Difference between revisions of "2013 USAJMO"
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===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: | + | Each cell of an <math>m\times n</math> board is filled with some nonnegative integer. Two numbers in the filling are said to be [i]adjacent[/i] 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 [i]garden[/i] if it satisfies the following two conditions: |
| − | (i) The difference between any two adjacent numbers is either or . | + | (i) The difference between any two adjacent numbers is either <math>0</math> or <math>1</math>. |
| − | (ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to . | + | (ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to <math>0</math>. |
| − | Determine the number of distinct gardens in terms of and . | + | Determine the number of distinct gardens in terms of <math>m</math> and <math>n</math>. |
[[2013 USAJMO Problems/Problem 2|Solution]] | [[2013 USAJMO Problems/Problem 2|Solution]] | ||
Revision as of 18:18, 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 [i]adjacent[/i] 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 [i]garden[/i] 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 | ||
