Difference between revisions of "2013 USAJMO"
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===Problem 5=== | ===Problem 5=== | ||
− | Quadrilateral is inscribed in the semicircle with diameter . Segments and meet at . Point is the foot of the perpendicular from to line . Point lies on such that line is perpendicular to line . Let be the intersection of segments and . Prove that | + | Quadrilateral <math>XABY</math> is inscribed in the semicircle <math>\omega</math> with diameter <math>XY</math>. Segments <math>AY</math> and <math>BX</math> meet at <math>P</math>. Point <math>Z</math> is the foot of the perpendicular from <math>P</math> to line <math>XY</math>. Point <math>C</math> lies on <math>\omega</math> such that line <math>XC</math> is perpendicular to line <math>AZ</math>. Let <math>Q</math> be the intersection of segments <math>AY</math> and <math>XC</math>. Prove that <cmath>\dfrac{BY}{XP}+\dfrac{CY}{XQ}=\dfrac{AY}{AX}.</cmath> |
[[2013 USAJMO Problems/Problem 5|Solution]] | [[2013 USAJMO Problems/Problem 5|Solution]] |
Revision as of 18:19, 11 May 2013
Contents
[hide]Day 1
Problem 1
Are there integers and such that and are both perfect cubes of integers?
Problem 2
Each cell of an board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a garden if it satisfies the following two conditions:
(i) The difference between any two adjacent numbers is either or . (ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to .
Determine the number of distinct gardens in terms of and .
Problem 3
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 .
Day 2
Problem 4
Let be the number of ways to write as a sum of powers of , where we keep track of the order of the summation. For example, because can be written as , , , , , and . Find the smallest greater than for which is odd.
Problem 5
Quadrilateral is inscribed in the semicircle with diameter . Segments and meet at . Point is the foot of the perpendicular from to line . Point lies on such that line is perpendicular to line . Let be the intersection of segments and . Prove that
Problem 6
Find all real numbers satisfying
See Also
2013 USAJMO (Problems • Resources) | ||
Preceded by 2012 USAJMO |
Followed by 2014 USAJMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |