Difference between revisions of "2017 USAJMO Problems"

m (Problem 3)
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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>.  
 
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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[[2017 USAJMO Problems/Problem 1|Solution]]
 
===Problem 2===
 
===Problem 2===
 
Consider the equation  
 
Consider the equation  
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(b) Describe all pairs <math>(x,y)</math> of positive integers satisfying the equation.  
 
(b) Describe all pairs <math>(x,y)</math> of positive integers satisfying the equation.  
  
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[[2017 USAJMO Problems/Problem 2|Solution]]
 
===Problem 3===
 
===Problem 3===
 
(<math>*</math>) Let <math>ABC</math> be an equilateral triangle and let <math>P</math> be a point on its circumcircle. Let lines <math>PA</math> and <math>PB</math> intersect at <math>D</math>; let lines <math>PB</math> and <math>CA</math> intersect at <math>E</math>; and let lines <math>PC</math> and <math>AB</math> intersect at <math>F</math>. Prove that the area of triangle <math>DEF</math> is twice the area of triangle <math>ABC</math>.
 
(<math>*</math>) Let <math>ABC</math> be an equilateral triangle and let <math>P</math> be a point on its circumcircle. Let lines <math>PA</math> and <math>PB</math> intersect at <math>D</math>; let lines <math>PB</math> and <math>CA</math> intersect at <math>E</math>; and let lines <math>PC</math> and <math>AB</math> intersect at <math>F</math>. Prove that the area of triangle <math>DEF</math> is twice the area of triangle <math>ABC</math>.
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[[2017 USAJMO Problems/Problem 3|Solution]]
  
 
==Day 2==
 
==Day 2==
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===Problem 6===
 
===Problem 6===
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{{MAA Notice}}
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{{USAJMO newbox|year= 2017 |before=[[2016 USAJMO]]|after=[[2018 USAJMO]]}}

Revision as of 19:07, 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 (ProblemsResources)
Preceded by
2016 USAJMO
Followed by
2018 USAJMO
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All USAJMO Problems and Solutions
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