2013年USAJMO 真题及答案 - American Mathematics Competitions

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?

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 $n$ .

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 $YX/XZ=BP/PC$ .

Day 2

Problem 4

Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$ , where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$ , $2+2$ , $2+1+1$ , $1+2+1$ , $1+1+2$ , and $1+1+1+1$ . Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd.

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