A convex polygon has the property that its vertices are coloured by three colors, each colour occurring at least once and any two adjacent vertices having different colours. Prove that the polygon can be divided into triangles by diagonals, no two of which intersect in the [b]interior[/b] of the polygon, in such a way that all the resulting triangles have vertices of all three colours.

Let $D_1$, $D_2$, ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$. Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$.

Let $ABCD$ be a square. Consider a variable point $P$ inside the square for which $\angle BAP \ge 60^\circ.$ Let $Q$ be the intersection of the line $AD$ and the perpendicular to $BP$ in $P$. Let $R$ be the intersection of the line $BQ$ and the perpendicular to $BP$ from $C$. [list] [*] [b](a)[/b] Prove that $|BP|\ge |BR|$ [*] [b](b)[/b] For which point(s) $P$ does the inequality in [b](a)[/b] become an equality?[/list]

Harry has $3$ sisters and $5$ brothers. His sister Harriet has $X$ sisters and $Y$ brothers. What is the product of $X$ and $Y$? $ \text{(A)}\ 8\qquad\text{(B)}\ 10\qquad\text{(C)}\ 12\qquad\text{(D)}\ 15\qquad\text{(E)}\ 18 $

Let $a, b, c, d$ be non-negative reals such that $\frac{1}{a+3}+\frac{1}{b+3}+\frac{1}{c+3}+\frac{1}{d+3}=1$. Show that there exists a permutation $(x_1, x_2, x_3, x_4)$ of $(a, b, c, d)$, such that $$x_1x_2+x_2x_3+x_3x_4+x_4x_1 \geq 4.$$

For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\{m, m+4, m+10, n+1, n+2, 2n\}$ are equal to $n$. What is $m+n$? $\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$

Let $a, b, x,$ and $y$ be real numbers with $a>4$ and $b>1$ such that $$\frac{x^2}{a^2}+\frac{y^2}{a^2-16}=\frac{(x-20)^2}{b^2-1}+\frac{(y-11)^2}{b^2}=1.$$ Find the least possible value of $a+b.$

Two right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral? [asy] size(4cm); fill((-5,0)--(0,12)--(0,6)--(8,0)--cycle, gray+opacity(0.3)); draw((0,0)--(0,12)--(-5,0)--cycle); draw((0,0)--(8,0)--(0,6)); label("5", (-2.5,0), S); label("13", (-2.5,6), dir(140)); label("6", (0,3), E); label("8", (4,0), S); [/asy] [i]Proposed by Nathan Xiong[/i]

A $20 \times 20 \times 20$ cube is composed of $2000$ bricks of size $2 \times 2 \times 1$ . Prove that it is possible to pierce the cube with a needle so that the needle passes through the cube without passing through a brick . (A . Andjans , Riga)

The function $\mu: \mathbb{N}\to \mathbb{C}$ is defined by \[\mu(n) = \sum^{}_{k \in R_{n}}\left( \cos \frac{2k\pi}{n}+i \sin \frac{2k\pi}{n}\right),\] where $R_{n}=\{ k \in \mathbb{N}\vert 1 \le k \le n, \gcd(k, n)=1 \}$. Show that $\mu(n)$ is an integer for all positive integer $n$.

Let $x$ and $y$ be positive real numbers. Which of the following expressions is larger than both $x$ and $y$? [b]A.[/b] $xy + 1$ [b]B.[/b] $(x + y)^2$ [b]C.[/b] $x^2 + y$ [b]D.[/b] $x(x + y)$ [b]E.[/b] $(x + y + 1)^2$

In $R^n$ is given $n-1$ vectors, the coordinates of each are zero-sum integers. Prove that the $(n-1)$-dimensional volume of an $(n-1)$-dimensional parallelepiped $P$ stretched by these vectors, is the product of an integer and $\sqrt(n)$.

Let $ZHAO$ be a square with area $2024$. Let $X$ be the center of this square and let $C$, $D$, $E$, $K$ be the centroids of $XZH$, $XHA$, $XAO$, and $XOZ$, respectively. Find $[ZHAO]$ $+$ $[CZHAO]$ $+$ $[DZHAO]$ $+$ $[EZHAO]$ $+$ $[KZHAO]$. (Here $[\mathcal P]$ denotes the area of the polygon $\mathcal P$.)

$3.$ Positive integers $a$ and $b$ are such that $a+b=\frac{a}{b}+\frac{b}{a}.$ What is the value of $a^2+b^2 ?$

Given trapezoid $ABCD$ with bases $AB \parallel CD$ ($AB < CD$). Let $O$ be the intersection of $AC$ and $BD$. Two straight lines from $D$ and $C$ are perpendicular to $AC$ and $BD$ intersect at $E$ , i.e. $CE \perp BD$ and $DE \perp AC$ . By analogy, $AF \perp BD$ and $BF \perp AC$ . Are three points $E , O, F$ located on the same line?

Solve for $x$: \begin{eqnarray*} v - w + x - y + z & = & 79 \\ v + w + x + y + z & = & -1 \\ v + 2w + 4x + 8y + 16z & = & -2 \\ v + 3w + 9x + 27y + 81z & = & -1 \\ v + 5w + 25x + 125y + 625z & = & 79. \end{eqnarray*}

Let $a \spadesuit b = \frac{a^2-b^2}{2b-2a}$ . Given that $3 \spadesuit x = -10$, compute $x$.

Let $I_a$ be the $A$-excenter of $\Delta ABC$ and the $A$-excircle of $\Delta ABC$ be tangent to the lines $AB,AC$ at $B',C'$, respectively. $ I_aB,I_aC$ meet $B'C'$ at $P,Q$, respectively. $M$ is the meet point of $BQ,CP$. Prove that the length of the perpendicular from $M$ to $BC$ is equal to $r$ where $r$ is the radius of incircle of $\Delta ABC$.

Let $k$ be a positive integer, and set $n=2^k$, $N=\{1, 2, \cdots, n\}$. For any bijective function $f:N\rightarrow N$, if a set $A\subset N$ contains an element $a\in A$ such that $\{a, f(a), f(f(a)), \cdots\} = A$, then we call $A$ as a cycle of $f$. Prove that: among all bijective functions $f:N\rightarrow N$, at least $\frac{n!}{2}$ of them have number of cycles less than or equal to $2k-1$. [i]Note: A function is bijective if and only if it is injective and surjective; in other words, it is 1-1 and onto.[/i] [i]Proposed by CSJL[/i]

Let $n$ and $k$ be positive integers such that $k\leq 2^n$. Banana and Corona are playing the following variant of the guessing game. First, Banana secretly picks an integer $x$ such that $1\leq x\leq n$. Corona will attempt to determine $x$ by asking some questions, which are described as follows. In each turn, Corona chooses $k$ distinct subsets of $\{1, 2, \ldots, n\}$ and, for each chosen set $S$, asks the question "Is $x$ in the set $S$?''. Banana picks one of these $k$ questions and tells both the question and its answer to Corona, who can then start another turn. Find all pairs $(n,k)$ such that, regardless of Banana's actions, Corona could determine $x$ in finitely many turns with absolute certainty. [i]Pitchayut Saengrungkongka, Thailand[/i]

Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.

Prove that for each positive integer $k $ there exists a triple $(a,b,c)$ of positive integers such that $abc = k(a+b+c)$. In all such cases prove that $a^3+b^3+c^3$ is not a prime.

Does there exist a positive integer $ n$ such that $ n$ has exactly 2000 prime divisors and $ n$ divides $ 2^n \plus{} 1$?

Let $x, y, z$ be real numbers from interval $(0, 1)$. Prove that $$\frac{1}{x(1-y)}+\frac{1}{y(1-x)}+\frac{1}{z(1-x)}\ge \frac{3}{xyz+(1-x)(1-y)(1-z)}$$

Two circles $C_1$ and $C_2$, with centres at $D$ and $E$ respectively, touch at $B$. The circle having $DE$ as diameter intersects the circle $C_1$ at $H$ and the circle $C_2$ at $K$. The points $H$ and $K$ both lie on the same side of the line $DE$. $HK$ extended in both directions meets the circle $C_1$ at $L$ and meets the circle $C_2$ at $M$. Prove that (a) $|LH| = |KM|$ (b) the line through $B$ perpendicular to $DE$ bisects $HK$.