Given a convex quadrilateral with $\angle B+\angle D<180$.Let $P$ be an arbitrary point on the plane,define $f(P)=PA*BC+PD*CA+PC*AB$. (1)Prove that $P,A,B,C$ are concyclic when $f(P)$ attains its minimum. (2)Suppose that $E$ is a point on the minor arc $AB$ of the circumcircle $O$ of $ABC$,such that$AE=\frac{\sqrt 3}{2}AB,BC=(\sqrt 3-1)EC,\angle ECA=2\angle ECB$.Knowing that $DA,DC$ are tangent to circle $O$,$AC=\sqrt 2$,find the minimum of $f(P)$.

Prove that in an isosceles trapezoid circumscibed around a circle, the segments connecting the points of tangency of opposite sides with the circle pass through the point of intersection of the diagonals.

\[ a\plus{}4b\plus{}9c\plus{}16d\plus{}25e\plus{}36f\plus{}49g\equal{}1\] \[ 4a\plus{}9b\plus{}16c\plus{}25d\plus{}36e\plus{}49f\plus{}64g\equal{}12\] \[ 9a\plus{}16b\plus{}25c\plus{}36d\plus{}49e\plus{}64f\plus{}81g\equal{}123\] Determine the value of $ 16a\plus{}25b\plus{}36c\plus{}49d\plus{}64e\plus{}81f\plus{}100g$.

Let a triangle $ABC$ inscribed in circle $\Gamma$ be given. Circle $\Theta$ lies in angle $Â$ of triangle and touches sides $AB, AC$ at $M_1, N_1$ and touches internally $\Gamma$ at $P_1$. The points $M_2, N_2, P_2$ and $M_3, N_3, P_3$ are defined similarly to angles $B$ and $C$ respectively. Show that $M_1N_1, M_2N_2$ and $M_3N_3$ intersect each other at their midpoints.

Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, \[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]

Let $ABC$ be a triangle. Let $A_1$ and $A_2$ be points on side $BC, B_1$ and $B_2$ be points on side $CA$ and $C_1$ and $C_2$ be points on side $AB$ such that $A_1A_2B_1B_2C_1C_2$ is a convex hexagon and that $B,A_1,A_2$ and $C$ are located in that order on side $BC$. We say that triangles $AB_2C_1, BA_1C_2$ and $CA_2B_1$ are glueable if there exists a triangle $PQR$ and there exist $X,Y$ and $Z$ on sides $QR, RP$ and $PQ$ respectively, such that triangle $AB_2C_1$ is congruent in that order to triangle $PYZ$, triangle $BA_1C_2$ is congruent in that order to triangle $QXZ$ and triangle $CA_2B_1$ is congruent in that order to triangle $RXY$. Prove that triangles $AB_2C_1, BA_1C_2$ and $CA_2B_1$ are glueable if and only if the centroids of triangles $A_1B_1C_1$ and $A_2B_2C_2$ coincide.

Find the sum of the roots of $\tan^2x-9\tan x+1=0$ that are between $x=0$ and $x=2\pi$ radians. $ \textbf{(A)}\ \frac{\pi}{2} \qquad\textbf{(B)}\ \pi \qquad\textbf{(C)}\ \frac{3\pi}{2} \qquad\textbf{(D)}\ 3\pi \qquad\textbf{(E)}\ 4\pi $

There are $100$ dwarfes with weight $1,2,...,100$. They sit on the left riverside. They can not swim, but they have one boat with capacity 100. River has strong river flow, so every dwarf has power only for one passage from right side to left as oarsman. On every passage can be only one oarsman. Can all dwarfes get to right riverside?

A square board with $8\text{cm}$ sides is divided into $64$ squares square with each side $1\text{cm}$. Each box can be painted white or black. Find the total number of ways to colour the board so that each square of side $2\text{cm}$ formed by four squares with a common vertex contains two white and two black squares.

Let $\triangle ABC$ be a triangle with $BA<AC$, $BC=10$, and $BA=8$. Let $H$ be the orthocenter of $\triangle ABC$. Let $F$ be the point on segment $AC$ such that $BF=8$. Let $T$ be the point of intersection of $FH$ and the extension of line $BC$. Suppose that $BT=8$. Find the area of $\triangle ABC$.

Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$.

A man was born on April 1st, [b]20[/b] BCE and died on April 1st, [b]24[/b] CE. How many years did he live? Clarification: Forget about the time he's born or died, assume he is born and died at the exact precise same time on each day

The country OMEC is divided in $5$ regions, each region is divided in $5$ districts, and, in each district, $1001$ people vote. Each person choose between $A$ or $B$. In a district, a candidate's letter wins if it's the letter with the most votes. In a region, a candidate's letter wins if it won in most districts. A candidate is the new president of OMEC if the candidate won in most regions. The candidate $A$ can rearrange the people of each district in each region (for example, A moves someone in District M to District N in region 1), but he can't change them to a different region. Find the minimum number of votes that the candidate $A$ needs to become the new president.

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

Find all positive integers $x$ and $n$ such that ${{x}^{3}}+3367={{2}^{n}}$.

The excircle of a triangle $ABC$ corresponding to $A$ touches the lines $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. The excircle corresponding to $B$ touches $BC,CA,AB$ at $A_2,B_2,C_2$, and the excircle corresponding to $C$ touches $BC,CA,AB$ at $A_3,B_3,C_3$, respectively. Find the maximum possible value of the ratio of the sum of the perimeters of $\triangle A_1B_1C_1$, $\triangle A_2B_2C_2$ and $\triangle A_3B_3C_3$ to the circumradius of $\triangle ABC$.

Given a prime $p$, consider integers $0<a<b<c<d<p$ such that $a^4\equiv b^4\equiv c^4\equiv d^4\pmod{p}$. Show that \[a+b+c+d\mid a^{2013}+b^{2013}+c^{2013}+d^{2013}\]

A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cub is $\tfrac12$ foot from the top face. The second cut is $\tfrac13$ foot below the first cut, and the third cut is $\tfrac1{17}$ foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet? [asy] import three; real d=11/102; defaultpen(fontsize(8)); defaultpen(linewidth(0.8)); currentprojection=orthographic(1,8/15,7/15); draw(unitcube, white, thick(), nolight); void f(real x) { draw((0,1,x)--(1,1,x)--(1,0,x)); } f(d); f(1/6); f(1/2); label("A", (1,0,3/4), W); label("B", (1,0,1/3), W); label("C", (1,0,1/6-d/4), W); label("D", (1,0,d/2), W); label("1/2", (1,1,3/4), E); label("1/3", (1,1,1/3), E); label("1/17", (0,1,1/6-d/4), E);[/asy] [asy] import three; real d=11/102; defaultpen(fontsize(8)); defaultpen(linewidth(0.8)); currentprojection=orthographic(2,8/15,7/15); int t=0; void f(real x) { path3 r=(t,1,x)--(t+1,1,x)--(t+1,1,0)--(t,1,0)--cycle; path3 f=(t+1,1,x)--(t+1,1,0)--(t+1,0,0)--(t+1,0,x)--cycle; path3 u=(t,1,x)--(t+1,1,x)--(t+1,0,x)--(t,0,x)--cycle; draw(surface(r), white, nolight); draw(surface(f), white, nolight); draw(surface(u), white, nolight); draw((t,1,x)--(t+1,1,x)--(t+1,1,0)--(t,1,0)--(t,1,x)--(t,0,x)--(t+1,0,x)--(t+1,1,x)--(t+1,1,0)--(t+1,0,0)--(t+1,0,x)); t=t+1; } f(d); f(1/2); f(1/3); f(1/17); label("D", (1/2, 1, 0), SE); label("A", (1+1/2, 1, 0), SE); label("B", (2+1/2, 1, 0), SE); label("C", (3+1/2, 1, 0), SE);[/asy] $\textbf{(A)}\:6\qquad \textbf{(B)}\:7\qquad \textbf{(C)}\:\frac{419}{51}\qquad \textbf{(D)}\:\frac{158}{17}\qquad \textbf{(E)}\:11$

If $S$ is the set of points $z$ in the complex plane such that $(3+4i)z$ is a real number, then $S$ is a $ \textbf{(A)}\text{ right triangle}\qquad\textbf{(B)}\text{ circle}\qquad\textbf{(C)}\text{ hyperbola}\qquad\textbf{(D)}\text{ line}\qquad\textbf{(E)}\text{ parabola} $

Jae and Yoon are playing SunCraft. The probability that Jae wins the $n$-th game is $\frac{1}{n+2}.$ What is the probability that Yoon wins the first six games, assuming there are no ties?

Let $ABCD$ be a parallelogram. Let $E$ be the midpoint of $AB$ and $F$ be the midpoint of $CD$. Points $P$ and $Q$ are on segments $EF$ and $CF$, respectively, such that $A, P$, and $Q$ are collinear. Given that $EP = 5$, $P F = 3$, and $QF = 12$, find $CQ$.

Consider the equation $x^2+(a+b+c)x+\lambda (ab+bc+ca)=0$ with $a,b,c>0$ and $\lambda\in \mathbb{R}$. Prove that: a)If $\lambda\le \frac{3}{4}$, the equation has real roots. b)If $a,b,c$ are the side lengths of a triangle and $\lambda\ge 1$, then the equation doesn't have real roots. [i]***[/i]

Given a finite string $S$ of symbols $X$ and $O$, we denote $\Delta(s)$ as the number of$X'$s in $S$ minus the number of $O'$s (For example, $\Delta(XOOXOOX)=-1$). We call a string $S$ [b]balanced[/b] if every sub-string $T$ of (consecutive symbols) $S$ has the property $-1\leq \Delta(T)\leq 2.$ (Thus $XOOXOOX$ is not balanced, since it contains the sub-string $OOXOO$ whose $\Delta$ value is $-3.$ Find, with proof, the number of balanced strings of length $n$.

Given two circles $k$ and $l$ which intersect at two points. One of their common tangents touches $k$ at point $K$, while the other common tangent touches $l$ at $L.$ Let $A$ and $B$ be the intersections of the line $KL$ with the circles $k$ and $l$, respectively. Prove that $\overline{AK} = \overline{BL}.$

Let $ABCD$ be a convex quadrilateral for which $DA = AB$ and $CA = CB$. Set $I_0 = C$ and $J_0 = D$, and for each nonnegative integer $n$, let $I_{n+1}$ and $J_{n+1}$ denote the incenters of $\triangle I_nAB$ and $\triangle J_nAB$, respectively. Suppose that \[ \angle DAC = 15^{\circ}, \quad \angle BAC = 65^{\circ} \quad \text{and} \quad \angle J_{2013}J_{2014}I_{2014} = \left( 90 + \frac{2k+1}{2^n} \right)^{\circ} \] for some nonnegative integers $n$ and $k$. Compute $n+k$. [i]Proposed by Evan Chen[/i]