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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Revision as of 19:18, 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

Let $P_1,....,P_{2n}$ 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 $n$ of them blue. Let $R_1,...,R_{n}$ 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 of the blue points $B_1,...,B_{n}$. 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 $R_1,...,R_{n}$ of the red points.

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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