Let $p>1,\frac1p+\frac1q=1$ and $r>1$. If $u(x,y),v(x,y)>0$, and $f(x,y),g(x,y)$ are continuous functions on $[a,b]\times[c,d]$, then prove $$\left(\frac{\left(\int^b_a\int^d_c(f(x,y)+g(x,y))^rdxdy\right)^{1/r}}{(u(x,y)+v(x,y))^{1/q}}\right)^p\le\left(\frac{\left(\int^b_a\int^d_cf(x,y)^rdxdy\right)^{1/r}}{u(x,y)^{1/q}}\right)^p+\left(\frac{\left(\int^b_a\int^d_cg(x,y)^rdxdy\right)^{1/r}}{v(x,y)^{1/q}}\right)^p,$$ with equality if and only if either $$\left(\lVert f(x,y)\rVert^r_r,\lVert g(x,y)\rVert^r_r\right)=\alpha\left(\lVert u(x,y)\rVert^r_r,\lVert v(x,y)\rVert^r_r\right)$$ for some $\alpha>0$ or $\lVert f(x,y)\rVert^r_r=\lVert g(x,y)\rVert^r_r=0$. [i]Proposed by Chang-Jian Zhao[/i]

Determine all Functions $f:\mathbb{Z} \to \mathbb{Z}$ such that $f(f(a)-b)+bf(2a)$ is a perfect square for all integers $a$ and $b$.

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

[b]4.[/b] Let $f(x)$ be a real- or complex-value integrable function on $(0,1)$ with $\mid f(x) \mid \leq 1 $. Set $ c_k = \int_0^1 f(x) e^{-2 \pi i k x} dx $ and construct the following matrices of order $n$: $ T= (t_{pq})_{p,q=0}^{n-1}, T^{*}= (t_{pq}^{*})_{p,q =0}^{n-1} $ where $t_{pq}= c_{q-p}, t^{*}= \overline {c_{p-q}}$ . Further, consider the following hyper-matrix of order $m$: $ S= \begin{bmatrix} E & T & T^2 & \dots & T^{m-2} & T^{m-1} \\ T^{*} & E & T & \dots & T^{m-3} & T^{m-2} \\ T^{*2} & T^{*} & E & \dots & T^{m-3} & T^{m-2} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ T^{*m-1} & T^{*m-2} & T^{*m-3} & \dots & T^{*} & E \end{bmatrix} $ ($S$ is a matrix of order $mn$ in the ordinary sense; E denotes the unit matrix of order $n$). Show that for any pair $(m , n) $ of positive integers, $S$ has only non-negative real eigenvalues. [b](R. 19)[/b]

A function $f$ is defined on the set of all real numbers except $0$ and takes all real values except $1$. It is also known that $\color{white}\ . \ \color{black}\ \quad f(xy)=f(x)f(-y)-f(x)+f(y)$ for any $x,y\not= 0$ and that $\color{white}\ . \ \color{black}\ \quad f(f(x))=\frac{1}{f(\frac{1}{x})}$ for any $x\not\in\{ 0,1\}$. Determine all such functions $f$.

The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $ 1$ to $ 15$ in clockwise order. Committee rules state that a Martian must occupy chair $ 1$ and an Earthling must occupy chair $ 15$. Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is $ N\cdot (5!)^3$. Find $ N$.

Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\] [i]Proposed by Denmark[/i]

Find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for all positive integers $m,n$ with $m\ge n$, $$f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3).$$ Here $\varphi(n)$ denotes the number of positive integers coprime to $n$ and not exceeding $n$.

Let $f,g,h:\mathbb{R}\rightarrow \mathbb{R}$ such that $f$ is differentiable, $g$ and $h$ are monotonic, and $f'=f+g+h$. Prove that the set of the points of discontinuity of $g$ coincides with the respective set of $h$.

Consider pairs $(f,g)$ of functions from the set of nonnegative integers to itself such that [list] [*]$f(0) \geq f(1) \geq f(2) \geq \dots \geq f(300) \geq 0$ [*]$f(0)+f(1)+f(2)+\dots+f(300) \leq 300$ [*]for any 20 nonnegative integers $n_1, n_2, \dots, n_{20}$, not necessarily distinct, we have $$g(n_1+n_2+\dots+n_{20}) \leq f(n_1)+f(n_2)+\dots+f(n_{20}).$$ [/list] Determine the maximum possible value of $g(0)+g(1)+\dots+g(6000)$ over all such pairs of functions. [i]Sean Li[/i]

Let $f$ be a bijection from $\mathbb N$ to itself. Prove that one can always find three natural number $a,b,c$ such that $a<b<c$ and $f(a)+f(c)=2f(b)$.

Let $n$ be a positive integer. Let $(a, b, c)$ be a random ordered triple of nonnegative integers such that $a + b + c = n$, chosen uniformly at random from among all such triples. Let $M_n$ be the expected value (average value) of the largest of $a$, $b$, and $c$. As $n$ approaches infinity, what value does $\frac{M_n}{n}$ approach?

If $n\heartsuit m=n^3m^2$, what is $\frac{2\heartsuit 4}{4\heartsuit 2}$? $\textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4$

let $f:R\to R$ be a continuous function tf(t)>0 for $t\neq 0$. Prove that there exists a non-zero differentiable function $y:[0,\infty)\to R$ such that $y'(t)=f(y(t-1))\,\forall t>1$ and the roots of y are bounded.

The function $f(n)$ is defined on the nonnegative integers $n$ by: $f(0) = 0, f(1) = 1$, and \[f(n) = f\left(n -\frac{1}{2}m(m - 1)\right)-f\left(\frac{1}{2}m(m+ 1)-n\right)\] for $\frac{1}{2}m(m - 1) < n \le \frac{1}{2}m(m+ 1), m \ge 2$. Find the smallest integer $n$ for which $f(n) = 5$.

Show that for nonnegative integers $m$ and $n$, $\frac{\dbinom{m}{0}}{n+1}-\frac{\dbinom{m}{1}}{n+2}+...+(-1)^m\frac{\dbinom{m}{m}}{n+m+1}$ $=\frac{\dbinom{n}{0}}{m+1}-\frac{\dbinom{n}{1}}{m+2}+...+(-1)^n\frac{\dbinom{n}{n}}{m+n+1}$.

Find all functions $ f : \mathbb{R}\to\mathbb{R}$ that satisfy \[ f (x^{3} \plus{} y^{3}) \equal{} x^{2}f (x) \plus{} yf (y^{2}) \] for all $ x, y \in\mathbb R.$

Find all functions $f:\mathbb R\to\mathbb R$ such that $$f\left( x^2+xf(y)\right)=xf(x+y)$$ for all reals $x,y$.

The lines $L$ and $K$ are symmetric to each other with respect to the line $y=x$. If the equation of the line $L$ is $y=ax+b$ with $a\neq 0$ and $b \neq 0$, then the equation of $K$ is $y=$ $\text{(A)}\ \frac 1ax+b \qquad \text{(B)}\ -\frac 1ax+b \qquad \text{(C)}\ \frac 1ax - \frac ba \qquad \text{(D)}\ \frac 1ax+\frac ba \qquad \text{(E)}\ \frac 1ax -\frac ba$

A real function $f$ is defined for $0\le x\le 1$, with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$. Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$, then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$.

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x, y, z,$ \[f(x+y^2+z)=f(f(x))+yf(y)+f(z). \]

Show that for any positive integer $m$, there exists a positive integer $n$ so that in the decimal representations of the numbers $5^{m}$ and $5^{n}$, the representation of $5^{n}$ ends in the representation of $5^{m}$.

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

Find all real-valued functions $f$ on the reals such that $f(-x) = -f(x)$, $f(x+1) = f(x) + 1$ for all $x$, and $f\left(\dfrac{1}{x}\right) = \dfrac{f(x)}{x^2}$ for $x \not = 0$.

Given a set $X$ and a function $f: X \rightarrow X$, for each $x \in X$ we define $f^1(x)=f(x)$ and, for each $j \ge 1$, $f^{j+1}(x)=f(f^j(x))$. We say that $a \in X$ is a fixed point of $f$ if $f(a)=a$. For each $x \in \mathbb{R}$, let $\pi (x)$ be the quantity of positive primes lesser or equal to $x$. Given an positive integer $n$, we say that $f: \{1,2, \dots, n\} \rightarrow \{1,2, \dots, n\}$ is [i]catracha[/i] if $f^{f(k)}(k)=k$, for every $k=1, 2, \dots n$. Prove that: (a) If $f$ is catracha, $f$ has at least $\pi (n) -\pi (\sqrt{n}) +1$ fixed points. (b) If $n \ge 36$, there exists a catracha function $f$ with exactly $ \pi (n) -\pi (\sqrt{n}) + 1$ fixed points.