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Difference between revisions of "2017 USAJMO Problems"
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===Problem 1===
 
===Problem 1===
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Prove that there are infinitely many distinct pairs <math>(a,b)</math> of relatively prime positive integers <math>a > 1</math> and <math>b > 1</math> such that <math>a^b + b^a</math> is divisible by <math>a + b</math>.
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Prove that there are infinitely many distinct pairs <math>(a,b)</math> of relatively prime positive integers <math>a > 1</math> and <math>b > 1</math> such that <math>a^b + b^a</math> is divisible by <math>a + b.</math>  
  
 
[[2017 USAJMO Problems/Problem 1|Solution]]
 
[[2017 USAJMO Problems/Problem 1|Solution]]
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===Problem 2===
 
===Problem 2===
 
Consider the equation  
 
Consider the equation  

Revision as of 18:24, 19 April 2017

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 $PB$ 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

Problem 4

Problem 5

Problem 6

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

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