Difference between revisions of "2013 USAJMO"
| Line 1: | Line 1: | ||
==Day 1== | ==Day 1== | ||
===Problem 1=== | ===Problem 1=== | ||
| + | |||
| + | Are there integers and such that and are both perfect cubes of integers? | ||
[[2013 USAMO Problems/Problem 1|Solution]] | [[2013 USAMO Problems/Problem 1|Solution]] | ||
===Problem 2=== | ===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 . | ||
[[2013 USAMO Problems/Problem 2|Solution]] | [[2013 USAMO Problems/Problem 2|Solution]] | ||
===Problem 3=== | ===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 . | |
[[2013 USAMO Problems/Problem 3|Solution]] | [[2013 USAMO Problems/Problem 3|Solution]] | ||
| Line 16: | Line 23: | ||
==Day 2== | ==Day 2== | ||
===Problem 4=== | ===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. | |
[[2013 USAMO Problems/Problem 4|Solution]] | [[2013 USAMO Problems/Problem 4|Solution]] | ||
| Line 22: | Line 29: | ||
===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 | ||
| + | [[2013 USAMO Problems/Problem 5|Solution]] | ||
| − | + | ===Problem 6=== | |
| − | + | Find all real numbers satisfying | |
[[2013 USAMO Problems/Problem 6|Solution]] | [[2013 USAMO Problems/Problem 6|Solution]] | ||
Revision as of 17:37, 11 May 2013
Contents
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 Solution
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 | ||
