# 2018 AIME I Problems

 2018 AIME I (Answer Key) Printable version | AoPS Contest Collections 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. 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

## Problem 1

Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is divided by $1000$.

## Problem 3

Kathy has $$5$$ red cards and $$5$$ green cards. She shuffles the $$10$$ cards and lays out $$5$$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders $$RRGGG, GGGGR,$$ or $$RRRRR$$ will make Kathy happy, but $$RRRGR$$ will not. The probability that Kathy will be happy is $$\dfrac{m}{n}$$, where $$m$$ and $$n$$ are relatively prime positive integers. Find $$m + n$$.

## Problem 4

In $$\triangle ABC, AB = AC = 10$$ and $$BC = 12$$. Point $$D$$ lies strictly between $$A$$ and $$B$$ on $$\overline{AB}$$ and point $$E$$ lies strictly between $$A$$ and $$C$$ on $$\overline{AC}$$ so that $$AD = DE = EC$$. Then $$AD$$ can be expressed in the form $$\dfrac{p}{q}$$, where $$p$$ and $$q$$ are relatively prime positive integers. Find $$p + q$$.

## Problem 9

Find the number of four-element subsets of $\{1,2,3,4,\dots, 20\}$ with the property that two distinct elements of a subset have a sum of $16$, and two distinct elements of a subset have a sum of $24$. For example, $\{3,5,13,19\}$ and $\{6,10,20,18\}$ are two such subsets.

## Problem 10

The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point $$A$$. At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a circular clockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path $$AJABCHCHIJA$$, which has $$10$$ steps. Let $$n$$ be the number of paths with $$15$$ steps that begin and end at point $$A$$. Find the remainder when $$n$$ is divided by $$1000$$.

## Problem 11

Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$.

## Problem 12

For every subset $T$ of $U = \{ 1,2,3,\ldots,18 \}$, let $s(T)$ be the sum of the elements of $T$, with $s(\emptyset)$ defined to be $0$. If $T$ is chosen at random among all subsets of $U$, the probability that $s(T)$ is divisible by $3$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.

## Problem 15

 2018 AIME I (Problems • Answer Key • Resources) Preceded by2017 AIME II Followed by2018 AIME II 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 All AIME Problems and Solutions