2013 USAJMO

Revision as of 19:18, 11 May 2013 by Mrdavid445 (talk | contribs) (Problem 2)

Day 1

Problem 1

Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?

Solution

Problem 2

Each cell of an $m\times n$ 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 $0$ or $1$. (ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to $0$.

Determine the number of distinct gardens in terms of $m$ and $n$.

Solution

Problem 3

In triangle $ABC$, points $P,Q,R$ lie on sides $BC,CA,AB$ respectively. Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively. Given the fact that segment $AP$ intersects $\omega_A$, $\omega_B$, $\omega_C$ again at $X,Y,Z$ respectively, prove that $YX/XZ=BP/PC$.

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