Difference between revisions of "2013 USAJMO"

Latest revision as of 19:05, 30 April 2014

2013 USAJMO problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.

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

@@ Line 1: / Line 1: @@
-==Day 1==
+'''2013 [[USAMO|USAJMO]]''' problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution.
-===Problem 1===
-Are there integers <math>a</math> and <math>b</math> such that <math>a^5b+3</math> and <math>ab^5+3</math> are both perfect cubes of integers?
+*[[2013 USAJMO Problems]]
+*[[2013 USAJMO Problems/Problem 1]]
+*[[2013 USAJMO Problems/Problem 2]]
+*[[2013 USAMO Problems/Problem 1|2013 USAJMO Problems/Problem 3]]
+*[[2013 USAJMO Problems/Problem 4]]
+*[[2013 USAJMO Problems/Problem 5]]
+*[[2013 USAMO Problems/Problem 4|2013 USAJMO Problems/Problem 6]]
-[[2013 USAJMO Problems/Problem 1|Solution]]
+{{USAJMO newbox|year= 2013 |before=[[2012 USAJMO]]|after=[[2014 USAJMO]]}}
-===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 USAJMO Problems/Problem 2|Solution]]
-===Problem 3===
-In triangle <math>ABC</math>, points <math>P,Q,R</math> lie on sides <math>BC,CA,AB</math> respectively.  Let <math>\omega_A</math>, <math>\omega_B</math>, <math>\omega_C</math> denote the circumcircles of triangles <math>AQR</math>, <math>BRP</math>, <math>CPQ</math>, respectively.  Given the fact that segment <math>AP</math> intersects <math>\omega_A</math>, <math>\omega_B</math>, <math>\omega_C</math> again at <math>X,Y,Z</math> respectively, prove that <math>YX/XZ=BP/PC</math>.
-[[2013 USAMO Problems/Problem 1|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.
-[[2013 USAJMO Problems/Problem 4|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
-[[2013 USAJMO Problems/Problem 5|Solution]]
-===Problem 6===
-Find all real numbers satisfying
-[[2013 USAMO Problems/Problem 4|Solution]]
-== See Also ==
-{{USAJMO newbox|year= 2013|before=[[2012 USAJMO]]|after=[[2014 USAJMO]]}}

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Difference between revisions of "2013 USAJMO"

Latest revision as of 19:05, 30 April 2014