Difference between revisions of "2013 USAJMO"

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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 USAJMO Problems/Problem 2|Solution]]
+
[[2013 USAMO Problems/Problem 1|Solution]]
  
 
===Problem 3===
 
===Problem 3===
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Find all real numbers satisfying  
 
Find all real numbers satisfying  
  
[[2013 USAJMO Problems/Problem 6|Solution]]
+
[[2013 USAMO Problems/Problem 4|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 17:51, 11 May 2013

Day 1

Problem 1

Are there integers and such that and are both perfect cubes of integers?

Solution

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 .

Solution

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 .

Solution

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.

Solution

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

Solution

See Also

2013 USAJMO (ProblemsResources)
Preceded by
2012 USAJMO
Followed by
2014 USAJMO
1 2 3 4 5 6
All USAJMO Problems and Solutions
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