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Difference between revisions of "2013 USAMO"

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===Problem 2===
 
===Problem 2===
 
For a positive integer <math>n\geq 3</math> plot <math>n</math> equally spaced points around a circle.  Label one of them <math>A</math>, and place a marker at <math>A</math>.  One may move the marker forward in a clockwise direction to either the next point or the point after that.  Hence there are a total of <math>2n</math> distinct moves available; two from each point.  Let <math>a_n</math> count the number of ways to advance around the circle exactly twice, beginning and ending at <math>A</math>, without repeating a move.  Prove that <math>a_{n-1}+a_n=2^n</math> for all <math>n\geq 4</math>.
 
For a positive integer <math>n\geq 3</math> plot <math>n</math> equally spaced points around a circle.  Label one of them <math>A</math>, and place a marker at <math>A</math>.  One may move the marker forward in a clockwise direction to either the next point or the point after that.  Hence there are a total of <math>2n</math> distinct moves available; two from each point.  Let <math>a_n</math> count the number of ways to advance around the circle exactly twice, beginning and ending at <math>A</math>, without repeating a move.  Prove that <math>a_{n-1}+a_n=2^n</math> for all <math>n\geq 4</math>.
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[[2013 USAMO Problems/Problem 2|Solution]]
 
[[2013 USAMO Problems/Problem 2|Solution]]
  

Revision as of 18:27, 11 May 2013

Day 1

Problem 1

In triangle $ABC$, points $P,Q,R$ lie on sides $BC,CA,AB$ respectively. Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively. Given the fact that segment $AP$ intersects $\omega_A$, $\omega_B$, $\omega_C$ again at $X,Y,Z$ respectively, prove that $YX/XZ=BP/PC$.

Solution

Problem 2

For a positive integer $n\geq 3$ plot $n$ equally spaced points around a circle. Label one of them $A$, and place a marker at $A$. One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of $2n$ distinct moves available; two from each point. Let $a_n$ count the number of ways to advance around the circle exactly twice, beginning and ending at $A$, without repeating a move. Prove that $a_{n-1}+a_n=2^n$ for all $n\geq 4$.

Solution

Problem 3

Let $n$ be a positive integer. There are $\tfrac{n(n+1)}{2}$ marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing $n$ marks. Initially, each mark has the black side up. An [i]operation[/i] is to choose a line parallel to the sides of the triangle, and flipping all the marks on that line. A configuration is called [i]admissible [/i] if it can be obtained from the initial configuration by performing a finite number of operations. For each admissible configuration $C$, let $f(C)$ denote the smallest number of operations required to obtain $C$ from the initial configuration. Find the maximum value of $f(C)$, where $C$ varies over all admissible configurations.

Solution

Day 2

Problem 4

Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]

Solution

Problem 5

Given postive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non-zero digit when written in base ten.

Solution

Problem 6

Let $ABC$ be a triangle. Find all points $P$ on segment $BC$ satisfying the following property: If $X$ and $Y$ are the intersections of line $PA$ with the common external tangent lines of the circumcircles of triangles $PAB$ and $PAC$, then \[\left(\frac{PA}{XY}\right)^2+\frac{PB\cdot PC}{AB\cdot AC}=1.\]

Solution

See Also

2013 USAMO (Problems • Resources)
Preceded by
2012 USAMO 		Followed by
2014 USAMO
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions
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