Difference between revisions of "2017 AIME I Problems"
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==Problem 2== | ==Problem 2== | ||
− | When each of 702, 787, and 855 is divided by the positive integer <math>m</math>, the remainder is always the positive integer <math>r</math>. When each of 412, 722, and 815 is divided by the positive integer <math>n</math>, the remainder is always the positive integer <math>s \neq r</math>. | + | When each of <math>702</math>, <math>787</math>, and <math>855</math> is divided by the positive integer <math>m</math>, the remainder is always the positive integer <math>r</math>. When each of <math>412</math>, <math>722</math>, and <math>815</math> is divided by the positive integer <math>n</math>, the remainder is always the positive integer <math>s \neq r</math>. Find <math>m+n+r+s</math>. |
[[2017 AIME I Problems/Problem 2 | Solution]] | [[2017 AIME I Problems/Problem 2 | Solution]] | ||
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==Problem 6== | ==Problem 6== | ||
− | A circle | + | A circle circumscribes an isosceles triangle whose two congruent angles have degree measure <math>x</math>. 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 <math>\frac{14}{25}</math>. Find the difference between the largest and smallest possible values of <math>x</math>. |
[[2017 AIME I Problems/Problem 6 | Solution]] | [[2017 AIME I Problems/Problem 6 | Solution]] | ||
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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 11== | ==Problem 11== | ||
+ | Consider arrangements of the <math>9</math> numbers <math>1, 2, 3, \dots, 9</math> in a <math>3 \times 3</math> array. For each such arrangement, let <math>a_1</math>, <math>a_2</math>, and <math>a_3</math> be the medians of the numbers in rows <math>1</math>, <math>2</math>, and <math>3</math> respectively, and let <math>m</math> be the median of <math>\{a_1, a_2, a_3\}</math>. Let <math>Q</math> be the number of arrangements for which <math>m = 5</math>. Find the remainder when <math>Q</math> is divided by <math>1000</math>. | ||
+ | |||
[[2017 AIME I Problems/Problem 11 | Solution]] | [[2017 AIME I Problems/Problem 11 | Solution]] | ||
==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, \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 mn</math>. Find the remainder when | ||
+ | <cmath> \sum_{m = 2}^{2017} Q(m) </cmath>is divided by <math>1000</math>. | ||
+ | |||
[[2017 AIME I Problems/Problem 13 | Solution]] | [[2017 AIME I Problems/Problem 13 | Solution]] | ||
==Problem 14== | ==Problem 14== | ||
+ | Let <math>a > 1</math> and <math>x > 1</math> satisfy <math>\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128</math> and <math>\log_a(\log_a x) = 256</math>. Find the remainder when <math>x</math> is divided by <math>1000</math>. | ||
+ | |||
[[2017 AIME I Problems/Problem 14 | Solution]] | [[2017 AIME I Problems/Problem 14 | Solution]] | ||
==Problem 15== | ==Problem 15== | ||
+ | |||
+ | The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths <math>2\sqrt3</math>, <math>5</math>, and <math>\sqrt{37}</math>, as shown, is <math>\tfrac{m\sqrt{p}}{n}</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, <math>m</math> and <math>n</math> are relatively prime, and <math>p</math> is not divisible by the square of any prime. Find <math>m+n+p</math>. | ||
+ | |||
+ | <asy> | ||
+ | size(5cm); | ||
+ | pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0); | ||
+ | real t = .385, s = 3.5*t-1; | ||
+ | pair R = A*t+B*(1-t), P=B*s; | ||
+ | pair Q = dir(-60) * (R-P) + P; | ||
+ | fill(P--Q--R--cycle,gray); | ||
+ | draw(A--B--C--A^^P--Q--R--P); | ||
+ | dot(A--B--C--P--Q--R); | ||
+ | </asy> | ||
+ | |||
[[2017 AIME I Problems/Problem 15 | Solution]] | [[2017 AIME I Problems/Problem 15 | Solution]] | ||
− | {{AIME box|year=2017|n=I|before=[[2016 AIME II]]|after=[[2017 AIME II]]}} | + | {{AIME box|year=2017|n=I|before=[[2016 AIME II Problems]]|after=[[2017 AIME II Problems]]}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 16:43, 2 June 2022
2017 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
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
.
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.