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Difference between revisions of "2017 AIME I Problems"

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==Problem 1==
 
==Problem 1==
 +
Fifteen distinct points are designated on <math>\triangle ABC</math>: the 3 vertices <math>A</math>, <math>B</math>, and <math>C</math>; <math>3</math> other points on side <math>\overline{AB}</math>; <math>4</math> other points on side <math>\overline{BC}</math>; and <math>5</math> other points on side <math>\overline{CA}</math>. Find the number of triangles with positive area whose vertices are among these <math>15</math> points.
 +
 
[[2017 AIME I Problems/Problem 1 | Solution]]
 
[[2017 AIME I Problems/Problem 1 | Solution]]
  
 
==Problem 2==
 
==Problem 2==
 +
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]]
  
 
==Problem 3==
 
==Problem 3==
 +
For a positive integer <math>n</math>, let <math>d_n</math> be the units digit of <math>1 + 2 + \dots + n</math>. Find the remainder when
 +
<cmath>\sum_{n=1}^{2017} d_n</cmath>is divided by <math>1000</math>.
 +
 
[[2017 AIME I Problems/Problem 3 | Solution]]
 
[[2017 AIME I Problems/Problem 3 | Solution]]
  
 
==Problem 4==
 
==Problem 4==
 +
A pyramid has a triangular base with side lengths <math>20</math>, <math>20</math>, and <math>24</math>. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length <math>25</math>. The volume of the pyramid is <math>m\sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>.
 +
 
[[2017 AIME I Problems/Problem 4 | Solution]]
 
[[2017 AIME I Problems/Problem 4 | Solution]]
  
 
==Problem 5==
 
==Problem 5==
 +
A rational number written in base eight is <math>\underline{a} \underline{b} . \underline{c} \underline{d}</math>, where all digits are nonzero. The same number in base twelve is <math>\underline{b} \underline{b} . \underline{b} \underline{a}</math>. Find the base-ten number <math>\underline{a} \underline{b} \underline{c}</math>.
 +
 
[[2017 AIME I Problems/Problem 5 | Solution]]
 
[[2017 AIME I Problems/Problem 5 | Solution]]
  
 
==Problem 6==
 
==Problem 6==
 +
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]]
  
 
==Problem 7==
 
==Problem 7==
 +
For nonnegative integers <math>a</math> and <math>b</math> with  <math>a + b \leq 6</math>, let <math>T(a, b) = \binom{6}{a} \binom{6}{b} \binom{6}{a + b}</math>. Let <math>S</math> denote the sum of all <math>T(a, b)</math>, where  <math>a</math> and <math>b</math> are nonnegative integers with <math>a + b \leq 6</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>.
 +
 
[[2017 AIME I Problems/Problem 7 | Solution]]
 
[[2017 AIME I Problems/Problem 7 | Solution]]
  
 
==Problem 8==
 
==Problem 8==
 +
Two real numbers <math>a</math> and <math>b</math> are chosen independently and uniformly at random from the interval <math>(0, 75)</math>. Let <math>O</math> and <math>P</math> be two points on the plane with <math>OP = 200</math>. Let <math>Q</math> and <math>R</math> be on the same side of line <math>OP</math> such that the degree measures of <math>\angle POQ</math> and <math>\angle POR</math> are <math>a</math> and <math>b</math> respectively, and <math>\angle OQP</math> and <math>\angle ORP</math> are both right angles. The probability that <math>QR \leq 100</math> is equal to <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
 +
 
[[2017 AIME I Problems/Problem 8 | Solution]]
 
[[2017 AIME I Problems/Problem 8 | Solution]]
  
 
==Problem 9==
 
==Problem 9==
 +
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]]
  
 
==Problem 10==
 
==Problem 10==
 +
Let <math>z_1 = 18 + 83i</math>, <math>z_2 = 18 + 39i, </math> and <math>z_3 = 78 + 99i,</math> where <math>i = \sqrt{-1}</math>. Let <math>z</math> be the unique complex number with the properties that <math>\frac{z_3 - z_1}{z_2 - z_1} \cdot \frac{z - z_2}{z - z_3}</math> is a real number and the imaginary part of <math>z</math> is the greatest possible. Find the real part of <math>z</math>.
 +
 
[[2017 AIME I Problems/Problem 10 | Solution]]
 
[[2017 AIME I Problems/Problem 10 | Solution]]
  
 
==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 15:43, 2 June 2022

2017 AIME I (Answer Key)
Printable version | AoPS Contest Collections • PDF

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

Fifteen distinct points are designated on $\triangle ABC$: the 3 vertices $A$, $B$, and $C$; $3$ other points on side $\overline{AB}$; $4$ other points on side $\overline{BC}$; and $5$ other points on side $\overline{CA}$. Find the number of triangles with positive area whose vertices are among these $15$ points.

Solution

Problem 2

When each of $702$, $787$, and $855$ is divided by the positive integer $m$, the remainder is always the positive integer $r$. When each of $412$, $722$, and $815$ is divided by the positive integer $n$, the remainder is always the positive integer $s \neq r$. Find $m+n+r+s$.

Solution

Problem 3

For a positive integer $n$, let $d_n$ be the units digit of $1 + 2 + \dots + n$. Find the remainder when \[\sum_{n=1}^{2017} d_n\]is divided by $1000$.

Solution

Problem 4

A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

Solution

Problem 5

A rational number written in base eight is $\underline{a} \underline{b} . \underline{c} \underline{d}$, where all digits are nonzero. The same number in base twelve is $\underline{b} \underline{b} . \underline{b} \underline{a}$. Find the base-ten number $\underline{a} \underline{b} \underline{c}$.

Solution

Problem 6

A circle circumscribes an isosceles triangle whose two congruent angles have degree measure $x$. 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 $\frac{14}{25}$. Find the difference between the largest and smallest possible values of $x$.

Solution

Problem 7

For nonnegative integers $a$ and $b$ with $a + b \leq 6$, let $T(a, b) = \binom{6}{a} \binom{6}{b} \binom{6}{a + b}$. Let $S$ denote the sum of all $T(a, b)$, where $a$ and $b$ are nonnegative integers with $a + b \leq 6$. Find the remainder when $S$ is divided by $1000$.

Solution

Problem 8

Two real numbers $a$ and $b$ are chosen independently and uniformly at random from the interval $(0, 75)$. Let $O$ and $P$ be two points on the plane with $OP = 200$. Let $Q$ and $R$ be on the same side of line $OP$ such that the degree measures of $\angle POQ$ and $\angle POR$ are $a$ and $b$ respectively, and $\angle OQP$ and $\angle ORP$ are both right angles. The probability that $QR \leq 100$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution

Problem 9

Let $a_{10} = 10$, and for each positive integer $n >10$ let $a_n = 100a_{n - 1} + n$. Find the least positive $n > 10$ such that $a_n$ is a multiple of $99$.

Solution

Problem 10

Let $z_1 = 18 + 83i$, $z_2 = 18 + 39i,$ and $z_3 = 78 + 99i,$ where $i = \sqrt{-1}$. Let $z$ be the unique complex number with the properties that $\frac{z_3 - z_1}{z_2 - z_1} \cdot \frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.

Solution

Problem 11

Consider arrangements of the $9$ numbers $1, 2, 3, \dots, 9$ in a $3 \times 3$ array. For each such arrangement, let $a_1$, $a_2$, and $a_3$ be the medians of the numbers in rows $1$, $2$, and $3$ respectively, and let $m$ be the median of $\{a_1, a_2, a_3\}$. Let $Q$ be the number of arrangements for which $m = 5$. Find the remainder when $Q$ is divided by $1000$.

Solution

Problem 12

Call a set $S$ product-free if there do not exist $a, b, c \in S$ (not necessarily distinct) such that $a b = c$. For example, the empty set and the set $\{16, 20\}$ are product-free, whereas the sets $\{4, 16\}$ and $\{2, 8, 16\}$ are not product-free. Find the number of product-free subsets of the set $\{1, 2, 3, 4, \ldots, 7, 8, 9, 10\}$.

Solution

Problem 13

For every $m \geq 2$, let $Q(m)$ be the least positive integer with the following property: For every $n \geq Q(m)$, there is always a perfect cube $k^3$ in the range $n < k^3 \leq mn$. Find the remainder when \[\sum_{m = 2}^{2017} Q(m)\]is divided by $1000$.

Solution

Problem 14

Let $a > 1$ and $x > 1$ satisfy $\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$ and $\log_a(\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$.

Solution

Problem 15

The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths $2\sqrt3$, $5$, and $\sqrt{37}$, as shown, is $\tfrac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$.

[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]

Solution

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

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