For $ \phi: \mathbb{N} \mapsto \mathbb{Z}$ let us define \[ M_{\phi} \equal{} \{f: \mathbb{N} \mapsto \mathbb{Z}, f(x) > f(\phi(x)), \forall x \in \mathbb{N} \}.\] Prove that if $ M_{\phi_1} \equal{} M_{\phi_2} \neq \emptyset,$ then $ \phi_1 \equal{} \phi_2.$ Does this property remain true if \[ M_{\phi} \equal{} \{f: \mathbb{N} \mapsto \mathbb{N}, f(x) > f(\phi(x)), \forall x \in \mathbb{N} \}?\]

The function $f$ is defined on the set of integers and satisfies \[ f(n)=\begin{cases} n-3 & \text{if } n\ge 1000 \\ f(f(n+5)) & \text{if } n<1000\end{cases} \] Find $f(84)$.

Suppose that $x$ is measured in radians. Find the maximum value of \[\frac{\sin2x+\sin4x+\sin6x}{\cos2x+\cos4x+\cos6x}\] for $0\le x\le \frac{\pi}{16}$

Five distinct points $A, B, C, D$ and $E$ lie in this order on a circle of radius $r$ and satisfy $AC = BD = CE = r$. Prove that the orthocentres of the triangles $ACD, BCD$ and $BCE$ are the vertices of a right-angled triangle.

Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.

Find all pairs of positive integers $(m,n)$ such that \[m+n-\frac{3mn}{m+n}=\frac{2011}{3}.\]

A function $f$ has its domain equal to the set of integers $0$, $1$, $\ldots$, $11$, and $f(n)\geq 0$ for all such $n$, and $f$ satisfies [list] [*]$f(0)=0$ [*]$f(6)=1$ [*]If $x\geq 0$, $y\geq 0$, and $x+y\leq 11$, then $f(x+y)=\tfrac{f(x)+f(y)}{1-f(x)f(y)}$.[/list] Find $f(2)^2+f(10)^2$.

Let $a$, $b$, $c$, $d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.

In $xyz$ space, find the volume of the solid expressed by $x^2+y^2\leq z\le \sqrt{3}y+1.$

The numbers from 1 to 1996 are written down ------ 12345678910111213.... How many zeros are written? A. 489 B. 699 C. 796 D. 996 E. None of these

The expression $\dfrac{\dfrac{a}{a+y}+\dfrac{y}{a-y}}{\dfrac{y}{a+y}-\dfrac{a}{a-y}}$, a real, $a\neq 0$, has the value $-1$ for: $\textbf{(A)}\ \text{all but two real values of }y \qquad \textbf{(B)}\ \text{only two real values of }y \qquad$ $\textbf{(C)}\ \text{all real values of }y \qquad \textbf{(D)}\ \text{only one real value of }y \qquad \textbf{(E)}\ \text{no real values of }y$

Let $A$ be the set of positive integers of the form $a^2 +2b^2$, where $a$ and $b$ are integers and $b \neq 0$. Show that if $p$ is a prime number and $p^2 \in A$, then $p \in A$.

Let's call $1 \times 2$ rectangle, which can be a rotated, a domino. Prove that there exists polygon, who can be covered by dominoes in exactly $2021$ different ways.

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.

Let $S$ be the set of positive real numbers. Find all functions $f\colon S^3 \to S$ such that, for all positive real numbers $x$, $y$, $z$ and $k$, the following three conditions are satisfied: (a) $xf(x,y,z) = zf(z,y,x)$, (b) $f(x, ky, k^2z) = kf(x,y,z)$, (c) $f(1, k, k+1) = k+1$. ([i]United Kingdom[/i])

Point $O$ is inside triangle $ABC$ such that $\angle AOB = \angle BOC = \angle COA = 120^\circ .$ Prove that \[\frac{AO^2}{BC}+\frac{BO^2}{CA}+\frac{CO^2}{AB} \geq \frac{AO+BO+CO}{\sqrt 3}.\]

Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find \[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \]

Find the mean value of the function $\ln r$ on the circle $(x - a)^2 + (y-b)^2 = R^2$ (of the function $1/r$ on the sphere).

Consider a latticelike packing of translates of a convex region $K$. Let $t$ be the area of the fundamental parallelogram of the lattice defining the packing, and let $t_{\min} (K)$ denote the minimal value of $t$ taken for all latticelike packings. Is there a natural number $N$ such that for any $n>N$ and for any $K$ different from a parallelogram, $nt_{\min} (K)$ is smaller that the area of any convex domain in which $n$ translates to $K$ can be placed without overlapping? (By a [i]latticelike packing[/i] of $K$ we mean a set of nonoverlapping translates of $K$ obtained from $K$ by translations with all vectors of a lattice.) [G. and L. Fejes-Toth]

Let $ A\subset \mathbb C$ be a closed and countable set. Prove that if the analytic function $ f: \mathbb C\backslash A\longrightarrow \mathbb C$ is bounded, then $ f$ is constant.

Let $A=\{1, 3, 5, 7, 9\}$ and $B=\{2, 4, 6, 8, 10\}$. Let $f$ be a randomly chosen function from the set $A\cup B$ into itself. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probablity that $f$ is a one-to-one function on $A\cup B$ given that it maps $A$ one-to-one into $A\cup B$ and it maps $B$ one-to-one into $A\cup B$. Find $m+n$.

For values of $ x$ less than $ 1$ but greater than $ \minus{}4$, the expression \[ \frac{x^2 \minus{} 2x \plus{} 2}{2x \minus{} 2} \] has: $ \textbf{(A)}\ \text{no maximum or minimum value}\qquad \\ \textbf{(B)}\ \text{a minimum value of }{\plus{}1}\qquad \\ \textbf{(C)}\ \text{a maximum value of }{\plus{}1}\qquad \\ \textbf{(D)}\ \text{a minimum value of }{\minus{}1}\qquad \\ \textbf{(E)}\ \text{a maximum value of }{\minus{}1}$

In the $xyz$-plane given points $P,\ Q$ on the planes $z=2,\ z=1$ respectively. Let $R$ be the intersection point of the line $PQ$ and the $xy$-plane. (1) Let $P(0,\ 0,\ 2)$. When the point $Q$ moves on the perimeter of the circle with center $(0,\ 0,\ 1)$ , radius 1 on the plane $z=1$, find the equation of the locus of the point $R$. (2) Take 4 points $A(1,\ 1,\ 1) , B(1,-1,\ 1), C(-1,-1,\ 1)$ and $D(-1,\ 1,\ 1)$ on the plane $z=2$. When the point $P$ moves on the perimeter of the circle with center $(0,\ 0,\ 2)$ , radius 1 on the plane $z=2$ and the point $Q$ moves on the perimeter of the square $ABCD$, draw the domain swept by the point $R$ on the $xy$-plane, then find the area.

Let $\mathbb{F}_p$ denote the field of integers modulo a prime $p,$ and let $n$ be a positive integer. Let $v$ be a fixed vector in $\mathbb{F}_p^n,$ let $M$ be an $n\times n$ matrix with entries in $\mathbb{F}_p,$ and define $G:\mathbb{F}_p^n\to \mathbb{F}_p^n$ by $G(x)=v+Mx.$ Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x)=G(x)$ and $G^{(k+1)}(x)=G(G^{(k)}(x)).$ Determine all pairs $p,n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0),$ $k=1,2,\dots,p^n$ are distinct.

Prove the following inequalities: (1) $ x\minus{}\sin x\leq \tan x\minus{}x\ \ \left(0\leq x<\frac{\pi}{2}\right)$ (2) $ \int_0^x \cos (\tan t\minus{}t)\ dt\leq \sin (\sin x)\plus{}\frac 12 \left(x\minus{}\frac{\sin 2x}{2}\right)\ \left(0\leq x\leq \frac{\pi}{3}\right)$