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2020 AIME I Problems

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2020 AIME I (Answer Key) Printable version \| AoPS Contest Collections • PDF
Instructions This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$ , inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
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Contents

1 Problem 1
2 Problem 2
3 Problem 3
4 Problem 4
5 Problem 5
6 Problem 6
7 Problem 7
8 Problem 8
9 Problem 9
10 Problem 10
11 Problem 11
12 Problem 12
13 Problem 13
14 Problem 14
15 Problem 15

Problem 1

Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule!

Problem 2

Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule!

Problem 3

Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule!

Problem 4

Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule!

Problem 5

Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule!

Problem 6

Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule!

Problem 7

Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule!

Problem 8

Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule! Pakistowners rule!

Problem 9

Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random with replacement from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m.$

Problem 10

Let $m$ and $n$ be positive integers satisfying the conditions

$\quad\bullet\ \gcd(m+n,210)=1,$

$\quad\bullet\ m^m$ is a multiple of $n^n,$ and

$\quad\bullet\ m$ is not a multiple of $n.$

Find the least possible value of $m+n.$

Problem 11

For integers $a,b,c$ and $d,$ let $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d.$ 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 $g(f(2))=g(f(4))=0.$

Problem 12

Let $n$ be the least positive integer for which $149^n-2^n$ is divisible by $3^3\cdot5^5\cdot7^7.$ Find the number of positive integer divisors of $n.$

Problem 13

Point $D$ lies on side $\overline{BC}$ of $\triangle ABC$ so that $\overline{AD}$ bisects $\angle BAC.$ The perpendicular bisector of $\overline{AD}$ intersects the bisectors of $\angle ABC$ and $\angle ACB$ in points $E$ and $F,$ respectively. Given that $AB=4,BC=5,$ and $CA=6,$ the area of $\triangle AEF$ can be written as $\tfrac{m\sqrt{n}}p,$ where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p.$

Problem 14

Let $P(x)$ be a quadratic polynomial with complex coefficients whose $x^2$ coefficient is $1.$ Suppose the equation $P(P(x))=0$ has four distinct solutions, $x=3,4,a,b.$ Find the sum of all possible values of $(a+b)^2.$

Problem 15

Let $\triangle ABC$ be an acute triangle with circumcircle $\omega,$ and let $H$ be the intersection of the altitudes of $\triangle ABC.$ Suppose the tangent to the circumcircle of $\triangle HBC$ at $H$ intersects $\omega$ at points $X$ and $Y$ with $HA=3,HX=2,$ and $HY=6.$ The area of $\triangle ABC$ can be written in the form $m\sqrt{n},$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n.$

2020 AIME I (Problems • Answer Key • Resources)
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