Difference between revisions of "2013 USAMO"

Revision as of 19:21, 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 be a positive integer. There are marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing marks. Initially, each mark has the black side up. An operation is to choose a line parallel to the sides of the triangle, and flipping all the marks on that line. A configuration is called admissible if it can be obtained from the initial configuration by performing a finite number of operations. For each admissible configuration , let denote the smallest number of operations required to obtain from the initial configuration. Find the maximum value of , where varies over all admissible configurations.

Solution

Day 2

Problem 4

Find all real numbers satisfying

Solution

Problem 5

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

Solution

Problem 6

Let be a triangle. Find all points on segment satisfying the following property: If and are the intersections of line with the common external tangent lines of the circumcircles of triangles and , then

Solution

@@ Line 6: / Line 6: @@
 ===Problem 2===
-For a positive integer plot equally spaced points around a circle. Label one of them , and place a marker at . 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 distinct moves available; two from each point. Let count the number of ways to advance around the circle exactly twice, beginning and ending at , without repeating a move. Prove that for all .
+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>.
 [[2013 USAMO Problems/Problem 2|Solution]]

2013 USAMO (Problems • Resources)
Preceded by 2012 USAMO	Followed by 2014 USAMO
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Difference between revisions of "2013 USAMO"

Revision as of 19:21, 11 May 2013

Contents

Day 1

Problem 1

Problem 2

Problem 3

Day 2

Problem 4

Problem 5

Problem 6

See Also