2018 AIME I Problems
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Instructions
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Contents
Problem 1
Let be the number of ordered pairs of integers
with
and
such that the polynomial
can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when
is divided by
.
Problem 2
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
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Find the number of four-element subsets of with the property that two distinct elements of a subset have a sum of
, and two distinct elements of a subset have a sum of
. For example,
and
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
Problem 12
Problem 13
Problem 14
Problem 15
2018 AIME I (Problems • Answer Key • Resources) | ||
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.