Difference between revisions of "2013 USAMO"
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===Problem 6=== | ===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 | + | Let <math>ABC</math> be a triangle. Find all points <math>P</math> on segment <math>BC</math> satisfying the following property: If <math>X</math> and <math>Y</math> are the intersections of line <math>PA</math> with the common external tangent lines of the circumcircles of triangles <math>PAB</math> and <math>PAC</math>, then <cmath>\left(\frac{PA}{XY}\right)^2+\frac{PB\cdot PC}{AB\cdot AC}=1.</cmath> |
[[2013 USAMO Problems/Problem 6|Solution]] | [[2013 USAMO Problems/Problem 6|Solution]] |
Revision as of 19:23, 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
.
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 [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
, 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
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
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 |