Difference between revisions of "2017 AIME I Problems"
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==Problem 9== | ==Problem 9== | ||
− | Let <math>a_{10} = 10</math>, and for each integer <math>n >10</math> let <math>a_n = 100a_{n - 1} + n</math>. Find the least <math>n > 10</math> such that <math>a_n</math> is a multiple of <math>99</math>. | + | Let <math>a_{10} = 10</math>, and for each positive integer <math>n >10</math> let <math>a_n = 100a_{n - 1} + n</math>. Find the least positive <math>n > 10</math> such that <math>a_n</math> is a multiple of <math>99</math>. |
[[2017 AIME I Problems/Problem 9 | Solution]] | [[2017 AIME I Problems/Problem 9 | Solution]] | ||
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==Problem 12== | ==Problem 12== | ||
− | Call a set <math>S</math> product-free if there do not exist <math>a, b, c \in S</math> (not necessarily distinct) such that <math>a b = c</math>. For example, the empty set and the set <math>\{16, 20\}</math> are product-free, whereas the sets <math>\{4, 16\}</math> and <math>\{2, 8, 16\}</math> are not product-free. Find the number of product-free subsets of the set <math>\{1, 2, 3, 4, | + | Call a set <math>S</math> product-free if there do not exist <math>a, b, c \in S</math> (not necessarily distinct) such that <math>a b = c</math>. For example, the empty set and the set <math>\{16, 20\}</math> are product-free, whereas the sets <math>\{4, 16\}</math> and <math>\{2, 8, 16\}</math> are not product-free. Find the number of product-free subsets of the set <math>\{1, 2, 3, 4, \ldots, 7, 8, 9, 10\}</math>. |
[[2017 AIME I Problems/Problem 12 | Solution]] | [[2017 AIME I Problems/Problem 12 | Solution]] | ||
==Problem 13== | ==Problem 13== | ||
− | For every <math>m \geq 2</math>, let <math>Q(m)</math> be the least positive integer with the following property: For every <math>n \geq Q(m)</math>, there is always a perfect cube <math>k^3</math> in the range <math>n < k^3 \leq | + | For every <math>m \geq 2</math>, let <math>Q(m)</math> be the least positive integer with the following property: For every <math>n \geq Q(m)</math>, there is always a perfect cube <math>k^3</math> in the range <math>n < k^3 \leq mn</math>. Find the remainder when |
<cmath> \sum_{m = 2}^{2017} Q(m) </cmath>is divided by 1000. | <cmath> \sum_{m = 2}^{2017} Q(m) </cmath>is divided by 1000. | ||
Latest revision as of 21:57, 22 September 2020
2017 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
Problem 1
Fifteen distinct points are designated on : the 3 vertices , , and ; other points on side ; other points on side ; and other points on side . Find the number of triangles with positive area whose vertices are among these points.
Problem 2
When each of , , and is divided by the positive integer , the remainder is always the positive integer . When each of , , and is divided by the positive integer , the remainder is always the positive integer . Find .
Problem 3
For a positive integer , let be the units digit of . Find the remainder when is divided by .
Problem 4
A pyramid has a triangular base with side lengths , , and . The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length . The volume of the pyramid is , where and are positive integers, and is not divisible by the square of any prime. Find .
Problem 5
A rational number written in base eight is , where all digits are nonzero. The same number in base twelve is . Find the base-ten number .
Problem 6
A circle circumscribes an isosceles triangle whose two congruent angles have degree measure . Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is . Find the difference between the largest and smallest possible values of .
Problem 7
For nonnegative integers and with , let . Let denote the sum of all , where and are nonnegative integers with . Find the remainder when is divided by .
Problem 8
Two real numbers and are chosen independently and uniformly at random from the interval . Let and be two points on the plane with . Let and be on the same side of line such that the degree measures of and are and respectively, and and are both right angles. The probability that is equal to , where and are relatively prime positive integers. Find .
Problem 9
Let , and for each positive integer let . Find the least positive such that is a multiple of .
Problem 10
Let , and where . Let be the unique complex number with the properties that is a real number and the imaginary part of is the greatest possible. Find the real part of .
Problem 11
Consider arrangements of the numbers in a array. For each such arrangement, let , , and be the medians of the numbers in rows , , and respectively, and let be the median of . Let be the number of arrangements for which . Find the remainder when is divided by .
Problem 12
Call a set product-free if there do not exist (not necessarily distinct) such that . For example, the empty set and the set are product-free, whereas the sets and are not product-free. Find the number of product-free subsets of the set .
Problem 13
For every , let be the least positive integer with the following property: For every , there is always a perfect cube in the range . Find the remainder when is divided by 1000.
Problem 14
Let and satisfy and . Find the remainder when is divided by .
Problem 15
The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths , , and , as shown, is , where , , and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find .
2017 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2016 AIME II Problems |
Followed by 2017 AIME II Problems | |
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.