Difference between revisions of "2013 USAJMO"
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Are there integers and such that and are both perfect cubes of integers? | Are there integers and such that and are both perfect cubes of integers? | ||
| − | [[2013 | + | [[2013 USAJMO Problems/Problem 1|Solution]] |
===Problem 2=== | ===Problem 2=== | ||
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Determine the number of distinct gardens in terms of and . | Determine the number of distinct gardens in terms of and . | ||
| − | [[2013 | + | [[2013 USAJMO 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 . | 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 | + | [[2013 USAJMO Problems/Problem 3|Solution]] |
==Day 2== | ==Day 2== | ||
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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. | 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 | + | [[2013 USAJMO Problems/Problem 4|Solution]] |
===Problem 5=== | ===Problem 5=== | ||
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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 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 | + | [[2013 USAJMO Problems/Problem 5|Solution]] |
===Problem 6=== | ===Problem 6=== | ||
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Find all real numbers satisfying | Find all real numbers satisfying | ||
| − | [[2013 | + | [[2013 USAJMO Problems/Problem 6|Solution]] |
== See Also == | == See Also == | ||
{{USAJMO newbox|year= 2013|before=[[2012 USAJMO]]|after=[[2014 USAJMO]]}} | {{USAJMO newbox|year= 2013|before=[[2012 USAJMO]]|after=[[2014 USAJMO]]}} | ||
Revision as of 17:50, 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
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 | ||
