Difference between revisions of "2018 AIME I Problems"
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+ | Let <math>N</math> be the number of complex numbers <math>z</math> with the properties that <math>|z|=1</math> and <math>z^{6!}-z^{5!}</math> is a real number. Find the remainder when <math>N</math> is divided by <math>1000</math>. | ||
Revision as of 16:03, 7 March 2018
2018 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
[hide]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
The number
Problem 3
Kathy has
Problem 4
In
Problem 5
Problem 6
Let be the number of complex numbers with the properties that and is a real number. Find the remainder when is divided by .
Problem 7
A right hexagonal prism has height . The bases are regular hexagons with side length . Any of the vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).
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
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
Let
Problem 14
Let be a heptagon. A frog starts jumping at vertex . From any vertex of the heptagon except , the frog may jump to either fo the two adjacent vertices. When it reaches vertex , the frog stops and stays there. Find the number of distinct sequences of jumps of no more than jumps that end at .
Problem 15
David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals,
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 |
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