# 2011 USAJMO Problems真题及答案

+答案解析请参考文末

## Day 1

### Problem 1

Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.

### Problem 2

Let $a$ $b$ $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \le 4$. Prove that $$\frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \ge 3.$$

### Problem 3

For a point $P = (a,a^2)$ in the coordinate plane, let $\ell(P)$ denote the line passing through $P$ with slope $2a$. Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2)$ $P_2 = (a_2, a_2^2)$ $P_3 = (a_3, a_3^2)$, such that the intersections of the lines $\ell(P_1)$ $\ell(P_2)$ $\ell(P_3)$ form an equilateral triangle $\Delta$. Find the locus of the center of $\Delta$ as $P_1 P_2 P_3$ ranges over all such triangles.

## Day 2

### Problem 4

word is defined as any finite string of letters. A word is a palindrome if it reads the same backwards as forwards. Let a sequence of words $W_0$ $W_1$ $W_2$ $\dots$ be defined as follows: $W_0 = a$ $W_1 = b$, and for $n \ge 2$ $W_n$ is the word formed by writing $W_{n - 2}$ followed by $W_{n - 1}$. Prove that for any $n \ge 1$, the word formed by writing $W_1$ $W_2$ $\dots$ $W_n$ in succession is a palindrome.

### Problem 5

Points $A$ $B$ $C$ $D$ $E$ lie on a circle $\omega$ and point $P$ lies outside the circle. The given points are such that (i) lines $PB$ and $PD$ are tangent to $\omega$, (ii) $P$ $A$ $C$ are collinear, and (iii) $\overline{DE} \parallel \overline{AC}$. Prove that $\overline{BE}$ bisects $\overline{AC}$.

### Problem 6

Consider the assertion that for each positive integer $n \ge 2$, the remainder upon dividing $2^{2^n}$ by $2^n - 1$ is a power of 4. Either prove the assertion or find (with proof) a counterexample.

### 2011 USAJMO真题参考答案及详解 