MathDB

2022 AIME Problems

Part of AIME Problems

Subcontests

(15)
11
2

Circle in a Parallelogram

Let ABCDABCD be a parallelogram with ∠BAD<90∘\angle BAD < 90^{\circ}. A circle tangent to sides DA‾\overline{DA}, AB‾\overline{AB}, and BC‾\overline{BC} intersects diagonal AC‾\overline{AC} at points PP and QQ with AP<AQAP < AQ, as shown. Suppose that AP=3AP = 3, PQ=9PQ = 9, and QC=16QC = 16. Then the area of ABCDABCD can be expressed in the form mnm\sqrt n, where mm and nn are positive integers, and nn is not divisible by the square of any prime. Find m+nm+n.
[asy] defaultpen(linewidth(0.6)+fontsize(11)); size(8cm); pair A,B,C,D,P,Q; A=(0,0); label("AA", A, SW); B=(6,15); label("BB", B, NW); C=(30,15); label("CC", C, NE); D=(24,0); label("DD", D, SE); P=(5.2,2.6); label("PP", (5.8,2.6), N); Q=(18.3,9.1); label("QQ", (18.1,9.7), W); draw(A--B--C--D--cycle); draw(C--A); draw(Circle((10.95,7.45), 7.45)); dot(A^^B^^C^^D^^P^^Q); [/asy]

MAA likes misplacing problems this year

Let ABCDABCD be a convex quadrilateral with AB=2,AD=7,AB=2, AD=7, and CD=3CD=3 such that the bisectors of acute angles ∠DAB\angle{DAB} and ∠ADC\angle{ADC} intersect at the midpoint of BC‾.\overline{BC}. Find the square of the area of ABCD.ABCD.
10
2
8
2

Geo(graphy) Reading Passage

Equilateral triangle △ABC\triangle ABC is inscribed in circle ω\omega with radius 18.18. Circle ωA\omega_A is tangent to sides AB‾\overline{AB} and AC‾\overline{AC} and is internally tangent to ω\omega. Circles ωB\omega_B and ωC\omega_C are defined analogously. Circles ωA\omega_A, ωB\omega_B, and ωC\omega_C meet in six points−-two points for each pair of circles. The three intersection points closest to the vertices of △ABC\triangle ABC are the vertices of a large equilateral triangle in the interior of △ABC\triangle ABC, and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of △ABC\triangle ABC. The side length of the smaller equilateral triangle can be written as a−b\sqrt{a}-\sqrt{b}, where aa and bb are positive integers. Find a+ba+b.

Which floor was this?

Find the number of positive integers n≤600n \le 600 whose value can be uniquely determined when the values of ⌊n4⌋\left\lfloor \frac n4\right\rfloor, ⌊n5⌋\left\lfloor\frac n5\right\rfloor, and ⌊n6⌋\left\lfloor\frac n6\right\rfloor are given, where ⌊x⌋\lfloor x \rfloor denotes the greatest integer less than or equal to the real number xx.
5
2

Fall Flashbacks Part 2

A straight river that is 264264 meters wide flows from west to east at a rate of 1414 meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of DD meters downstream from Sherry. Relative to the water, Melanie swims at 8080 meters per minute, and Sherry swims at 6060 meters per minute. At the same time, Melanie and Sherry begin swimming in straight lines to a point on the north bank of the river that is equidistant from their starting positions. The two women arrive at this point simultaneously. Find DD.

Triangles from Prime Numbers

Twenty distinct points are marked on a circle and labeled 11 through 2020 in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original 2020 points.
4
2
14
2

Beauty and the Bash

Given △ABC\triangle ABC and a point PP on one of its sides, call line ℓ\ell the splitting line of △ABC\triangle ABC through PP if ℓ\ell passes through PP and divides △ABC\triangle ABC into two polygons of equal perimeter. Let △ABC\triangle ABC be a triangle where BC=219BC = 219 and ABAB and ACAC are positive integers. Let MM and NN be the midpoints of AB‾\overline{AB} and AC‾\overline{AC}, respectively, and suppose that the splitting lines of △ABC\triangle ABC through MM and NN intersect at 30∘30^{\circ}. Find the perimeter of △ABC\triangle ABC.

I got money on my mind

For positive integers aa, bb, and cc with a<b<ca < b < c, consider collections of postage stamps in denominations aa, bb, and cc cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole number of cents up to 10001000 cents, let f(a,b,c)f(a, b, c) be the minimum number of stamps in such a collection. Find the sum of the three least values of cc such that f(a,b,c)=97f(a, b, c) = 97 for some choice of aa and bb.
3
2
15
2

Circumradii???

Let xx, yy, and zz be positive real numbers satisfying the system of equations \begin{align*} \sqrt{2x - xy} + \sqrt{2y - xy} & = 1\\ \sqrt{2y - yz} + \hspace{0.1em} \sqrt{2z - yz} & = \sqrt{2}\\ \sqrt{2z - zx\vphantom{y}} + \sqrt{2x - zx\vphantom{y}} & = \sqrt{3}. \end{align*}Then [(1−x)(1−y)(1−z)]2\big[ (1-x)(1-y)(1-z) \big] ^2 can be written as mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.

Another Hexagon Problem?

Two externally tangent circles ω1\omega_1 and ω2\omega_2 have centers O1O_1 and O2O_2, respectively. A third circle Ω\Omega passing through O1O_1 and O2O_2 intersects ω1\omega_1 at BB and CC and ω2\omega_2 at AA and DD, as shown. Suppose that AB=2AB = 2, O1O2=15O_1O_2 = 15, CD=16CD = 16, and ABO1CDO2ABO_1CDO_2 is a convex hexagon. Find the area of this hexagon. [asy] import geometry; size(10cm); point O1=(0,0),O2=(15,0),B=9*dir(30); circle w1=circle(O1,9),w2=circle(O2,6),o=circle(O1,O2,B); point A=intersectionpoints(o,w2)[1],D=intersectionpoints(o,w2)[0],C=intersectionpoints(o,w1)[0]; filldraw(A--B--O1--C--D--O2--cycle,0.2*red+white,black); draw(w1); draw(w2); draw(O1--O2,dashed); draw(o); dot(O1); dot(O2); dot(A); dot(D); dot(C); dot(B); label("ω1\omega_1",8*dir(110),SW); label("ω2\omega_2",5*dir(70)+(15,0),SE); label("O1O_1",O1,W); label("O2O_2",O2,E); label("BB",B,N+1/2*E); label("AA",A,N+1/2*W); label("CC",C,S+1/4*W); label("DD",D,S+1/4*E); label("1515",midpoint(O1--O2),N); label("1616",midpoint(C--D),N); label("22",midpoint(A--B),S); label("Ω\Omega",o.C+(o.r-1)*dir(270)); [/asy]
2
2

I am based

Find the three-digit positive integer a‾ b‾ c‾\underline{a} \ \underline{b} \ \underline{c} whose representation in base nine is b‾ c‾ a‾nine\underline{b} \ \underline{c} \ \underline{a}_{\hspace{.02in}\text{nine}}, where aa, bb, and cc are (not necessarily distinct) digits.

Another year another tournament

Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability 23\frac23. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability 34\frac34. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is pq\frac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp+q.
1
2
7
2
9
2

RBBYGGYROPPO GANGNAM STYLE

Ellina has twelve blocks, two each of red (R),\left({\bf R}\right), blue (B),\left({\bf B}\right), yellow (Y),\left({\bf Y}\right), green (G),\left({\bf G}\right), orange (O),\left({\bf O}\right), and purple (P).\left({\bf P}\right). Call an arrangement of blocks even if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement R B B Y G G Y R O P P O {\text {\bf R B B Y G G Y R O P P O}}is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is mn\frac mn, where mm and nn are relatively prime positive integers. Find m+nm+n.

When MAA can't decide between geo and combo

Let ℓA\ell_A and ℓB\ell_B be two distinct parallel lines. For positive integers mm and nn, distinct points A1,A2,A3,…,AmA_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m lie on ℓA\ell_A, and distinct points B1,B2,B3,…,BnB_1, B_2, B_3, \ldots, B_n lie on ℓB\ell_B. Additionally, when segments AiBj‾\overline{A_iB_j} are drawn for all i=1,2,3,…,mi=1,2,3,\ldots, m and j=1,2,3,…,nj=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n, no point strictly between ℓA\ell_A and ℓB\ell_B lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when m=7m=7 and n=5n=5. The figure shows that there are 8 regions when m=3m=3 and n=2n=2. [asy] import geometry; size(10cm); draw((-2,0)--(13,0)); draw((0,4)--(10,4)); label("ℓA\ell_A",(-2,0),W); label("ℓB\ell_B",(0,4),W); point A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2); draw(B1--A1--B2); draw(B1--A2--B2); draw(B1--A3--B2); label("A1A_1",A1,S); label("A2A_2",A2,S); label("A3A_3",A3,S); label("B1B_1",B1,N); label("B2B_2",B2,N); label("1",centroid(A1,B1,I1)); label("2",centroid(B1,I1,I3)); label("3",centroid(B1,B2,I3)); label("4",centroid(A1,A2,I1)); label("5",(A2+I1+I2+I3)/4); label("6",centroid(B2,I2,I3)); label("7",centroid(A2,A3,I2)); label("8",centroid(A3,B2,I2)); dot(A1); dot(A2); dot(A3); dot(B1); dot(B2); [/asy]
6
2

imagine not sillying this

Find the number of ordered pairs of integers (a,b)(a, b) such that the sequence 3,4,5,a,b,30,40,503, 4, 5, a, b, 30, 40, 50 is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.

Maximum Difference

Let x1≤x2≤⋯≤x100x_1\leq x_2\leq \cdots\leq x_{100} be real numbers such that ∣x1∣+∣x2∣+⋯+∣x100∣=1|x_1| + |x_2| + \cdots + |x_{100}| = 1 and x1+x2+⋯+x100=0x_1 + x_2 + \cdots + x_{100} = 0. Among all such 100100-tuples of numbers, the greatest value that x76−x16x_{76} - x_{16} can achieve is mn\tfrac mn, where mm and nn are relatively prime positive integers. Find m+nm+n.
13
2

The Quintessential Quadruplets

Let SS be the set of all rational numbers that can be expressed as a repeating decimal in the form 0.abcd‾,0.\overline{abcd}, where at least one of the digits a,b,c,a, b, c, or dd is nonzero. Let NN be the number of distinct numerators when numbers in SS are written as fractions in lowest terms. For example, both 44 and 410410 are counted among the distinct numerators for numbers in SS because 0.3636‾=4110.\overline{3636} = \frac{4}{11} and 0.1230‾=4103333.0.\overline{1230} = \frac{410}{3333}. Find the remainder when NN is divided by 1000.1000.

looks like roots of unity filter!

There is a polynomial P(x)P(x) with integer coefficients such that P(x)=(x2310−1)6(x105−1)(x70−1)(x42−1)(x30−1)P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)} holds for every 0<x<1.0<x<1. Find the coefficient of x2022x^{2022} in P(x)P(x)
12
2

It's not like I share elements with you or anything, baka!

For any finite set XX, let ∣X∣|X| denote the number of elements in X.X. Define Sn=∑∣A∩B∣,S_n = \sum |A \cap B|, where the sum is taken over all ordered pairs (A,B)(A, B) such that AA and BB are subsets of {1,2,3,…,n}\{1, 2, 3, …, n\} with ∣A∣=∣B∣.|A| = |B|. For example, S2=4S_2 = 4 because the sum is taken over the pairs of subsets (A,B)∈{(∅,∅),({1},{1}),({1},{2}),({2},{1}),({2},{2}),({1,2},{1,2})},(A, B) \in \{ (\emptyset, \emptyset), (\{1\}, \{1\}), (\{1\}, \{2\}), (\{2\}, \{1\}), (\{2\}, \{2\}), (\{1, 2\}, \{1, 2\})\}, giving S2=0+1+0+0+1+2=4.S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4. Let S2022S2021=pq,\frac{S_{2022}}{S_{2021}} = \frac{p}{q}, where pp and qq are relatively prime positive integers. Find the remainder when p+qp + q is divided by 1000.1000.

conics ew

Let a,b,x,a, b, x, and yy be real numbers with a>4a>4 and b>1b>1 such that x2a2+y2a2−16=(x−20)2b2−1+(y−11)2b2=1.\frac{x^2}{a^2}+\frac{y^2}{a^2-16}=\frac{(x-20)^2}{b^2-1}+\frac{(y-11)^2}{b^2}=1. Find the least possible value of a+b.a+b.