of shadings with this property.

(1997 AIME) How many different 4x4 arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0?

pascal's identity

(2020 AIME I) A club consisting of 11 men and 12 women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as 1 member or as many as 23 members. Find the number of such committees that can be formed.

recursion

mail carrier on elm street

(2001 AIME I) A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are possible?

system of recursions

(2008 AIME I) Consider sequences that consist entirely of A's and B's and that have the property that every run of consecutive A's has even length, and every run of consecutive B's has odd length. Examples of such sequences are AA, B, and AABAA, while BBAB is not such a sequence. How many such sequences have length 14?

gaming strategy - fix a letter each step

maxwell's socks

(1994 AIME) A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. Find the probability that the bag will be emptied.

(2015 AIME I) In a drawer Maxwell has 5 pairs of socks, each pair a different color. On Monday, Maxwell selects two individual socks at random from the 10 socks in the drawer. On Tuesday Maxwell selects 2 of the remaining 8 socks at random, and on Wednesday two of the remaining 6 socks at random. Find the probability that Wednesday is the first day Maxwell selects matching socks.

(2017 AIME II) A special deck of cards contains 49 cards, each labeled with a number from 1 to 7 and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, find the probability that Sharon can discard one of her cards and still have at least one card of each color and at least one card with each number.

gaming strategy - string partition

guessing price of 3 prizes

(2009 AIME I) A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $1 to 9999 inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were 1,1,1,1,3,3,3. Find the total number of possible guesses for all three prizes consistent with the hint.

gaming strategy - symmetry

pick cards based on shape/color/shades

(1997 AIME) Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the following statements are true: (i). Either each of the three cards has a different shape or all three of the card have the same shape. (ii). Either each of the three cards has a different color or all three of the cards have the same color. (iii). Either each of the three cards has a different shade or all three of the cards have the same shade. How many different complementary three-card sets are there?

ABSTRACT ALGEBRA

pie

(2021 AIME 2) For any finite set S, let |S| denote the number of elements in S. Find the number of ordered pairs (A,B) such that A and B are (not necessarily distinct) subsets of {1,2,3,4,5} that satisfy |A||B|=|A∩B||A∪B|

restricted subsets

(2017 AIME I) Call a set S product-free if there do not exist a,b,c in S (not necessarily distinct) such that ab=c. For example, the empty set and the set {16,20} are product-free, whereas the sets {4,16} and {2,8,16} are not product-free. Find the number of product-free subsets of the set {1,2,3,4,5,6,7,8,9,10}.

injection/surjection/bijection

(2014 AIME I) Let A={1,2,3,4}, and f and g be randomly chosen (not necessarily distinct) functions from A to A. Find the probability that the range of f and the range of g are disjoint.

symmetry

(2022 12B) A cube is constructed from 4 white unit cubes and 4 blue unit cubes. How many different ways are there to construct the 2x2x2 cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)

(2022 12B) Regular polygons with 5, 6, 7, and 8 sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?

rings - handshaking polygons

facebook friends

(2012 12A) Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?

(2012 AIME II) In a group of nine people each person shakes hands with exactly two of the other people from the group. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the number of ways this handshaking can occur.

burnside - octahedron

8-color octahedron with rotation

(2000 12) Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)

burnside - polynomial

integer coefficients interger zeros

(2010 AIME II) Find the number of second-degree polynomials f(x) with integer coefficients and integer zeros for which f(0)=2010.

graph theory - nodes

(2017 AIME II) Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).

EUCLIDEAN GEOMETRY

pythagorean theorem

(2001 AIME II) Square ABCD is inscribed in a circle. Square EFGH has vertices E and F on CD and vertices G and H on the circle. Find the ratio of the area of square EFGH to the area of square ABCD.

(2006 AIME I) In quadrilateral ABCD, the angle B is a right angle, diagonal AC is perpendicular to CD, AB=18, BC=21, and CD=14. Find the perimeter of ABCD.

similar triangle

(1998 AIME) Let ABCD be a parallelogram. Extend DA through A to a point P and let PC meet AB at Q and DB at R. Given that PQ=735 and QR=112, find RC.

angle chasing - same arc same angle

(2016 AIME I) In triangle ABC let I be the center of the inscribed circle, and let the bisector of the angle ACB intersect AB at L. The line through C and L intersects the circumscribed circle of ABC at the two points C and D. If LI=2 and LD=3, find IC.

(2020 AIME I) In triangle ABC with AB=AC, point D lies strictly between A and C on side AC, and point E lies strictly between A and B on side AB such that AE=ED=DB=BC. Find the degree measure of the angle ABC.

power of a point

(Canadian MO 1971) DEB is a chord of a circle such that DE=3 and EB=5. Let O be the center of the circle. Join OE and extend OE to cut the circle at C Given EC=1, find the radius of the circle.

(2013 12A) In triangle ABC, AB=86, and AC=97. A circle with center A and radius AB intersects BC at points B and X. Moreover, BX and CX have integer lengths. What is BC?

similar triangle + power of a point

(2022 12B) Triangle ABC has side lengths AB=11, BC=24, and CA=20. The bisector of angle BAC intersects BC in point D, and intersects the circumcircle of triangle ABC in point E (not A). The circumcircle of triangle BED intersects the line AB in points B and F (not B). Find CF.

10 ways to find area of a triangle

(2002 AIME I) In triangle ABC, angle B is a right angle. Point D is on BC, and AD bisects angle A. Points E and F are on AB and AC, respectively, so that AE=3 and AF=10. Given that EB=9 and FC=27, and EF intersects AD at G, find the area of quadrilateral DCFG.

(2003 AIME II) Triangle ABC is a right triangle with AC=7, BC=24, and right angle at C. Point M is the midpoint of AB and D is on the same side of line AB as C so that AD=BD=15. Find the area of triangle CDM.

(2021 AIME 2) For positive real numbers a, let T(s) denote the set of all obtuse triangles that have area s and two sides with lengths 4 and 10. The set of all s for which T(s) is nonempty, but all triangles in T(s) are congruent, is an interval [a,b). Find a and b.

triangle centroid

union after 180-degree rotation

(2003 AIME II) In triangle ABC, AB=13,BC=14,AC=15, and point G is the intersection of the medians. Points A’,B’ and C’ are the images of A,B,C, respectively, after a 180 degree rotation about G. What is the area of the union of the two regions enclosed by the triangles ABC and A’B’C’.

construct special triangle

rotation

(2012 AIME I) Three concentric circles have radii 3, 4 and 5. An equilateral triangle with one vertex on each circle. Find the largest possible area of the triangle.

reflection

(2014 12A) In triangle BAC, the angle BAC is 40 degree, AB=10, and AC=6. Points D and E lie on AB and AC respectively. What is the minimum possible value of BE+DE+CD?

tangent circles

(1994 AIME) A circle with diameter PQ of length 10 is internally tangent at P to a circle of radius 20. Square ABCD is constructed with A and B on the larger circle, CD tangent at Q to the smaller circle, and the smaller circle outside ABCD. Find The length of AB.

(1995 AIME) Circles of radius 3 and 6 are externally tangent to each other and are internally tangent to a circle of radius 9. The circle of radius 9 has a chord that is a common external tangent of the other two circles. Find the length of this chord.

(1997 AIME) Circles of radii 5, 5, 8, and x are mutually externally tangent, find x.

(2004 AIME II) Let ABCD be an isosceles trapezoid, whose dimensions are AB=6, BC=DA=5, and CD=4. Draw circles of radius 3 centered at A and B and circles of radius 2 centered at C and D. A circle contained within the trapezoid is tangent to all four of these circles. Find its radius.

(2014 AIME II) Circle C with radius 2 has diameter AB. Circle D is internally tangent to circle C at A. Circle E is internally tangent to circle C, externally tangent to circle D, and tangent to AB. The radius of circle D is three times the radius of circle E. Find the radious of circle D.

(2020 AIME I) A flat board has a circular hole with radius 1 and a circular hole with radius 2 such that the distance between the centers of the two holes is 7. Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. Find the square of the radius of the spheres.

(2021 AIME I) Circles w¹ and w² with radii 961 and 625, respectively, intersect at distinct points A and B. A third circle w is externally tangent to both w¹ and w². Suppose line AB intersects w at two points P and Q such that the measure of minor arc PQ is 120 degree. Find the distance between the centers of w¹ and w².

stewart's theorem

(2002 12B) In triangle ABC, we have AB=1 and AC=2. Side BC and the median from A to BC have the same length. What is BC?

(2003 AIME I) Point B is on AC with AB=9 and BC=21. Point D is not on AC so that AD=CD, and AD and BD are integers. Find the sum of all possible perimeters of triangle ACD.

(2005 AIME I) Triangle ABC has BC=20. The incircle of the triangle evenly trisects the median AD. Find the area of the triangle.

(2009 AIME II) Four lighthouses are located at points A, B, C and D. The lighthouse at A is 5 kilometers from the lighthouse at B, the lighthouse at B is 12 kilometers from the lighthouse at C, and the lighthouse at A is 13 kilometers from the lighthouse at C. To an observer at A the angle determined by the lights at B and D and the angle determined by the lights at C and D are equal. To an observer at C, the angle determined by the lights at A and B and the angle determined by the lights at D and B are equal. Find the distance from A to D.

(2021 12B) Triangle ABC has AB=13,BC=14,AC=15, Let P be the point on AC such that PC=10. There are exactly two points D and E on line BP such that quadrilaterals ABCD and ABCD are trapezoids. What is the distance DE?

angle bisector theorem

(2011 AIME I) In triangle ABC, AB=125, AC=117, and BC=120. The angle bisector of angle A intersects BC at point L, and the angle bisector of angle B intersects AC at point K. Let M and N be the feet of the perpendiculars from C to BK and AL, respectively. Find MN.

incircle

(1993 AIME) Let CH be an altitude of ABC. Let R and S be the points where the circles inscribed in the triangles ACH and BCH are tangent to CH. If AB=1995, AC=1994 and BC=1993. Find RS.

(2001 AIME I) Triangle ABC has AB=21, AC=22 and BC=20. Points D and E are located on AB and AC, respectively, such that DE is parallel to BC and contains the center of the inscribed circle of triangle ABC. Find DE.

(2007 AIME II) Rectangle ABCD is given with AB=63 and BC=448. Points E and F lie on AD and BC, respectively, such that AE=CF=84. The inscribed circle of triangle BEF is tangent to EF at point P, and the inscribed circle of triangle DEF is tangent to EF at point Q. Find PQ.

(2009 AIME I) In right triangle ABC with hypotenuse AB, AC=12, BC=35, and CD is the altitude to AB. Let w be the circle having CD as a diameter. Let I be a point outside ABC such that AI and BI are both tangent to circle w. Find the ratio of the perimeter of triangle ABI to the length AB.

(2012 12A) Triangle ABC has AB=27, AC=26, and BC=25. Let I be the intersection of the internal angle bisectors of ABC. What is BI?

(2020 AIME I) Point D lies on side BC of triangle ABC so that AD bisects angle BAC. The perpendicular bisector of AD intersects the bisectors of angles ABC and ACB in points E and F respectively. Given that AB=4, BC=5 and CA=6, find the area of triangle AEF.

circumcircle

(2002 AIME I) In triangle ABC the medians AD and CE have lengths 18 and 27, respectively, and AB=24. Extend CE to intersect the circumcircle of ABC at F. Find the area of triangle AFB.

cube hangs on cycinder

(2015 AIME II) A cylindrical barrel with radius 4 feet and height 10 feet is full of water. A solid cube with side length 8 feet is set into the barrel so that the diagonal of the cube is vertical. Find the volume of water displaced.

circumcircle - central angle theorem

(2015 AIME II) The circumcircle of acute triangle ABC has center O. The line passing through point O perpendicular to OB intersects lines AB and BC at P and Q, respectively. Also AB=5, BC=4, BQ=4.5. Find BP.

cyclic quadrilateral

(2011 AIME II) A circle with center O has radius 25. Chord AB of length 30 and chord CD of length 14 intersect at point P. The distance between the midpoints of the two chords is 12. Find OP.

(2021 AIME I) Let ABCD be a cyclic quadrilateral with AB=4, BC=5, CD=6 and DA=7. Let A¹ and C¹ be the feet of the perpendiculars from A and C respectively, to line BD; and let B¹ and D¹ be the feet of the perpendiculars from B and D respectively, to line AC. Find the perimeter of A¹B¹C¹D¹.

ptolemy

(2016 AIME II) Triangle ABC is inscribed in circle w. Points P and Q are on side AB with AP < AQ. Rays CP and CQ meet w again at S and T (other than C), respectively. If AP=4, PQ=3, QB=6, BT=5, and AS=7, find ST.

trapezoid

(2000 AIME II) One base of a trapezoid is 100 units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio 2:3. Find the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area.

(1992 AIME) Trapezoid ABCD has sides AB=92, BC=50, CD=19, and AD=70 with AB parallel to CD. A circle with center P on AB is drawn tangent to BC and AD. Find AP.

(2008 AIME I) Let ABCD be an isosceles trapezoid with AD//BC whose angle at the longer base AD is 60 degree. The diagonals have length 10$\sqrt{\mathrm{21}}$, and point E is at distances 10$\sqrt{7}$ and 30$\sqrt{7}$ from vertices A and D, respectively. Let F be the foot of the altitude from C to AD. Find the distance EF.

special trapezoid

trapezoid with sum of two base angles 90 degree

(2008 AIME II) In trapezoid ABCD, BC//AD, let BC=1000 and AD=2008. Let angles A=37 degree and D=53 degree, and M and N be the midpoints of BC and AD, respectively. Find the length MN.

polygons

(2022 12B) Three identical square sheets of paper each with side length 6 are stacked on top of each other. The middle sheet is rotated clockwise 30 degree about its center and the top sheet is rotated clockwise 60 degree about its center, resulting in the 24-sided polygon. Find the area of this polygon.

rhombus

built by 4 congruent right triangles

(2003 AIME II) Find the area of rhombus ABCD given that the radii of the circles circumscribed around triangles ABD and ACD are 12.5 and 25, respectively.

(1991 AIME) Rhombus PQRS is inscribed in rectangle ABCD so that vertices P, Q, R, and S are interior points on sides AB, BC, CD, and DA, respectively. It is given that PB=15, BQ=20, PR=30, and QS=40. Find the perimeter of ABCD.

geometric maxima/minima

(2005 AIME I) A semicircle with diameter d is contained in a square whose sides have length 8. Find the maximum value of d.

(2013 AIME II) Given a circle of radius $\sqrt{\mathrm{13}}$, let A be a point at a distance 4+$\sqrt{\mathrm{13}}$} from the center O of the circle. Let B be the point on the circle nearest to point A. A line passing through the point A intersects the circle at points K and L. Find the maximum possible area for triangle BKL.

(1993 AIME) A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, find the smallest perimeter.

PLATONIC SOLID AND 3D SHAPES

dihedral

(2016 AIME II) Equilateral ABC has side length 600. Points P and Q lie outside the plane of ABC and are on opposite sides of the plane. Furthermore, PA=PB=PC and QA=QB=QC, and the planes of PAB and QAB form a 120 degree dihedral angle (the angle between the two planes). There is a point O whose distance from each of A,B,C,P and Q is the same, find this distance.

tetrahedron

(1992 AIME) Faces ABC and BCD of tetrahedron ABCD meet at an angle of 30 degree. The area of face ABC is 120, the area of face BCD is 80, and BC=10. Find the volume of the tetrahedron.

(2001 AIME II) Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra P_i is defined recursively as follows: P₀ is a regular tetrahedron whose volume is 1. To obtain P_i+1, replace the midpoint triangle of every face of P_i by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. Find the volume of P₃.

cube

(2003 AIME I) Find the sum of the areas of all triangles whose vertices are also vertices of a 1 by 1 by 1 cube.

octahedron

cut octahedron in half

(2009 12A) A regular octahedron has side length 1. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. What’s the area of polygon formed by the intersection of the plane and the octahedron?

(2005 AIME II) Given that O is a regular octahedron, that C is the cube whose vertices are the centers of the faces of O. Find the ratio of the volume of O to that of C.

euler's formula

space diagonals of polyhedron

(2004 AIME I) A convex polyhedron P has 26 vertices, 60 edges, and 36 faces, 24 of which are triangular, and 12 of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does P have?

cone

(2020 AIME II) Two congruent right circular cones each with base radius 3 and height 8 have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance 3 from the base of each cone. A sphere with radius r lies within both cones. Find the maximum possible value of r.

(2004 AIME II) A right circular cone has a base with radius 600 and height 200$\sqrt{7}$. A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is 125, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is 375$\sqrt{2}$. Find the least distance that the fly could have crawled.

ANALYTICAL GEOMETRY

line function

(2017 AIME II) Rectangle ABCD has side lengths AB=84 and AD=42. Point M is the midpoint of AD, point N is the trisection point of AB closer to A, and point O is the intersection of CM and DN. Point P lies on the quadrilateral BCON, and BP bisects the area of BCON. Find the area of triangle CDP.

circle function

(2002 AIME II) Circles A and B intersect at two points, one of which is (9,6), and the product of the radii is 68. The x-axis and the line y=mx, where m>0, are tangent to both circles. Find m.

parabola function

(2011 AIME I) Suppose that a parabola has vertex (1/4, -9/8) and equation y=ax²+bx+c, where a > 0 and a+b+c is an integer. Find the minimum possible value of a.

(2021 AIME I) Let S be the set of positive integers k such that the two parabolas y=x²-k and x=2(y-20)²-k intersect in four distinct points, and these four points lie on a circle with radius at most 21. Find the sum of the least element of S and the greatest element of S.

mass point

(2011 AIME II) In triangle ABC, AB=20, AC=11. The angle bisector of angle A intersects BC at point D, and point M is the midpoint of AD. Let P be the point of intersection of AC and the line BM. Find the ratio of CP to PA.

(2016 12B) In triangle ABC, AB=7, BC=8, CA=9, and AH is an altitude. Points D and E lie on sides AC and AB, respectively, so that BD and CE are angle bisectors, intersecting AH at Q and P, respectively. What is PQ?

split mass

(2009 AIME I) In parallelogram ABCD, point M is on AB so that AM:AB=17:1000 and point N is on AD so that AN:AD=17:2009. Let P be the point of intersection of AC and MN. Find the ratio of AC to AP.

plane function

(1998 AIME) Three of the edges of a cube are AB, BC and CD; and AD is an interior diagonal. Points P,Q,R are on AB, BC and CD respectively, so that AP=5,PB=15,BQ=15 and CR=10 What is the area of the polygon that is the intersection of plane PQR and the cube?

(2007 AIME I) A square pyramid with base ABCD and vertex E has eight edges of length 4. A plane passes through the midpoints of AE, BC, and CD. Find the area of the plane's intersection with the pyramid.

(2012 AIME I) Four edges of a cube AB, BC, CD and DA are on a plane with edge length of 1; and AG, BH, CE and DF are interior diagonals. The cube is cut by a plane passing through vertex D and the midpoints M and N of AB and CG respectively. The plane divides the cube into two solids. Find the volume of the larger of the two solids.

distance formula

(1987 AIME) What is the largest possible distance between two points, one on the sphere of radius 19 with center (-2,-10,5) and the other on the sphere of radius 87 with center (12,8,-16)?

(2022 12B) Right triangle ABC has side lengths BC=6, AC=8, and AB=10. A circle centered at O is tangent to line BC at B and passes through A. A circle centered at P is tangent to line AC at A and passes through B. Find OP.

shoelace

(2007 AIME I) In isosceles triangle ABC, A is located at the origin and B is located at (20,0). Point C is in the first quadrant with AC=BC and angle BAC=75 degree. If triangle ABC is rotated counterclockwise about point A until the image of C lies on the positive y-axis, find the area of the region common to the original and the rotated triangle.

barycentric coordinate

(1990 AIME) A triangle has vertices P=(-8,5), Q=(-15,-19), and R=(1,-7). Solve the equation of the bisector of the angle P.

(2013 AIME II) In triangle ABC, AC=BC, and point D is on BC so that CD=3BD. Let E be the midpoint of AD. Given that CE= $\sqrt{7}$ and BE=3, find the area of the triangle ABC.

point to line/plane formula

(2001 AIME I) A sphere is inscribed in the tetrahedron whose vertices are A=(6,0,0), B=(0,4,0),C=(0,0,2), and D=(0,0,0). Find the radius of the sphere.

(2010 AIME I) Let R be the region consisting of the set of points in the coordinate plane that satisfy both |8-x|+y ≤ 10 and 3y-x ≥ 15. When R is revolved around the line whose equation is 3y-x=15, find the volume of the resulting solid.

(2011 AIME I ) Let L be the line with slope 5/12 that contains the point A=(24,-1), and let M be the line perpendicular to line L that contains the point B=(5,6). The original coordinate axes are erased, and line L is made the x-axis and line M the y-axis. In the new coordinate system, point A is on the positive x-axis, and point B is on the positive y-axis. Find the new coordinates for the point P with original coordinates of (-14,27).

(2021 AIME I) Let ABCD be an isosceles trapezoid with AD=BC and AB < CD. Suppose that the distances from A to the lines BC, CD, and BD are 15, 18, and 10, respectively. Find the area of ABCD.

non-linear transformation

(1999 AIME) A transformation of the first quadrant of the coordinate plane maps each point (x,y) to the point ($\sqrt{x}$,$\sqrt{y}$) The vertices of quadrilateral ABCD are A=(900,300),B(1800,600),C(600,1800),and D(300,900). Find the area of the region enclosed by the image of quadrilateral ABCD.

trapezoid

(2000 AIME II) The coordinates of the vertices of isosceles trapezoid ABCD are all integers, with A=(20,100), and D=(21,107). The trapezoid has no horizontal or vertical sides, and AB and CD are the only parallel sides. Find the sum of the absolute values of all possible slopes for AB.

rotation matrix

(2020 AIME II) Triangles ABC and A’B’C’ lie in the coordinate plane with vertices A(0,0), B(0,12), C(16,0), A’(24,18), B’(36,18), C’(24,2). A rotation of m degrees clockwise around the point (x,y) where m=(0,180), will transform ABC to A’B’C’. Find m+x+y.

TRIGNOMETRY

definition of sine/cosine/tangent

length of dodecagon diagonals

(1990 AIME) A regular 12-gon is inscribed in a circle of radius 12. Find the sum of the lengths of all sides and diagonals of the 12-gon.

product-to-sum/sum-to-product

(2006 AIME I) Find the sum of the values of x such that cos³3x+cos³5x=8cos³4xcos³x, where x is measured in degrees and 100 < x < 200.

tan(a+b)

(2000 AIME II) A circle is inscribed in quadrilateral ABCD, tangent to AB at P and to CD at Q. Given that AP=19, PB=26, CQ=37, and QD=23, find the square of the radius of the circle.

(2005 AIME II) Square ABCD has center O, AB=900, E and F are on AB with AE < BF and E between A and F, and angle EOF is 45 degree, and EF=400. Find BF.

double/triple identity

(1995 AIME) Triangle ABC is isosceles, with AB=AC and altitude AM=11 Suppose that there is a point D on AM with AD=10 and angles BDC=3BAC. Find the perimeter of the triangle ABC.

(2001 AIME I) In a certain circle, the chord of a d-degree arc is 22 centimeters long, and the chord of a 2d-degree arc is 20 centimeters longer than the chord of a 3d-degree arc, where d < 200. Find the length of the chord of a 3d-degree arc.

(2012 AIME II) Let x and y be real numbers such that sinx/siny=3 and cosx/cosy=1/2. Find the value of (sin2x/sin2y)+(cox2x/cos2y).

(2013 AIME II) In equilateral triangle ABC let points D and E trisect BC. Find sin(DAE).

(2015 AIME I) Triangle ABC has positive integer side lengths with AB=AC. Let I be the intersection of the bisectors of angles B and C. Suppose BI=8. Find the smallest possible perimeter of the triangle ABC.

law of cosine

fermet point

(1987 AIME) Triangle ABC has right angle at B,and contains a point P for which PA=10, PB=6, and angles APB=BPC=CPA. Find PC.

(1989 AIME) Two skaters, Allie and Billie, are at points A and B, respectively, on a flat, frozen lake. The distance between A and B is 100 meters. Allie leaves A and skates at a speed of 8 meters per second on a straight line that makes a 60 degree angle with AB. At the same time Allie leaves A, Billie leaves B at a speed of 7 meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?

(1995 AIME) In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lengths, and find the area of each of the region.

(2005 AIME I) In quadrilateral ABCD, BC=8, CD=12, AD=10, and the angles A and B are both 60 degree. Find AB.

(2013 AIME I) A paper equilateral triangle ABC has side length 12. The paper triangle is folded so that vertex A touches a point on side BC, a distance 9 from point B. Find the length of the line segment along which the triangle is folded.

(2013 AIME II) A hexagon that is inscribed in a circle has side lengths 22,22,20,22,22,and 20 in that order. Find the radius of the circle.

(2014 AIME I) A disk with radius 1 is externally tangent to a disk with radius 5. Let A be the point where the disks are tangent, C be the center of the smaller disk, and E be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until the smaller disk has turned through an angle of 360 degree. That is, if the center of the smaller disk has moved to the point D, and the point on the smaller disk that began at A has now moved to point B, then AC is parallel to BD. Find the angle BEA.

(2014 AIME II) In triangle RED, angles DRE and RED are 75 and 45 degree, respectively. RD=1. Let M be the midpoint of segment RD. Point C lies on side ED such that RC is perpendicular to EM. Extend segment DE through E to point A such that CA=AR. Find AE.

law of sine

(2003 AIME I) Triangle ABC is isosceles with AC=BC and the angle ACB is 106 degree. Point M is in the interior of the triangle so that the angle MAC is 7 degree and the angle MCA is 23 degree. Find the number of degrees in the angle CMB.

(2013 AIME I) Let triangle PQR be a triangle with angles P and Q are 75 and 60 degree, respectively. A regular hexagon ABCDEF with side length 1 is drawn inside PQR so that side AB lies on PQ, side CD lies on QR, and one of the remaining vertices lies on RP. Find the area of PQR.

graphing

(2007 12A) For each integer n>1, let F(n) be the number of solutions to the equation sin(x)=sin(nx) on the interval [0,Π]. Find F(2)+F(3)+...+F(2007)?

brocard angle

(1999 AIME) Point P is located inside triangle ABC so that angles PAB, PBC, and PCA are all congruent. The sides of the triangle have lengths AB=13, BC=14, and CA=15. Find the tangent of angle PAB.

rigid transformation

(2013 AIME II) In the Cartesian plane let A=(1,0) and b=(2,2$\sqrt{3}$). Equilateral triangle ABC is constructed so that C lies in the first quadrant. Find the center of ABC.

trig geometric sequence

(1997 AIME) Find (cos1+cos2+…+cos44)/(sin1+sin2+…+sin44). Number in degree.

30-60-90 triangle

(2012 AIME I) Let triangle ABC be a right triangle with right angle at C. Let D and E be points on AB with D between A and E such that CD and CE trisect C. If DE/BE=8/15, find tanB.

15-75-90 triangle

(2006 AIME II) Square ABCD has sides of length 1. Points E and F are on BC and CD respectively, so that the triangle AEF is equilateral. A square with vertex B has sides that are parallel to those of ABCD and a vertex on AE. Find the length of a side of this smaller square.

ELEMENTARY ALGEBRA

speed - storm chasing

mini cooper chasing super duper storm

(1997 AIME) A mini cooper travels due east at 2/3 miles per minute on a long, straight road. At the same time, a circular storm Super Duper, whose radius is 51 miles, moves southeast at $\sqrt{2}$/2 miles per minute. At time t=0, the center of the storm is 110 miles due north of the car. At time t=t1 minutes, the car enters the storm circle, and at time t=t2 minutes, the car leaves the storm circle. Find t1+t2.

speed - train chasing

2 guys chasing 2 trains

(2014 AIME I) Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running the east/west direction. Jon rides east at 20 miles per hour, and Steve rides west at 20 miles per hour. Two trains of equal length, traveling in opposite directions at constant but different speeds each pass the two riders. Each train takes exactly 1 minute to go past Jon. The westbound train takes 10 times as long as the eastbound train to go past Steve. Find the length of each train.

logarithm - base change

(2020 AIME I) There is a unique positive real number x such that the three numbers log₈2x, log₄x, and log₂x, in that order, form a geometric progression with positive common ratio. Find x.

logarithm - exponent manipulation

(2017 AIME I) Let a>1 and x>1 satisfy log_a(log_a(log_a2)+log_a24-128)=128 and log_a(log_ax)=256. Find the remainder when x is divided by 1000.

logarithm - serial equations

(2012 AIME I) Let x, y, and z be positive real numbers that satisfy 2log_x(2y)=2log_2x(4z)= log_2x⁴(8yz), which is not zero. Find the value of xy⁵z.

logarithm - graphing

Find the number of integer values of k in the closed interval [-500,500] for which the equation log(kx)=2log(x+2) has exactly one real solution.

x+y and xy manipulation

(2010 AIME I) Let (a,b,c) be a real solution of the system of equations x³-xyz=2, y³-xyz=6, z³-xyz=20. Find the greatest possible value of a³+b³+c³.

absolute function

(1998 AIME) The graph of y²+2xy+40|x|=400 partitions the plane into several regions. What is the area of the bounded region?

(2021 AIME I) Find the number of integers c such that the equation ||20|x|-x²|-c|=21 has 12 distinct real solutions.

cyclic equation

(1984 AIME) A function f is defined for all real numbers and satisfies f(2+x)=f(2-x) and f(7+x)=f(7-x) for all x. If x=0 is a root for f(x)=0, what is the least number of roots f(x)=0 must have in the interval [-1000,1000]?

polynomial - fta

(2022 12B) Suppose that P(z), Q(z), and R(z) are polynomials with real coefficients, having degrees 2,3, and 6, respectively, and constant terms 1,2, and 3, respectively. Find the minimum number of distinct complex numbers a that satisfy the equation P(z)*Q(z)=R(z).

polynomial - vertex form

(2007 AIME I) The polynomial P(x) is cubic. What is the largest value of k for which the polynomials Q(x)=x²+(k-29)x-k and R(x)=2x²+(2k-43)x+k are both factors of P(x)?

polynomial - substitution

(2005 AIME II) Let P(x) be a polynomial with integer coefficients that satisfies P(17)=10 and P(24)=17. Given that P(n)=n+3 has two distinct integer solutions n₁ and n₂ find the product n₁ and n₂.

polynomial - integer coefficients

(2013 AIME II) Let S be the set of all polynomials of the form z³+az²+bz+c, where a,b,c are integers. Find the number of polynomials in S such that each of its roots z satisfies either |z|=20 or |z|=13.

(2020 AIME I) For integers a,b,c and d, let f(x)=x²+ax+b and g(x)=x²+cx+d. Find the number of ordered triples (a,b,c) of integers with absolute values not exceeding 10 for which there is an integer d such that g(f(2))=g(f(4))=0.

polynomial - vieta

(2014 AIME I) The graphs y=3(x-h)²+j and y=2(x-h)²+k and have y-intercepts of 2013 and 2014, respectively, and each graph has two positive integer x-intercepts. Find h.

(2014 AIME I) Let x₁ < x₂ < x₃ be the three real roots of the equation $\sqrt{\mathrm{2014}}$x³-4029x²+2=0. Find x₂(x₁+x₃).

polynomial - nested

(2020 AIME I) Let P(x) be a quadratic polynomial with complex coefficients whose x2 coefficient is 1. Suppose the equation P(P(x)) has four distinct solutions, x=3,4,a,b. Find the sum of all possible values of (a+b)2.

function - u-sub

(2001 AIME II) A certain function f has the properties that f(3x)=3f(x) for all positive real values of x, and that f(x)=1-|x-2| for x in [1,3]. Find the smallest x for which f(x)=f(2001).

function - graphing

(2015 AIME I) Let f(x) be a third-degree polynomial with real coefficients satisfying |f(1)|= |f(2)|= |f(3)|= |f(5)|= |f(6)|= |f(7)|=12. Find |f(0)|.

sequence - operations

(1989 AIME) If the integer k is added to each of the numbers 36, 300, and 596, one obtains the squares of three consecutive terms of an arithmetic series. Find k.

(2002 AIME II) Two distinct, real, infinite geometric series each have a sum of 1 and have the same second term. The third term of one of the series is 1/8, find all possible second terms.

sequence - self recursive

(2016 AIME I) Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line y=24. A fence is located at the horizontal line y=0. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where y=0, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where y is negative. Freddy starts his search at the point (0,21) and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.

(1998 AIME) Except for the first two terms, each term of the sequence 1000,x,1000-x,… is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer x produces a sequence of maximum length?

sequence - construct by special number

(2017 AIME I) Let a₁₀=10, and for each positive integer n>10 let a_n=100a_n-1+n. Find the least positive n>10 such that an is a multiple of 99.

sequence - believe there's a pattern

(2020 AIME II) Define a sequence recursively by t₁=20, t₂=21, and t_n=(5t_n-1+1)/(25t_n-2) for all n=3 and larger. Find t₂₀₂₀.

COMPLEX ANALYSIS

complex number operations

(2002 AIME I) Let F(z)=(z+i)/(z-i) for all complex numbers z not equal to i, and let z_n=F(z_n-1) for all positive integers n. Given that z₀=(1/137) + i and z₂₀₀₂=a+bi, find z₂₀₀₂.

(2012 AIME II) The complex numbers z and w satisfy the system z+(20i/w)=5+i; and w+(12i/z)=-4+10i. Find the smallest possible value of |zw|².

conjugates

(1992 AIME) Find the area of the region in the complex plane that consists of all points z such that both z/40 and 40/(z bar) have real and imaginary parts between 0 and 1, inclusive.

(2022 12B) For real numbers x, let P(x)=1+cos(x)+isin(x)-cos(2x)-isin(2x)+cos(3x)+isin(3x). For how many values of x in [0,2π) does P(x)=0.

de moivre's

(2005 AIME II) For how many positive integers n less than or equal to 1000 is (sin(t)+icos(t))ⁿ=sin(nt)+icos(nt) true for all real t?

de moivre's - nth root of unity

(1990 AIME) The sets A={z:z¹⁸=1} and B{w:w⁴⁸=1} are both sets of complex roots of unity. The set C={zw:z in A and w in B} is also a set of complex roots of unity. How many distinct elements are in C?

(1996 AIME) Let P be the product of the roots of z⁶+z⁴+z³+z²+1=0 that have a positive imaginary part, and suppose that P=rcisx, where r is positive and x is [0,360). Find x.

(2014 AIME II) Let z be a complex number with |z|=2014. Let P be the polygon in the complex plane whose vertices are z and every w such that 1/(z+w)=1/z + 1/w. Find the area enclosed by P.

de moivre's - maximize distance of two points

(2012 AIME II) Let z=a+bi be the complex number with |z|=5 and b>0 such that the distance between (1+2i)z³ and z⁵ is maximized. Find z⁴.

de moivre's - equations

(2000 AIME II) Given that z is a complex number such that z+(1/z)=2cos3, find the least integer that is greater than z²⁰⁰⁰+(1/z²⁰⁰⁰). Number in degree.

(2011 AIME II) Let z₁,z₂,…,z₁₂ be the 12 zeroes of the polynomial z¹²=2³⁶. For each j, let w_j be one of z_j or iz_j. Find the maximum possible value of the real part of w₁+w₂+...+w₁₂.

(2012 AIME I) The complex numbers z and w satisfy z¹³=w, w¹¹=z, find the imaginary part of z.

re/im

(2020 AIME II) Convex pentagon ABCDE has side lengths AB=5,BC=CD=DE=6, and EA=7. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of ABCDE.

angles between vectors

(2017 AIME I) Let z₁=18+83i, z₂=18+39i, and z₃=78+99i. Let z be the unique complex number with the properties that [(z₃-z₁)/(z₂-z₁)][(z-z₂)/(z-z₃)] is a real number and the imaginary part of z is the greatest possible. Find the real part of z.

NUMBER THEORY

operations - division manipulation

(2021 WOOT) Let n be a positive integer. Find the number of factors of 2 in (n+1)(n+2)...(2n−1)(2n).

operations - carry

(2015 AIME I) For positive integer n, let s(n) denote the sum of the digits of n. Find the smallest positive integer satisfying s(n)=s(n+864)=20.

prime numbers

(1995 AIME) What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?

(WOOT 2021) The number 9⁶+1 is the product of three primes. Compute the largest of these primes.

(WOOT 2021) The number 85⁹-21⁹+6⁹ is divisible by an integer between 2000 and 3000. Compute that integer.

prime factorization

(2003 12B) Let x and y be positive integers such that 7x⁵=11y¹³. Find the minimum possible value of x.

(2020 AIME II) Find the number of ordered pairs of positive integers (m,n), such that m²n=20²⁰.

simon's favorite factoring trick

(2001 British MO) Find all positive integers m, n, where n is odd, that satisfy (1/m)+(4/n)=(1/12)

diophantine - complete squares

(2017 AIME II) Find the sum of all positive integers n such that $\sqrt{\mathrm{n^2+85n+2017}}$ is an integer.

diophantine - (a+b)(a-b)

(1994 AIME) In triangle ABC angle C is a right angle and the altitude from C meets AB at D. The lengths of the sides of ABC are integers, BD=29³, find cosB.

diophantine - number simplication

(2008 AIME I) Ten identical crates each of dimensions 3 ft x 4 ft x 6 ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Find the probability that the stack of crates is exactly 41 ft tall.

(1994 AIME) Ninety-four bricks, each measuring 4x10x19 inch are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes 4 or 10 or 19 to the total height of the tower. How many different tower heights can be achieved using all 94 of the bricks?

diophantine - symmetry

(2009 AIME II) Let m be the number of solutions in positive integers to the equation 4x+3y+2z=2009, and let n be the number of solutions in positive integers to the equation 4x+3y+2z=2000. Find the remainder when m-n is divided by 1000.

remainder

(2022 12B) For n a positive integer, let R(n) be the sum of the remainders when n is divided by 2,3,4,5,6,7,8,9 and 10. For example, R(15)=1+0+3+0+3+1+7+6+5=26. How many two-digit positive integers n satisfy R(n)=R(n+1)?

modular

(1986 AIME) In a parlor game, the magician asks one of the participants to think of a three digit number (abc) where a, b, and c represent digits in base 10 in the order indicated. The magician then asks this person to form the numbers (acb), (bca), (bac), (cab), and (cba), to add these five numbers, and to reveal their sum, N. If told the value of N, the magician can identify the original number, (abc). Play the role of the magician and determine the (abc) if N=3194.

(2001 AIME II) How many positive integer multiples of 1001 can be expressed in the form 10j-10i, where I and j are integers and 0<=i

(2021 AIME I) Find the number of pairs (m,n) of positive integers with 1 ≤ m < n ≤ 30 such that there exists a real number x satisfying sin(mx)+sin(nx)=2.

base conversion

(1986 AIME) The increasing sequence 1,3,4,9,10,12,13… consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the 100th term of this sequence.

(2001 AIME I) Call a positive integer N a 7-10 double if the digits of the base-7 representation of N form a base-10 number that is twice N. For example, 51 is a 7-10 double because its base-7 representation is 102. What is the largest 7-10 double?

(2012 AIME II) Let S be the increasing sequence of positive integers whose binary representation has exactly 8 ones. Find the 1000th number in S.

(2017 AIME II) Find the number of positive integers less than or equal to 2017 whose base-three representation contains no digit equal to 0.

(2020 AIME I) A positive integer N has base-eleven representation abc and base-eight representation 1bca, where a, b, and c represent (not necessarily distinct) digits. Find the least such N expressed in base ten.

(2020 AIME II) For each positive integer n, let f(n) be the sum of the digits in the base-four representation of n and let g(n) be the sum of the digits in the base-eight representation of f(n). For example, f(2020)=10 and g(2020)=3. Let N be the least value of n such that the base-sixteen representation of g(n) cannot be expressed using only the digits 0 through 9. Find the remainder when N is divided by 1000.

counting numbers

(2006 AIME II) Find the number of ordered pairs of positive integers (a,b) such that a+b=1000 and neither a nor b has a zero digit.

counting by illustration

(2020 AIME I) Six cards numbered 1 through 6 are to be lined up in a row. Find the number of arrangements of these six cards where one of the cards can be removed leaving the remaining five cards in either ascending or descending order.

cube roots

(1987 AIME) Let m be the smallest integer whose cube root is of the form n+r, where n is a positive integer and r is a positive real number less than 1/1000. Find n.

(2017 AIME I) For every m≥2, let Q(m) be the least positive integer with the following property: For every n≥Q(m), there is always a perfect cube k3 in the range n<k3≤mn. Find Q(2)+Q(3)+…+Q(2017).

counting divisors – exponent plus 1

(1996 AHSME) If n is a positive integer such that 2n has 28 positive divisors and 3n has 30 positive divisors, then how many positive divisors does 6n have?

(2020 AIME I) Let S be the set of positive integers N with the property that the last four digits of N are 2020 and when the last four digits are removed, the result is a divisor of N. For example, 42020 is in S because 4 is a divisor of 42020. Find the sum of all the digits of all the numbers in S. For example, the number 42020 contributes 4+2+0+2+0=8 to this total.

counting divisors – even vs odd divisors

(2000 AIME II) What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?

counting divisors – factorial floor function

(2006 AIME II) Let P be the product of the first 100 positive odd integers. Find the largest integer k such that P is divisible by 3^k.

counting divisors – pairing

(1987 AIME) By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?

counting divisors – sum

(2000 AIME I) Let S be the sum of all numbers of the form a/b, where a and b are relatively prime positive divisors of 1000. What is the greatest integer that does not exceed S/10?

lcm – separate prime factors

lcm of 3 pairs of variables

(2016 12A) How many ordered triples (x,y,z) of positive integers satisfy lcm(x,y)=72, lcm(x,z)=600, and lcm(y,z)=900?

gcd

(2020 AIME I) Let m and n be positive integers satisfying the conditions
1. gcd(m+n,210)=1
2. m^m is a multiple of nⁿ
3. m is not a multiple of n Find the least possible value of m+n

(2022 12B) Suppose a,b,c are positive integers such that a+b+c=23 and gcd(a,b)+gcd(b,c)+gcd(c,a)=9. What is the sum of all possible distinct values of a²+b²+c²?

euclidean algorithm

divisors containing n

(1986 AIME) What is that largest positive integer n for which n³+100 is divisible by n+10?

binomial theorem

(2020 AIME I) Let n be the least positive integer for which 149ⁿ-2ⁿ is divisible by 3³5⁵7⁷. Find the number of positive integer divisors of n.

chinese remainder theorem

(2013 AIME I) Ms. Math's kindergarten class has 16 registered students. The classroom has a very large number, N, of play blocks which satisfies the conditions: (a) If 16, 15, or 14 students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and (b) There are three integers 0 < x < y < z < 14 such that when x,y,or z students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over. Find the sum of the distinct prime divisors of the least possible value of N satisfying the above conditions.

consecutive integers - sum

(2006 AIME II) How many integers N less than 1000 can be written as the sum of j consecutive positive odd integers from exactly 5 values of j>=1?

consecutive integers - product

(1990 AIME) Someone observed that 6!=8x9x10. Find the largest positive integer n for which n! can be expressed as the product of n-3 consecutive positive integers.

totient

(1992 AIME) Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms.

floor function

(2012 AIME II) Find the number of positive integers n less than 1000 for which there exists a positive real number x such that n is product of x and the greatest integer less than or equal to x.

(2009 AIME I) How many positive integers N less than 1000 are there such that the equation x^floor(x) has a solution for x?

greedy algorithm

(1986 AIME) Let the sum of a set of numbers be the sum of its elements. Let S be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of S have the same sum. What is the largest sum a set S with these properties can have?

CALCULUS

polynomial differentiation

(2016 AIME I) Let P(x) be a nonzero polynomial such that (x-1)P(x+1)=(x+2)P(x)for every real x, and (P(2))²=P(3). Find P(7/2).

maxima/minima

(2016 AIME I) Triangle ABC has AB=40, AC=31 and sinA=1/5. This triangle is inscribed in rectangle AQRS with B on QR and C on RS. Find the maximum possible area of AQRS.

minimize ax+by=1

(2017 AIME I) Find the area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths 2$\sqrt{3}$, 5, and $\sqrt{\mathrm{37}}$.

LINEAR ALGEBRA

vector

(1986 AIME) Let triangle ABC be a right triangle in the xy-plane with a right angle at C. Given that the length of the hypotenuse AB is 60, and that the medians through A and B lie along the lines y=x+3 and y=2x+4, respectively, find the area of triangle ABC.

(2000 AIME II) The points A, B and C lie on the surface of a sphere with center O and radius 20. It is given that AB=13, BC=14, CA=15. Find the distance from O to triangle ABC.

(2005 AIME I) Triangle ABC lies in the cartesian plane and has an area of 70. The coordinates of B and C are (12,19) and (23,20) respectively. The line containing the median to side BC has slope -5. Find the coordinates of A.

(2013 AIME I) A rectangular box has width 12 inches and length 16 inches. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of 30 square inches. Find the height of the box.

(2020 AIME I) A bug walks all day and sleeps all night. On the first day, it starts at point O, faces east, and walks a distance of 5 units due east. Each night the bug rotates 60 degree counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to the point P. Find OP.

dot product

(2000 AIME II) In trapezoid ABCD, leg BC is perpendicular to bases AB and CD, and diagonals AC and BD are perpendicular. Given that AB=$\sqrt{\mathrm{11}}$ and AD=$\sqrt{\mathrm{1001}}$, find BC.

cross product

(2009 AIME I) Consider the set of all triangles OPQ where O is the origin and P and Q are distinct points in the plane with nonnegative integer coordinates (x,y) such that 41x+y=2009. Find the number of such distinct triangles whose area is a positive integer.

vector in complext analysis

(2014 AIME I) Let w and z be complex numbers such that |w|=1 and |z|=10. Let θ=arg((w-z)/z). Find the maximum possible value of tan²θ

linear system

(2003 AIME II) Find the eighth term of the sequence 1440, 1716, 1848… whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.

STATISTICS

sampling

(2003 AIME II) The members of a distinguished committee were choosing a president, and each member gave one vote to one of the 27 candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least 1 than the number of votes for that candidate. What is the smallest possible number of members of the committee?

linearity/mean

(2022 12B) What is the average number of pairs of consecutive integers in a randomly selected subset of 5 distinct integers chosen from the set {1,2,3,…,30}? (For example the set {1,17,18,19,30} has 2 pairs of consecutive integers.)

probability – a moving target

rains on saturday rains on sunday

(2016 AIME II) There is a 40% chance of rain on Saturday and a 30% chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. Find the probability that it rains at least one day this weekend.

combinatorial probability

biased coins

(1989 AIME) When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to 0 and is the same as that of getting heads exactly twice. Find the probability that the coin comes up heads in exactly 3 out of 5 flips.

alex's poker team

(2010 AIME II) The 52 cards in a deck are numbered 1,2,…,52. Alex, Blair, Corey, and Dylan each pick a card from the deck randomly and without replacement. The two people with lower numbered cards form a team, and the two people with higher numbered cards form another team. Let p(a) be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards a and a+9, and Dylan picks the other of these two cards. Find the minimum value of p(a) for which p(a)≥0.5.

probability - endgame scenario

(1995 AIME) Let p be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails.

probability - geometric

(2005 AIME I) Twenty-seven unit cubes are painted orange on a set of four faces so that the two unpainted faces share an edge. The 27 cubes are then randomly arranged to form a 3x3x3 cube. Find the probability that the entire surface of the larger cube is orange.

(2011 AIME I) The probability that a set of three distinct vertices chosen at random from among the vertices of a regular n-gon determine an obtuse triangle is 93/125. Find the sum of all possible values of n.

(2017 AIME I) A circle circumscribes an isosceles triangle whose two congruent angles have degree measure x. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is 14/25. Find the difference between the largest and smallest possible values of x.

(2017 AIME I) Two real numbers a and b are chosen independently and uniformly at random from the interval (0,75). Let O and P be two points on the plane with OP=200. Let Q and R be on the same side of line OP such that the degree measures of angles POQ and POR are a and b, respectively, and angles OQP and ORP are both right angles. Find the probability that QR <=100.

(2020 AIME II) Let P be a point chosen uniformly at random in the interior of the unit square with vertices at (0,0),(1,0),(1,1), and (0,1). Find the probability that the slope of the line determined by P and the point (5/8,3/8) is greater than or equal to 1/2.

probability - cyclic

(2021 AIME I) Let A₁A₂A₃…A₁₂ be a dodecagon (12-gon). Three frogs initially sit at A₄,A₈ and A₁₂. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. Find the expected number of minutes until the frogs stop jumping.

(2022 12B) A bug starts at a vertex of a grid made of equilateral triangles of side length 1. At each step the bug moves in one of the 6 possible directions along the grid lines randomly and independently with equal probability. What is the probability that after 5 moves the bug never will have been more than unit away from the starting position?