Difference between revisions of "2017 USAJMO Problems"
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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>. | ||
+ | [[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. | ||
+ | [[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
Contents
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 of relatively prime positive integers and such that is divisible by .
Problem 2
Consider the equation
(a) Prove that there are infinitely many pairs of positive integers satisfying the equation.
(b) Describe all pairs of positive integers satisfying the equation.
Problem 3
() Let be an equilateral triangle and let be a point on its circumcircle. Let lines and intersect at ; let lines and intersect at ; and let lines and intersect at . Prove that the area of triangle is twice the area of triangle .
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.
2017 USAJMO (Problems • Resources) | ||
Preceded by 2016 USAJMO |
Followed by 2018 USAJMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |