Art of Problem Solving

AoPS Online

2017 AIME I Problems

Revision as of 14:32, 8 March 2017 by MSmathlete1018 (talk | contribs) (→‎Problem 9)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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

Contents

1 Problem 1
2 Problem 2
3 Problem 3
4 Problem 4
5 Problem 5
6 Problem 6
7 Problem 7
8 Problem 8
9 Problem 9
10 Problem 10
11 Problem 11
12 Problem 12
13 Problem 13
14 Problem 14
15 Problem 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.

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 $s \neq r$ . Fine $m+n+r+s$ .

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 $1000$ .

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 $m+n$ .

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 $\underline{a} \underline{b} \underline{c}$ .

Problem 6

A circle is circumscribed around 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 $x$ .

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 $1000$ .

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 $m + n$ .

Problem 9

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

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

2017 AIME I (Problems • Answer Key • Resources)
Preceded by 2016 AIME II	Followed by 2017 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.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2017_AIME_I_Problems&oldid=84451"

© 2024 AoPS Incorporated