Find the smallest $n \in \mathbb{N}$ such that if any 5 vertices of a regular $n$-gon are colored red, there exists a line of symmetry $l$ of the $n$-gon such that every red point is reflected across $l$ to a non-red point.

Let $M = \{1, 2, \ldots, n\}$ and $\phi : M \to M$ be a bijection. (i) Prove that there exist bijections $\phi_1, \phi_2 : M \to M$ such that $\phi_1 \cdot \phi_2 = \phi , \phi_1^2 =\phi_2^2=E$, where $E$ is the identity mapping. (ii) Prove that the conclusion in (i) is also true if $M$ is the set of all positive integers.

Show that every integer greater than $1$ can be written as a sum of two square-free integers.

Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.

Three metals, $A, B $and $C$, with solutions of their respective cations are tested in a voltaic cell with the following results: $A$ and $B$: $A$ is the cathode $B$ and $C$: $C$ is the cathode $A$ and $C$: $A$ is the anode What is the order of the reduction potentials from highest to lowest for the cations of these metals? $ \textbf{(A)}\ A>B>C \qquad\textbf{(B)}\ B>C>A\qquad$ ${\textbf{(C)}\ C>A>B\qquad\textbf{(D}}\ B>A>C\qquad$

(1) Find the smallest positive integer $a$ such that $221|3^a -2^a$, (2) Let $A=\{n\in N^*: 211|1+2^n+3^n+4^n\}$. Are there infinitely many numbers $n$ such that both $n$ and $n+1$ belong to set $A$?

On a blackboard, there are $17$ integers not divisible by $17$. Alice and Bob play a game. Alice starts and they alternately play the following moves: $\bullet$ Alice chooses a number $a$ on the blackboard and replaces it with $a^2$ $\bullet$ Bob chooses a number $b$ on the blackboard and replaces it with $b^3$. Alice wins if the sum of the numbers on the blackboard is a multiple of $17$ after a finite number of steps. Prove that Alice has a winning strategy. (Daniel Holmes)

Circles $A,B,C,D$ are given on the plane such that circles $A$ and $B$ are externally tangent at $P, B$ and $C$ at $Q, C$ and $D$ at $R$, and $D$ and $A$ at $S$. Circles $A$ and $C$ do not meet, and so do not $B$ and $D$. (a) Prove that the points $P,Q,R,S$ lie on a circle. (b) Suppose that $A$ and $C$ have radius $2, B$ and $D$ have radius $3$, and the distance between the centers of $A$ and $C$ is $6$. Compute the area of the quadrilateral $PQRS$.

Let $A=\{1,2,\cdots ,100\}$. Let $S$ be a subset of power set of $A$ such that any two elements of $S$ has nonzero intersection (Note that elements of $S$ are actually some subsets of $A$). Then the maximum possible cardinality of $S$ is [list=1] [*] $2^{99}$ [*] $2^{99}+1$ [*] $2^{99}+2^{98}$ [*] None of these [/list]

Prove that there is no $1989$-digit natural number at least three of whose digits are equal to $5$ and such that the product of its digits equals their sum.

A differentiable function $f$ with $f(0) = f(1) = 0$ is defined on the interval $[0,1]$. Prove that there exists a point $y \in [0,1]$ such that $| f' (y)| = 4 \int _0^1 | f(x)|dx$.

Find the number of sequences of $2005$ terms with the following properties: (i) No three consecutive terms of the sequence are equal, (ii) Every term equals either $1$ or $-1$, (iii) The sum of all terms of the sequence is at least $666$.

Function $F(x)=|\cos^2x+2\sin x\cos x-\sin^2x+Ax+B|$, where $A,B$ are two real numbers, $x\in[0,\frac{3}{2}\pi]$. $M$ is the maximun value of $F(x)$. Find the minumum value of $M$.

There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$, and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group.

Does there exist a set $ M$ of points in space such that every plane intersects $ M$ at a finite but nonzero number of points?

A finite set $S{}$ of $n{}$ points is given in the plane. No three points lie on the same line. The number of non-self-intersecting closed $n{}$-link polylines with vertices at these points will be denoted by $f(S).$ Prove that [list=a] [*]$f(S)>0$ for all sets $S{};$ [*]$f(S)=1$ if and only if all the points of $S{}$ lie on the convex hull of $S{};$ [*]if $f(S)>1$ then $f(S)\geqslant n-1$, with equality if and only if one point of $S$ lies inside the convex hull; [*]if exactly two points of $S{}$ lie inside the convex hull, then\[f(S)\geqslant\frac{(n-2)(n-3)}{2}.\] [/list]Let $n\geqslant 3.$ Denote by $F(n)$ the largest possible value of the function $f(S)$ over all admissible sets $S{}$ of $n{}$ points. Prove that \[F(n)\geqslant3\cdot 2^{(n-8)/3}.\][i]Proposed by E. Bakaev and D. Magzhanov[/i]

Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$, \[f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y).\]

Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs.

Let be three distinct squares $ ABCD, BCEF, EFGH. $ Show that $ \angle EDF +\angle HDG =45^{\circ } . $

Three unit vectors $a,b,c$ are given on the plane. Prove that one can choose the signs in the expression $x=\pm a\pm b\pm c$ so as to obtain a vector $x$ with $|x|\le\sqrt2$.

If $ a$, $ b$, $ c$, and $ d$ are the solutions of the equation $ x^4 \minus{} bx \minus{} 3 \equal{} 0$, then an equation whose solutions are \[ \frac {a \plus{} b \plus{} c}{d^2}, \frac {a \plus{} b \plus{} d}{c^2}, \frac {a \plus{} c \plus{} d}{b^2}, \frac {b \plus{} c \plus{} d}{a^2} \]is $ \textbf{(A)}\ 3x^4 \plus{} bx \plus{} 1 \equal{} 0\qquad \textbf{(B)}\ 3x^4 \minus{} bx \plus{} 1 \equal{} 0\qquad \textbf{(C)}\ 3x^4 \plus{} bx^3 \minus{} 1 \equal{} 0$ $ \textbf{(D)}\ 3x^4 \minus{} bx^3 \minus{} 1 \equal{} 0\qquad \textbf{(E)}\ \text{none of these}$

Let $a_{1}, a_{2}, \cdots, a_{k}$ be relatively prime positive integers. Determine the largest integer which cannot be expressed in the form \[x_{1}a_{2}a_{3}\cdots a_{k}+x_{2}a_{1}a_{3}\cdots a_{k}+\cdots+x_{k}a_{1}a_{2}\cdots a_{k-1}\] for some nonnegative integers $x_{1}, x_{2}, \cdots, x_{k}$.

Let $a$ and $b$ be real numbers. It is known that the graph of the parabola $y =ax^2 +b$ cuts the graph of the curve $y = x+1/x$ in exactly three points. Prove that $3ab < 1$.

New license plates consist of two letters, three digits, and two letters (from the English alphabet of$ 26$ letters). What is the largest possible number of such license plates if it is required that every two of them differ at no less than two positions?

Let $a, b, c$ be real numbers, and define $S_n = a^n + b^n + c^n$ for positive integers $n$. Suppose that $S_1, S_2, S_3$ are integers satisfying $6 | 5S_1 - 3S_2 - 2S_3$. Show that $S_n$ is an integer for all positive integers $n$.