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Difference between revisions of "2017 USAJMO Problems"

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===Problem 6===
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Let <math>P_1,....,P_{2n}</math> be <math>2n</math> distinct points on the unit circle <math>x^2+y^2=1</math> other than <math>(1,0)</math>. Each point is colored either red or blue, with exactly <math>n</math> of them red and <math>n</math> of them blue. Let <math>R_1,...,R_{n}</math> be any ordering of the red points. Let <math>B_1</math> be the nearest blue point to <math>R_1</math> traveling counterclockwise around the circle starting from <math>R_1</math>. Then let <math>B_2</math> be the nearest of the remaining blue points to <math>R_2</math> traveling counterclockwise around the circle from <math>R_2</math>, and so on, until we have labeled all of the blue points <math>B_1,...,B_{n}</math>. Show that the number of counterclockwise arcs of the form <math>R_{i} \rightarrow B_{i}</math> that contain the point <math>(1,0)</math> is independent of the way we chose the ordering <math>R_1,...,R_{n}</math> of the red points.
 
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{{USAJMO newbox|year= 2017 |before=[[2016 USAJMO]]|after=[[2018 USAJMO]]}}

Revision as of 18:18, 20 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 $BC$ 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

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
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All USAJMO Problems and Solutions
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