2015 USAMO Problems真题及答案

2015 USAMO Problems真题及答案

真题下载请前往【纯真题】小程序

纯真题小程序

完整版真题免费下载

+答案解析请参考文末

Day 1

Problem 1

Solve in integers the equation\[x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3.\]

 

Problem 2

Quadrilateral $APBQ$ is inscribed in circle $\omega$ with $\angle P = \angle Q = 90^{\circ}$ and $AP = AQ < BP$. Let $X$ be a variable point on segment $\overline{PQ}$. Line $AX$ meets $\omega$ again at $S$ (other than $A$). Point $T$ lies on arc $AQB$ of $\omega$ such that $\overline{XT}$ is perpendicular to $\overline{AX}$. Let $M$ denote the midpoint of chord $\overline{ST}$As $X$ varies on segment $\overline{PQ}$, show that $M$ moves along a circle.

 

Problem 3

Let $S = \{1, 2, ..., n\}$, where $n \ge 1$. Each of the $2^n$ subsets of $S$ is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set $T \subseteq S$, we then write $f(T)$ for the number of subsets of T that are blue.

Determine the number of colorings that satisfy the following condition: for any subsets $T_1$ and $T_2$ of $S$,\[f(T_1)f(T_2) = f(T_1 \cup T_2)f(T_1 \cap T_2).\]

 

Day 2

Problem 4

Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k), (i, l), (j, k), (j, l)$ for some $1\leq i, j, k, l\leq n$, such that $i<j$ and $k<l$. A stone move consists of either removing one stone from each of $(i, k)$ and $(j, l)$ and moving them to $(i, l)$ and $(j, k)$ respectively, or removing one stone from each of $(i, l)$ and $(j, k)$ and moving them to $(i, k)$ and $(j, l)$ respectively.

Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.

How many different non-equivalent ways can Steve pile the stones on the grid?

 

Problem 5

Let $a, b, c, d, e$ be distinct positive integers such that $a^4 + b^4 = c^4 + d^4 = e^5$. Show that $ac + bd$ is a composite number.

 

Problem 6

Consider $0<\lambda<1$, and let $A$ be a multiset of positive integers. Let $A_n=\{a\in A: a\leq n\}$. Assume that for every $n\in\mathbb{N}$, the set $A_n$contains at most $n\lambda$ numbers. Show that there are infinitely many $n\in\mathbb{N}$ for which the sum of the elements in $A_n$ is at most $\frac{n(n+1)}{2}\lambda$. (A multiset is a set-like collection of elements in which order is ignored, but repetition of elements is allowed and multiplicity of elements is significant. For example, multisets $\{1, 2, 3\}$ and $\{2, 1, 3\}$ are equivalent, but $\{1, 1, 2, 3\}$ and $\{1, 2, 3\}$ differ.)

 

完整版2015USAMO真题pdf版本免费下载

注册登录可见:

2015USAMO真题参考答案及详解

翰林学院公众号
请输入验证码查看内容
验证码:
   
请关注“翰林国际教育Linstitute”官方微信公众号,回复关键字“usamo”,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

免费领取备考资料,更有讲座不定时和你见面,和一群志同道合的朋友一起努力!一起学习!

还可领取国际竞赛最新年份真题及解析!

翰林国际教育资讯二维码