Difference between revisions of "2013 USAMO"
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==Day 1== | ==Day 1== | ||
===Problem 1=== | ===Problem 1=== | ||
| − | In triangle <math> ABC</math>, points <math>P,Q,R</math> | + | In triangle <math> ABC</math>, points <math>P,Q,R</math> lie on sides respectively. Let , , denote the circumcircles of triangles , , , respectively. Given the fact that segment intersects , , again at respectively, prove that . |
[[2013 USAMO Problems/Problem 1|Solution]] | [[2013 USAMO Problems/Problem 1|Solution]] | ||
===Problem 2=== | ===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 . | |
[[2013 USAMO Problems/Problem 2|Solution]] | [[2013 USAMO Problems/Problem 2|Solution]] | ||
===Problem 3=== | ===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. | ||
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===Problem 4=== | ===Problem 4=== | ||
| − | + | Find all real numbers satisfying | |
[[2013 USAMO Problems/Problem 4|Solution]] | [[2013 USAMO Problems/Problem 4|Solution]] | ||
===Problem 5=== | ===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. | |
[[2013 USAMO Problems/Problem 5|Solution]] | [[2013 USAMO Problems/Problem 5|Solution]] | ||
| − | ===Problem | + | ===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 | |
[[2013 USAMO Problems/Problem 6|Solution]] | [[2013 USAMO Problems/Problem 6|Solution]] | ||
== See Also == | == See Also == | ||
{{USAMO newbox|year= 2013|before=[[2012 USAMO]]|after=[[2014 USAMO]]}} | {{USAMO newbox|year= 2013|before=[[2012 USAMO]]|after=[[2014 USAMO]]}} | ||
Revision as of 18:35, 11 May 2013
Contents
Day 1
Problem 1
In triangle 
, points 
lie on sides respectively. Let , , denote the circumcircles of triangles , , , respectively. Given the fact that segment intersects , , again at respectively, prove that .
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 .
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.
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.
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
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 | ||
