2011 USAJMO Problems真题及答案

2011 USAJMO Problems真题及答案

完整版真题免费下载

+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.

完整版2011 USAJMO真题pdf版本免费下载

注册登录可见:

2011 USAJMO真题参考答案及详解

翰林学院公众号
此处内容已经被作者无情的隐藏,请输入验证码查看内容
验证码:
请关注“上海翰林学院”官方微信公众号,回复关键字“usajmo”,获取验证码。如失效请联系我们任意一位客服或小助手。

AMC8/AMC10/AMC12/AIME

报名及辅导请联系