MathDB

2019 AIME Problems

Part of AIME Problems

Subcontests

(15)
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Soccer substitutions

A soccer team has 22 available players. A fixed set of 11 players starts the game, while the other 11 are available as substitutes. During the game, the coach may make as many as 3 substitutions, where any one of the 11 players in the game is replaced by one of the substitutes. No player removed from the game may reenter the game, although a substitute entering the game may be replaced later. No two substitutions can happen at the same time. The players involved and the order of the substitutions matter. Let nn be the number of ways the coach can make substitutions during the game (including the possibility of making no substitutions). Find the remainder when nn is divided by 1000.

Perfect Square Dice

A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is mn\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
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so what's an excircle?

In △ABC\triangle ABC, the sides have integers lengths and AB=ACAB=AC. Circle ω\omega has its center at the incenter of △ABC\triangle ABC. An excircle of △ABC\triangle ABC is a circle in the exterior of △ABC\triangle ABC that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to BC‾\overline{BC} is internally tangent to ω\omega, and the other two excircles are both externally tangent to ω\omega. Find the minimum possible value of the perimeter of △ABC\triangle ABC.

angle chase or bash

Triangle ABCABC has side lengths AB=7,BC=8,AB=7, BC=8, and CA=9.CA=9. Circle ω1\omega_1 passes through BB and is tangent to line ACAC at A.A. Circle ω2\omega_2 passes through CC and is tangent to line ABAB at A.A. Let KK be the intersection of circles ω1\omega_1 and ω2\omega_2 not equal to A.A. Then AK=mn,AK=\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+n.m+n.
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8
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Particle Pathing

A moving particle starts at the point (4,4)\left(4,4\right) and moves until it hits one of the coordinate axes for the first time. When the particle is at the point (a,b)\left(a,b\right), it moves at random to one of the points (a−1,b)\left(a-1,b\right), (a,b−1)\left(a,b-1\right), or (a−1,b−1)\left(a-1,b-1\right), each with probability 13\tfrac{1}{3}, independently of its previous moves. The probability that it will hit the coordinate axes at (0,0)\left(0,0\right) is m3n\tfrac{m}{3^n}, where mm and nn are positive integers, and mm is not divisible by 33. Find m+nm+n.

Ambassador's Round Table

Four ambassadors and one advisor for each of them are to be seated at a round table with 1212 chairs numbered in order from 11 to 1212. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are NN ways for the 88 people to be seated at the table under these conditions. Find the remainder when NN is divided by 10001000.
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high quality system

There are positive integers xx and yy that satisfy the system of equations \begin{align*} \log_{10} x + 2 \log_{10} (\gcd(x,y)) &= 60 \\ \log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) &= 570. \end{align*} Let mm be the number of (not necessarily distinct) prime factors in the prime factorization of xx, and let nn be the number of (not necessarily distinct) prime factors in the prime factorization of yy. Find 3m+2n3m+2n.

Similar Triangles

Triangle ABCABC has side lengths AB=120AB=120, BC=220BC=220, and AC=180AC=180. Lines ℓA\ell_{A}, ℓB\ell_{B}, and ℓC\ell_{C} are drawn parallel to BC‾\overline{BC}, AC‾\overline{AC}, and AB‾\overline{AB}, respectively, such that the intersection of ℓA\ell_{A}, ℓB\ell_{B}, and ℓC\ell_{C} with the interior of △ABC\triangle ABC are segments of length 5555, 4545, and 1515, respectively. Find the perimeter of the triangle whose sides lie on ℓA\ell_{A}, ℓB\ell_{B}, and ℓC\ell_{C}.
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Circumcircle Intersections

Triangle ABCABC has side lengths AB=4AB=4, BC=5BC=5, and CA=6CA=6. Points DD and EE are on ray ABAB with AB<AD<AEAB<AD<AE. The point F≠CF \neq C is a point of intersection of the circumcircles of △ACD\triangle ACD and △EBC\triangle EBC satisfying DF=2DF=2 and EF=7EF=7. Then BEBE can be expressed as a+bcd\tfrac{a+b\sqrt{c}}{d}, where aa, bb, cc, and dd are positive integers such that aa and dd are relatively prime, and cc is not divisible by the square of any prime. Find a+b+c+da+b+c+d.

Point Inside an Octagon

Regular octagon A1A2A3A4A5A6A7A8A_1A_2A_3A_4A_5A_6A_7A_8 is inscribed in a circle of area 11. Point PP lies inside the circle so that the region bounded by PA1‾\overline{PA_1}, PA2‾\overline{PA_2}, and the minor arc A1A2^\widehat{A_1A_2} of the circle has area 17\tfrac17, while the region bounded by PA3‾\overline{PA_3}, PA4‾\overline{PA_4}, and the minor arc A3A4^\widehat{A_3A_4} of the circle has area 19\tfrac 19. There is a positive integer nn such that the area of the region bounded by PA6‾\overline{PA_6}, PA7‾\overline{PA_7}, and the minor arc A6A7^\widehat{A_6A_7} is equal to 18−2n\tfrac18 - \tfrac{\sqrt 2}n. Find nn.
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2019 Roots

For distinct complex numbers z1,z2,…,z673z_1,z_2,\dots,z_{673}, the polynomial (x−z1)3(x−z2)3⋯(x−z673)3 (x-z_1)^3(x-z_2)^3 \cdots (x-z_{673})^3 can be expressed as x2019+20x2018+19x2017+g(x)x^{2019} + 20x^{2018} + 19x^{2017}+g(x), where g(x)g(x) is a polynomial with complex coefficients and with degree at most 20162016. The value of ∣∑1≤j<k≤673zjzk∣ \left| \sum_{1 \le j <k \le 673} z_jz_k \right| can be expressed in the form mn\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.

0, 1, 2, 3, 0, ...

There is a unique angle θ\theta between 0∘0^{\circ} and 90∘90^{\circ} such that for nonnegative integers nn, the value of tan⁡(2nθ)\tan{\left(2^{n}\theta\right)} is positive when nn is a multiple of 33, and negative otherwise. The degree measure of θ\theta is pq\tfrac{p}{q}, where pp and qq are relatively prime integers. Find p+qp+q.
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AIME I Problem 2

Jenn randomly chooses a number JJ from 1,2,3,…,19,201, 2, 3,\ldots, 19, 20. Bela then randomly chooses a number BB from 1,2,3,…,19,201, 2, 3,\ldots, 19, 20 distinct from JJ. The value of B−JB - J is at least 22 with a probability that can be expressed in the form mn\frac{m}{n} where mm and nn are relatively prime positive integers. Find m+nm+n.

Lily pads and a Frog

Lily pads 1,2,3,…1,2,3,\ldots lie in a row on a pond. A frog makes a sequence of jumps starting on pad 11. From any pad kk the frog jumps to either pad k+1k+1 or pad k+2k+2 chosen randomly and independently with probability 12\tfrac12. The probability that the frog visits pad 77 is pq\tfrac pq, where pp and qq are relatively prime positive integers. Find p+qp+q.
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a wild symmedian appears

Let AB‾\overline{AB} be a chord of a circle ω\omega, and let PP be a point on the chord AB‾\overline{AB}. Circle ω1\omega_1 passes through AA and PP and is internally tangent to ω\omega. Circle ω2\omega_2 passes through BB and PP and is internally tangent to ω\omega. Circles ω1\omega_1 and ω2\omega_2 intersect at points PP and QQ. Line PQPQ intersects ω\omega at XX and YY. Assume that AP=5AP=5, PB=3PB=3, XY=11XY=11, and PQ2=mnPQ^2 = \tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.

Line Passing through Feet of Altitudes Intersects Circumcircle

In acute triangle ABCABC points PP and QQ are the feet of the perpendiculars from CC to AB‾\overline{AB} and from BB to AC‾\overline{AC}, respectively. Line PQPQ intersects the circumcircle of △ABC\triangle ABC in two distinct points, XX and YY. Suppose XP=10XP=10, PQ=25PQ=25, and QY=15QY=15. The value of AB⋅ACAB\cdot AC can be written 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.