2017 USAJMO Problems

Day 1

Note: For any geometry problem whose statement begins with an asterisk ( $*$ ), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.

Problem 1

Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime positive integers $a > 1$ and $b > 1$ such that $a^b + b^a$ is divisible by $a + b.$

Solution

Problem 2

Consider the equation $\left(3x^3 + xy^2 \right) \left(x^2y + 3y^3 \right) = (x-y)^7.$

(a) Prove that there are infinitely many pairs $(x,y)$ of positive integers satisfying the equation.

(b) Describe all pairs $(x,y)$ of positive integers satisfying the equation.

Solution

Problem 3

( $*$ ) Let $ABC$ be an equilateral triangle and let $P$ be a point on its circumcircle. Let lines $PA$ and $BC$ intersect at $D$ ; let lines $PB$ and $CA$ intersect at $E$ ; and let lines $PC$ and $AB$ intersect at $F$ . Prove that the area of triangle $DEF$ is twice the area of triangle $ABC$ .

Solution

Day 2

Note: For any geometry problem whose statement begins with an asterisk ( $*$ ), the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.

Problem 4

Are there any triples $(a,b,c)$ of positive integers such that $(a-2)(b-2)(c-2) + 12$ is a prime that properly divides the positive number $a^2 + b^2 + c^2 + abc - 2017$ ?

Solution

Problem 5

( $*$ ) Let $O$ and $H$ be the circumcenter and the orthocenter of an acute triangle $ABC$ . Points $M$ and $D$ lie on side $BC$ such that $BM = CM$ and $\angle BAD = \angle CAD$ . Ray $MO$ intersects the circumcircle of triangle $BHC$ in point $N$ . Prove that $\angle ADO = \angle HAN$ .

Solution

Problem 6

Let $P_1, \ldots, P_{2n}$ be $2n$ distinct points on the unit circle $x^2 + y^2 = 1$ other than $(1,0)$ . Each point is colored either red or blue, with exactly $n$ of them red and exactly $n$ of them blue. Let $R_1, \ldots, R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$ . Then let $B_2$ be the nearest of the remaining blue points to $R_2$ traveling counterclockwise around the circle from $R_2$ , and so on, until we have labeled all the blue points $B_1, \ldots, B_n$ . Show that the number of counterclockwise arcs of the form $R_i \rightarrow B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \ldots, R_n$ of the red points.

Solution

2017 USAJMO (Problems • Resources)
Preceded by 2016 USAJMO Problems	Followed by 2018 USAJMO Problems
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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2017 USAJMO Problems

Contents

Day 1

Problem 1

Problem 2

Problem 3

Day 2

Problem 4

Problem 5

Problem 6