Difference between revisions of "2018 AIME I Problems"
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==Problem 12== | ==Problem 12== | ||
| − | + | For every subset <math>T</math> of <math>U = \{ 1,2,3,\ldots,18 \}</math>, let <math>s(T)</math> be the sum of the elements of <math>T</math>, with <math>s(\emptyset)</math> defined to be <math>0</math>. If <math>T</math> is chosen at random among all subsets of <math>U</math>, the probability that <math>s(T)</math> is divisible by <math>3</math> is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m</math>. | |
[[2018 AIME I Problems/Problem 12 | Solution]] | [[2018 AIME I Problems/Problem 12 | Solution]] | ||
Revision as of 15:59, 7 March 2018
| 2018 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
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Instructions
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| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.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
Find the least positive integer 
such that when 
is written in base 
, its two right-most digits in base are 
.
Problem 12
For every subset 
of 
, let 
be the sum of the elements of , with 
defined to be 
. If is chosen at random among all subsets of 
, the probability that is divisible by 
is 
, where 
and are relatively prime positive integers. Find .
Problem 13
Problem 14
Problem 15
| 2018 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by 2017 AIME II |
Followed by 2018 AIME II | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
