The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$. [i]Proposed by Estonia[/i]

In the year $ 2001$, the United States will host the International Mathematical Olympiad. Let $ I$, $ M$, and $ O$ be distinct positive integers such that the product $ I\cdot M \cdot O \equal{} 2001$. What's the largest possible value of the sum $ I \plus{} M \plus{} O$? $ \textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\ 111 \qquad \textbf{(E)}\ 671$

Suppose that $a, b, c,d$ are pairwise distinct positive integers such that $a+b = c+d = p$ for some odd prime $p > 3$ . Prove that $abcd$ is not a perfect square.

For natural number $n$, define the function $f(n)$ to be the number you get by $f(n)$ adding the digits of the number $n$. For example, $f(16)=7$, $f(f(78))=6$, and $f(f(f(5978)))=2$. Find the least natural number $n$ such that $f(f(f(n)))$ is not a one-digit number.

Given an equilateral triangle, find all points inside the triangle such that the distance from the point to one of the sides is equal to the geometric mean of the distances from the point to the other two sides of the triangle.

Find the index of the singular point $0$ of the vector field with components \[(x^4+y^4+z^4,x^3y-xy^3,xyz^2)\]

Santa Claus has at least $n$ gifts for $n$ children. For $i\in\{1,2, ... , n\}$, the $i$-th child considers $x_i > 0$ of these items to be desirable. Assume that \[\dfrac{1}{x_1}+\cdots+\dfrac{1}{x_n}\le1.\] Prove that Santa Claus can give each child a gift that this child likes.

Let $A$ be a point on the circle $\omega$ centred at $B$ and $\Gamma$ a circle centred at $A$. For $i=1,2,3$, a chord $P_iQ_i$ of $\omega$ is tangent to $\Gamma$ at $S_i$ and another chord $P_iR_i$ of $\omega$ is perpendicular to $AB$ at $M_i$. Let $Q_iT_i$ be the other tangent from $Q_i$ to $\Gamma$ at $T_i$ and $N_i$ be the intersection of $AQ_i$ with $M_iT_i$. Prove that $N_1,N_2,N_3$ are collinear.

Show that given any $n+1$ diagonals of a convex $n$-gon, one can always find two which have no common point.

A positive integer has exactly $50$ divisors. Is it possible that no difference of two different divisors is divisible by $100$? [i]Proposed by A. Chironov[/i]

Let $d_1,d_2,\ldots,d_k$ be all factors of a positive integer $n$ including $1$ and $n$. If $d_1+d_2+\ldots+d_k=72$ then $\frac1{d_1}+\frac1{d_2}+\ldots+\frac1{d_k}$ is $\textbf{(A)}~\frac{k^2}{72}$ $\textbf{(B)}~\frac{72}k$ $\textbf{(C)}~\frac{72}n$ $\textbf{(D)}~\text{None of the above}$

Points $O_1$ and $O_2$ are the centers of the circumscribed and inscribed circles of an isosceles triangle $ABC$ ($AB = BC$). The circumcircles of triangles $ABC$ and $O_1O_2A$ intersect at points $A$ and $D$. Prove that line $BD$ is tangent to the circumcircle of the triangle $O_1O_2A$.

Let $n$ be a positive integer. Consider $2n$ distinct lines on the plane, no two of which are parallel. Of the $2n$ lines, $n$ are colored blue, the other $n$ are colored red. Let $\mathcal{B}$ be the set of all points on the plane that lie on at least one blue line, and $\mathcal{R}$ the set of all points on the plane that lie on at least one red line. Prove that there exists a circle that intersects $\mathcal{B}$ in exactly $2n - 1$ points, and also intersects $\mathcal{R}$ in exactly $2n - 1$ points. [i]Proposed by Pakawut Jiradilok and Warut Suksompong, Thailand[/i]

A laticial cycle of length $n$ is a sequence of lattice points $(x_k, y_k)$, $k = 0, 1,\cdots, n$, such that $(x_0, y_0) = (x_n, y_n) = (0, 0)$ and $|x_{k+1} -x_{k}|+|y_{k+1} - y_{k}| = 1$ for each $k$. Prove that for all $n$, the number of latticial cycles of length $n$ is a perfect square.

Let $N$ be a positive integer. Prove that at least one of the numbers $N$ of $3N$ contains at least one of the digits $1,2,9$. [i]Proposed by Evan Chang (squareman), USA[/i]

Given a positive integer $n$, let $[n] = \{1,2,\dots,n\}$. Let [list] [*] $a_n$ denote the number of functions $f: [n] \to [n]$ such that $f(f(i))\ge i$ for all $i$; and [*] $b_n$ denote the number of ordered set partitions of $[n]$, i.e., the number of ways to pick an integer $k$ and an ordered $k$-tuple of pairwise disjoint nonempty sets $(A_1,\dots,A_k)$ whose union is $[n]$. [/list] Prove that $a_n=b_n$. [i]Derek Liu[/i]

In $\triangle ABC$ with $\angle BAC = 60^{\circ}$ and circumcircle $\omega$, the angle bisector of $\angle BAC$ intersects side $\overline{BC}$ at point $D$, and line $AD$ is extended past $D$ to a point $A'$. Let points $E$ and $F$ be the feet of the perpendiculars of $A'$ onto lines $AB$ and $AC$, respectively. Suppose that $\omega$ is tangent to line $EF$ at a point $P$ between $E$ and $F$ such that $\tfrac{EP}{FP} = \tfrac{1}{2}$. Given that $EF=6$, the area of $\triangle ABC$ can be written as $\tfrac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p$. [i]Proposed by Taiki Aiba[/i]

$ABC$ is a triangle with $AB < AC$ and $\angle A = 2 \angle C$. $D$ is the point on $AC$ such that $CD = AB$. Let L be the line through $B$ parallel to $AC$. Let $L$ meet the external bisector of $\angle A$ at $M$ and the line through $C$ parallel to $AB$ at $N$. Show that $MD = ND$.

$\triangle ABC$ is an isosceles triangle with $AC = BC$ and $\angle ACB < 60^{\circ}$. $I$ and $O$ are the incenter and circumcenter of $\triangle ABC$. The circumcircle of $\triangle BIO$ intersects $BC$ at $D \neq B$. (a) Do the lines $AC$ and $DI$ intersect? Give a proof. (b) What is the angle of intersection between the lines $OD$ and $IB$?

Let triangle ABC has semiperimeter $ p$. E,F are located on AB such that $ CE\equal{}CF\equal{}p$. Prove that the C-excircle of triangle ABC touches the circumcircle (EFC).

Show that there exists a real $C > 1$ that satisfy the following property: if $n > 1$ and $a_0 < a_1 < \cdots < a_n$ are positive integers such that $\frac{1}{a_0},\frac{1}{a_1},\dots,\frac{1}{a_n}$ are in arithmetic progression, then $a_0 > C^n.$

et $ABCDEF$ be a convex hexagon which has an inscribed circle and a circumcribed. Denote by $\omega_{A}, \omega_{B},\omega_{C},\omega_{D},\omega_{E}$ and $\omega_{F}$ the inscribed circles of the triangles $FAB, ABC, BCD, CDE, DEF$ and $EFA$, respecitively. Let $l_{AB}$, be the external of $\omega_{A}$ and $\omega_{B}$; lines $l_{BC}$, $l_{CD}$, $l_{DE}$, $l_{EF}$, $l_{FA}$ are analoguosly defined. Let $A_1$ be the intersection point of the lines $l_{FA}$ and $l_{AB}$, $B_1, C_1, D_1, E_1, F_1$ are analogously defined. Prove that $A_1D_1, B_1E_1, C_1F_1$ are concurrent.

Fix a positive integer $n\geq 3$. Does there exist infinitely many sets $S$ of positive integers $\lbrace a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n\rbrace$, such that $\gcd (a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n)=1$, $\lbrace a_i\rbrace _{i=1}^n$, $\lbrace b_i\rbrace _{i=1}^n$ are arithmetic progressions, and $\prod_{i=1}^n a_i = \prod_{i=1}^n b_i$?

An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?