In the $x$-$y$ plane, for the origin $ O$, given an isosceles triangle $ OAB$ with $ AO \equal{} AB$ such that $ A$ is on the first quadrant and $ B$ is on the $ x$ axis. Denote the area by $ s$. Find the area of the common part of the traingle and the region expressed by the inequality $ xy\leq 1$ to give the area as the function of $ s$.

Find all functions $ f:\mathbb R \to \mathbb R$ such that $xf(y)+yf(x)=(x+y)f(x)f(y)$ for all reals $x$ and $y$.

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$. Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that \[ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right). \] [hide="comment"] [i]Edited by Orl.[/i] [/hide] [i]Proposed by Reid Barton, USA[/i]

The $2^{nd}$ order differentiable function $f:\mathbb R \longrightarrow \mathbb R$ is in such a way that for every $x\in \mathbb R$ we have $f''(x)+f(x)=0$. [b]a)[/b] Prove that if in addition, $f(0)=f'(0)=0$, then $f\equiv 0$. [b]b)[/b] Use the previous part to show that there exist $a,b\in \mathbb R$ such that $f(x)=a\sin x+b\cos x$.

Let $ f_i (x), i \equal{} 1,2,3 \cdots$ be defined by $ f_1 (x) \equal{} \frac{1}{1 \minus{} x}$ and $ f_{i\plus{}1} (x) \equal{} f_i (f_1 (x))$. Then $ f_{1998} (1998)$ equals A. 0 B. 1998 C. -1/1997 D. 1997/1998 E. None of these

Find the sum of all real numbers $x$ such that $5x^4-10x^3+10x^2-5x-11=0$.

Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number? [i]Nanang Susyanto, Jogjakarta[/i]

Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

Let $f (x) = 2x^2 + x - 1$, $f^0(x) = x$ and $f^{n + 1}(x) = f (f^n(x))$ for all real $x$ and $n \ge 0$ integer . (a) Determine the number of real distinct solutions of the equation of $f^3(x) = x$. (b) Determine, for each integer $n \ge 0$, the number of real distinct solutions of the equation $f^n(x) = 0$.

Let $f\colon \mathbb{R}^1\rightarrow \mathbb{R}^2$ be a continuous function such that $f(x)=f(x+1)$ for all $x$, and let $t\in [0,\frac14]$. Prove that there exists $x\in\mathbb{R}$ such that the vector from $f(x-t)$ to $f(x+t)$ is perpendicular to the vector from $f(x)$ to $f(x+\frac12)$. (translated by Miklós Maróti)

For any two nonnegative integers $n$ and $k$ satisfying $n\geq k$, we define the number $c(n,k)$ as follows: - $c\left(n,0\right)=c\left(n,n\right)=1$ for all $n\geq 0$; - $c\left(n+1,k\right)=2^{k}c\left(n,k\right)+c\left(n,k-1\right)$ for $n\geq k\geq 1$. Prove that $c\left(n,k\right)=c\left(n,n-k\right)$ for all $n\geq k\geq 0$.

The function $f(x)$ is defined on the reals such that $$f\left(\frac{1-4x}{4-x}\right) = 4-xf(x)$$ for all $x \ne 4$. There exists two distinct real numbers $a, b \ne 4$ such that $f(a) = f(b) = \frac{5}{2}$. $a+b$ can be represented as $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $10p + q$.

a) Find the value of $\sum_{n=1}^{\infty}\frac{\phi(n)}{2^n-1}$; b) Show that $\sum_k {m\choose k}{{n+k}\choose m}=\sum_k {m\choose k} {n\choose k} 2^k$ for $m,n\geq 0$; c) Using the identity $(1-x)^{-\frac 12}(1-x)^{-\frac 12}=(1-x)^{-1}$ derive a combinatorial identity! d) Express the value of $\sum (2^{a_1}-1)\ldots (2^{a_k}-1)$ where the sum is over all $2^{n-1}$ ways of choosing $(a_1,a_2,\ldots,a_k)$ such that $a_1+a_2+\ldots +a_k=n$, as a function of some Fibonacci term.

If $F(n)$ is a function such that $F(1)=F(2)=F(3)=1$, and such that $F(n+1)=\dfrac{F(n)\cdot F(n-1)+1}{F(n-2)}$ for $n\ge 3$, then $F(6)$ is equal to $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }7\qquad\textbf{(D) }11\qquad \textbf{(E) }26$

Given a sphere of unit radius with the big circle (i.e of unit radius) that will be called "equator". We shall use the words "pole", "parallel","meridian" as self-explanatory. a) Let $g(x)$, where $x$ is a point on the sphere, be the distance from this point to the equator plane. Prove that $g(x)$ has the property if $x_1, x_2, x_3$ are the ends of the pairwise orthogonal radiuses, then $$g(x_1)^2 + g(x_2)^2 + g(x_3)^2 = 1 \,\,\,\, (*)$$ Let function $f(x)$ be an arbitrary nonnegative function on a sphere that satisfies (*) property. b) Let $x_1$ and $x_2$ points be on the same meridian between the north pole and equator, and $x_1$ is closer to the pole than $x_2$. Prove that $f(x_1) > f(x_2)$. c) Let $y_1$ be closer to the pole than $y_2$. Prove that $f(y_1) > f(y_2)$. d) Let $z_1$ and $z_2$ be on the same parallel. Prove that $f(z_1) = f(z_2)$. e) Prove that for all $x , f(x) = g(x)$.

Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$. [i]Carl Lian and Brian Hamrick.[/i]

Let $ x,y,z$ be real numbers greater than or equal to $ 1.$ Prove that \[ \prod(x^{2} \minus{} 2x \plus{} 2)\le (xyz)^{2} \minus{} 2xyz \plus{} 2.\]

Let $f : \mathbb{N} \longrightarrow \mathbb{N}$ be a function such that $(x + y)f (x) \le x^2 + f (xy) + 110$, for all $x, y$ in $\mathbb{N}$. Determine the minimum and maximum values of $f (23) + f (2011)$.

Let $ I,J $ be two intervals, $ \varphi :J\longrightarrow\mathbb{R} $ be a continuous function whose image doesn't contain $ 0, $ and $ f,g:I\longrightarrow J $ be two differentiable functions such that $ f'=\varphi\circ f,g'=\varphi\circ g $ and such that the image of $ f-g $ contains $ 0. $ Show that $ f $ and $ g $ are the same function.

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

Let $S$ be the domain surrounded by the two curves $C_1:y=ax^2,\ C_2:y=-ax^2+2abx$ for constant positive numbers $a,b$. Let $V_x$ be the volume of the solid formed by the revolution of $S$ about the axis of $x$, $V_y$ be the volume of the solid formed by the revolution of $S$ about the axis of $y$. Find the ratio of $\frac{V_x}{V_y}$.

Suppose we have some proteins that each protein is a sequence of 7 "AMINO-ACIDS" $A,\ B,\ C,\ H,\ F,\ N$. For example $AFHNNNHAFFC$ is a protein. There are some steps that in each step an amino-acid will change to another one. For example with the step $NA\rightarrow N$ the protein $BANANA$ will cahnge to $BANNA$("in Persian means workman"). We have a set of allowed steps that each protein can change with these steps. For example with the set of steps: $\\ 1)\ AA\longrightarrow A\\ 2)\ AB\longrightarrow BA\\ 3)\ A\longrightarrow \mbox{null}$ Protein $ABBAABA$ will change like this: $\\ ABB\underline{AA}BA\\ \underline{AB}BABA\\ B\underline{AB}ABA\\ BB\underline{AA}BA\\ BB\underline{AB}A\\ BBB\underline{AA}\\ BBB\underline{A}\\ BBB$ You see after finite steps this protein will finish it steps. Set of allowed steps that for them there exist a protein that may have infinitely many steps is dangerous. Which of the following allowed sets are dangerous? a) $NO\longrightarrow OONN$ b) $\left\{\begin{array}{c}HHCC\longrightarrow HCCH\\ CC\longrightarrow CH\end{array}\right.$ c) Design a set of allowed steps that change $\underbrace{AA\dots A}_{n}\longrightarrow\underbrace{BB\dots B}_{2^{n}}$ d) Design a set of allowed steps that change $\underbrace{A\dots A}_{n}\underbrace{B\dots B}_{m}\longrightarrow\underbrace{CC\dots C}_{mn}$ You see from $c$ and $d$ that we acn calculate the functions $F(n)=2^{n}$ and $G(M,N)=mn$ with these steps. Find some other calculatable functions with these steps. (It has some extra mark.)

Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.

(a) Let $a,x,y$ be positive integers. Prove: if $x\ne y$, the also \[ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).\] (b) Show that there are no two positive integers $a$ and $b$ such that \[ab+\gcd(a,b)+\text{lcm}(a,b)=2014.\]

Find a function $f(x)$ defined for all real values of $x$ such that for all $x$, \[f(x+ 2) - f(x) = x^2 + 2x + 4,\] and if $x \in [0, 2)$, then $f(x) = x^2.$