Find all functions $f : N \to N$ such that $f(m)+f(n)$ divides $m+n$ for all positive integers $m$ and $n$.

Find the largest integer $n$ for which $2^n$ divides \[ \binom 21 \binom 42 \binom 63 \dots \binom {128}{64}. \][i]Proposed by Evan Chen[/i]

Find all functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, such that 1) $f(0,x)$ is non-decreasing ; 2) for any $x,y \in \mathbb{R}$, $f(x,y)=f(y,x)$ ; 3) for any $x,y,z \in \mathbb{R}$, $(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0$ ; 4) for any $x,y,a \in \mathbb{R}$, $f(x+a,y+a)=f(x,y)+a$ .

Define $x\star y$ to be $x^y+y^x$.Compute $2\star (2\star 2)$.

Prove that the function \[ f(\nu)= \int_1^{\frac{1}{\nu}} \frac{dx}{\sqrt{(x^2-1)(1-\nu^2x^2)}}\] (where the positive value of the square root is taken) is monotonically decreasing in the interval $ 0<\nu<1$. [P. Turan]

A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn. How many ways are there to put the cards in the three boxes so that the trick works?

Find all $ (x,y,z) \in \mathbb{Z}^3$ such that $ x^3 \minus{} 5x \equal{} 1728^{y}\cdot 1733^z \minus{} 17$. [i](Paolo Leonetti)[/i]

What is the value of the sum \[ \sum_z \frac{1}{{\left|1 - z\right|}^2} \, , \] where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?

Hi guys, I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this: 1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though. 2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary. 3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions: A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh? B. Do NOT go back to the previous problem(s). This causes too much confusion. C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for. 4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving! Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D

[b]3.[/b] Let the density function $f(x)$ of the random variable $\xi$ bean even function; let further $f(x)$ be monotonically non-increasing for $x>0$. Suppose that $D^{2}= \int_{-\infty }^{\infty }x^{2}f(x) dx$ exists. Prove that for every non negative $\lambda $ $P(\left |\xi \right |\geq \lambda D)\leq \frac{1}{1+\lambda ^{2}}$. [b](P. 7)[/b]

Let $f:[0,1]\to\mathbb{R}$ be a function for which there exists a constant $K>0$ such that $|f(x)-f(y)|\le K|x-y|$ for all $x,y\in [0,1].$ Suppose also that for each rational number $r\in [0,1],$ there exist integers $a$ and $b$ such that $f(r)=a+br.$ Prove that there exist finitely many intervals $I_1,\dots,I_n$ such that $f$ is a linear function on each $I_i$ and $[0,1]=\bigcup_{i=1}^nI_i.$

A real valued function $f$ is defined on the interval $(-1,2)$. A point $x_0$ is said to be a fixed point of $f$ if $f(x_0)=x_0$. Suppose that $f$ is a differentiable function such that $f(0)>0$ and $f(1)=1$. Show that if $f'(1)>1$, then $f$ has a fixed point in the interval $(0,1)$.

Let $f : Z_{\ge 0} \to Z_{\ge 0}$ satisfy the functional equation $$f(m^2 + n^2) =(f(m) - f(n))^2 + f(2mn)$$ for all nonnegative integers $m, n$. If $8f(0) + 9f(1) = 2006$, compute $f(0)$.

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(2f(x)) = f(x - f(y)) + f(x) + y$$ for all $x, y \in \mathbb{R}$.

Let $\mathbb{Z}^+$ denote the set of positive integers. Find all functions $f: \mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that [list=i] [*] $f(m,m)=m$ [*] $f(m,k) = f(k,m)$ [*] $f(m, m+k) = f(m,k)$[/list] , for each $m,k \in \mathbb{Z}^+$.

Find all the functions $f:\mathbb{N}\to\mathbb{R}$ that satisfy \[ f(x+y)=f(x)+f(y) \] for all $x,y\in\mathbb{N}$ satisfying $10^6-\frac{1}{10^6} < \frac{x}{y} < 10^6+\frac{1}{10^6}$. Note: $\mathbb{N}$ denotes the set of positive integers and $\mathbb{R}$ denotes the set of real numbers.

On the domain $0\leq \theta \leq 2\pi:$ (a) Prove that $\sin^{2}\theta \cdot \sin 2\theta$ takes its maximum at $\frac{\pi}{3}$ and $\frac{4 \pi}{3}$ (and hence its minimum at $\frac{2 \pi}{3}$ and $\frac{ 5 \pi}{3}$). (b) Show that $$| \sin^{2} \theta \cdot \sin^{3} 2\theta \cdot \sin^{3} 4 \theta \cdots \sin^{3} 2^{n-1} \theta \cdot \sin 2^{n} \theta |$$ takes its maximum at $\frac{4 \pi}{3}$ (the maximum may also be attained at other points). (c) Derive the inequality: $$ \sin^{2} \theta \cdot \sin^{2} 2\theta \cdots \sin^{2} 2^{n} \theta \leq \left( \frac{3}{4} \right)^{n}.$$

Find all functions $f:R \rightarrow R$ such that $$(x+y^2)f(yf(x))=xyf(y^2+f(x))$$, where $x,y \in \mathbb{R}$

Find all functions $f$ from the set of real numbers to itself satisfying \[ f(x(1+y)) = f(x)(1 + f(y)) \] for all real numbers $x, y$.

For a positive integer $n$ denote $A = \{1,2,..., n\}$. Suppose that $g : A\to A$ is a fixed function with $g(k) \ne k$ and $g(g(k)) = k$ for $k \in A$. How many functions $f: A \to A$ are there such that $f(k)\ne g(k)$ and $f(f(f(k))= g(k)$ for $k \in A$?

Let the function $f:N^*\to N^*$ such that [b](1)[/b] $(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*$; [b](2)[/b] $n\le f(n)\le n+2014 , \forall n\in N^*$ Show that: there exists the positive integers $N$ such that $ f(n)=n $, for each integer $n \ge N$. (High School Affiliated to Nanjing Normal University )

$(FRA 5)$ Let $\alpha(n)$ be the number of pairs $(x, y)$ of integers such that $x+y = n, 0 \le y \le x$, and let $\beta(n)$ be the number of triples $(x, y, z)$ such that$ x + y + z = n$ and $0 \le z \le y \le x.$ Find a simple relation between $\alpha(n)$ and the integer part of the number $\frac{n+2}{2}$ and the relation among $\beta(n), \beta(n -3)$ and $\alpha(n).$ Then evaluate $\beta(n)$ as a function of the residue of $n$ modulo $6$. What can be said about $\beta(n)$ and $1+\frac{n(n+6)}{12}$? And what about $\frac{(n+3)^2}{6}$? Find the number of triples $(x, y, z)$ with the property $x+ y+ z \le n, 0 \le z \le y \le x$ as a function of the residue of $n$ modulo $6.$What can be said about the relation between this number and the number $\frac{(n+6)(2n^2+9n+12)}{72}$?

Find the twice-differentiable functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that have the property that $$ f'(x)+F(x)=2f(x)+x^2/2, $$ for any real numbers $ x; $ where $ F $ is a primitive of $ f. $ [i]Carmen Botea[/i]

A function $ f$ from the integers to the integers is defined as follows: \[ f(n) \equal{} \begin{cases} n \plus{} 3 & \text{if n is odd} \\ n/2 & \text{if n is even} \end{cases} \]Suppose $ k$ is odd and $ f(f(f(k))) \equal{} 27$. What is the sum of the digits of $ k$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 15$

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers for which \[ a_{n+2} = \left[\frac{2a_n}{a_{n+1}}\right]+\left[\frac{2a_{n+1}}{a_n}\right]. \] Prove that there exists a positive integer $m$ such that $a_m=4$ and $a_{m+1} \in\{3,4\}$. [b]Note.[/b] $[x]$ is the greatest integer not exceeding $x$.