A square $ABCD$ is inscribed in a circle. If $M$ is a point on the shorter arc $AB$, prove that \[MC \cdot MD > 3\sqrt{3} \cdot MA \cdot MB.\]

Points $A_1$, $A_2$, $A_3$, $A_4$ are the vertices of a regular tetrahedron of edge length $1$. The points $B_1$ and $B_2$ lie inside the figure bounded by the plane $A_1A_2A_3$ and the spheres of radius $1$ and centres $A_1$, $A_2$, $A_3$. Prove that $B_1B_2 < \max\{B_1A_1, B_1A_2, B_1A_3, B_1A_4\}$. [i] A. Kupavsky [/i]

In a triangle $ ABC$ incircle touches the sides $ AB$, $ AC$ and $ BC$ at $ C_1$, $ B_1$ and $ A_1$ respectively. Prove that $ \sqrt {\frac {AB_1}{AB}} \plus{} \sqrt {\frac {BC_1}{BC}} \plus{} \sqrt {\frac {CA_1}{CA}}\leq\frac {3}{\sqrt {2}}$ is true.

Given $\triangle ABC$ with incenter $I$. Line $AI$ meets $BC$ at $D$. The incenter of $\triangle ABD, \triangle ADC$ are $E,F$, respectively. Line $DE$ meets the circumcircle of $\triangle BCE$ at$ P(\neq E)$ and line $DF$ meets the circumcircle of $\triangle BCF$ at$ Q(\neq F)$. Show that the midpoint of $BC$ lies on the circumcircle of $\triangle DPQ$.

Consider the cube $ABCDA'B'C'D'$ (with face $ABCD$ directly above face $A'B'C'D'$). a) Find the locus of the midpoints of the segments $XY$, where $X$ is any point of $AC$ and $Y$ is any piont of $B'D'$; b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with $ZY=2XZ$.

Given a convex, $n$-sided polygon $P$, form a $2n$-sided polygon $\text{clip}(P)$ by cutting off each corner of $P$ at the edges' trisection points. In other words, $\text{clip}(P)$ is the polygon whose vertices are the $2n$ edge trisection points of $P$, connected in order around the boundary of $P$. Let $P_1$ be an isosceles trapezoid with side lengths $13,13,13,$ and $3$, and for each $i\geq 2$, let $P_i=\text{clip}(P_{i-1}).$ This iterative clipping process approaches a limiting shape $P_\infty=\lim_{i\to\infty}P_i$. If the difference of the areas of $P_{10}$ and $P_\infty$ is written as a fraction $\tfrac xy$ in lowest terms, calculate the number of positive integer factors of $x\cdot y$.

Prove that if a triangle has integer side lengths and the area (in square units) equals the perimeter (in units), then the perimeter is not a prime number.

Let $W,X,Y$ and $Z$ be points on a circumference $\omega$ with center $O$, in that order, such that $WY$ is perpendicular to $XZ$; $T$ is their intersection. $ABCD$ is the convex quadrilateral such that $W,X,Y$ and $Z$ are the tangency points of $\omega$ with segments $AB,BC,CD$ and $DA$ respectively. The perpendicular lines to $OA$ and $OB$ through $A$ and $B$, respectively, intersect at $P$; the perpendicular lines to $OB$ and $OC$ through $B$ and $C$, respectively, intersect at $Q$, and the perpendicular lines to $OC$ and $OD$ through $C$ and $D$, respectively, intersect at $R$. $O_1$ is the circumcenter of triangle $PQR$. Prove that $T,O$ and $O_1$ are collinear. [i]Proposed by CDMX[/i]

Two identical squares havind a side length of $ 5\text{cm} $ are each divided separately into $ 5 $ regions through intersection with some lines. Show that we can color the regions of the first square with five colors and the regions of the second with the same five colors such that the sum of the areas of the resultant regions that have the same colors at superpositioning the two squares is at least $ 5\text{cm}^2. $

A circular disk is partitioned into $ 2n$ equal sectors by $ n$ straight lines through its center. Then, these $ 2n$ sectors are colored in such a way that exactly $ n$ of the sectors are colored in blue, and the other $ n$ sectors are colored in red. We number the red sectors with numbers from $ 1$ to $ n$ in counter-clockwise direction (starting at some of these red sectors), and then we number the blue sectors with numbers from $ 1$ to $ n$ in clockwise direction (starting at some of these blue sectors). Prove that one can find a half-disk which contains sectors numbered with all the numbers from $ 1$ to $ n$ (in some order). (In other words, prove that one can find $ n$ consecutive sectors which are numbered by all numbers $ 1$, $ 2$, ..., $ n$ in some order.) [hide="Problem 8 from CWMO 2007"]$ n$ white and $ n$ black balls are placed at random on the circumference of a circle.Starting from a certain white ball,number all white balls in a clockwise direction by $ 1,2,\dots,n$. Likewise number all black balls by $ 1,2,\dots,n$ in anti-clockwise direction starting from a certain black ball.Prove that there exists a chain of $ n$ balls whose collection of numbering forms the set $ \{1,2,3\dots,n\}$.[/hide]

A circle passing through $A, B$ and the orthocenter of triangle $ABC$ meets sides $AC, BC$ at their inner points. Prove that $60^o < \angle C < 90^o$ . (A. Blinkov)

Let $A_1A_2A_3$ be a triangle and, for $1 \leq i \leq 3$, let $B_i$ be an interior point of edge opposite $A_i$. Prove that the perpendicular bisectors of $A_iB_i$ for $1 \leq i \leq 3$ are not concurrent.

A spherical shell of mass $M$ and radius $R$ is completely filled with a frictionless fluid, also of mass M. It is released from rest, and then it rolls without slipping down an incline that makes an angle $\theta$ with the horizontal. What will be the acceleration of the shell down the incline just after it is released? Assume the acceleration of free fall is $g$. The moment of inertia of a thin shell of radius $r$ and mass $m$ about the center of mass is $I = \frac{2}{3}mr^2$; the momentof inertia of a solid sphere of radius r and mass m about the center of mass is $I = \frac{2}{5}mr^2$. $\textbf{(A) } g \sin \theta \\ \textbf{(B) } \frac{3}{4} g \sin \theta\\ \textbf{(C) } \frac{1}{2} g \sin \theta\\ \textbf{(D) } \frac{3}{8} g \sin \theta\\ \textbf{(E) } \frac{3}{5} g \sin \theta$

Let $ABC $ be a triangle and $D $ be the feet of $A $-altitude.\\ $E,F $ are defined on segments $AD,BC $,respectively such that $\frac {AE}{DE}=\frac{BF}{CF} $.\\ Assume that $G $ lies on $AF $ such that $BG\perp AF $.Prove that $EF $ is tangent to the circumcircle of $CFG $. [i]Proposed by Mehdi Etesami Fard[/i]

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

Let $ABCDEF$ be a convex hexagon such that $AB=BC,CD=DE, EF=FA.$ Prove that $\frac{BC}{BE}+\frac{DE}{DA}+\frac{FA}{FC}\ge\frac{3}{2}$ and find when equality holds.

Let $\triangle ABC$ be a right triangle at the vertex $A$ such that the side $AB$ is shorter than the side $AC$. Let $D$ be the foot of the altitude from $A$ to $BC$ and $M$ the midpoint of $BC$. Let $E$ be a point on the ray $AB$, outside of the segment $AB$. Line $ED$ intersects the segment $AM$ at the point $F$. Point $H$ is on the side $AC$ such that $\angle EFH=90^{\circ}$. Suppose that $ED=FH$. Find the measure of the angle $\angle AED$.

Let positive integers $a,\ b,\ c$ be pairwise coprime. Denote by $g(a, b, c)$ the maximum integer not representable in the form $xa+yb+zc$ with positive integral $x,\ y,\ z$. Prove that \[ g(a, b, c)\ge \sqrt{2abc}\] [i](M. Ivanov)[/i] [hide="Remarks (containing spoilers!)"] 1. It can be proven that $g(a,b,c)\ge \sqrt{3abc}$. 2. The constant $3$ is the best possible, as proved by the equation $g(3,3k+1,3k+2)=9k+5$. [/hide]

Show that it is impossible to dissect an arbitary tetrahedron into six parts by planes or portions thereof so that each of the parts has a plane of symmetry.

Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ that passes through $M$. Suppose $ABCDEF$ is a cyclic hexagon such that $l(A, BDF), l(B, ACE), l(D, ABF), l(E, ABC)$ intersect at a single point. Prove that $CDEF$ is a rectangle. [color=blue]Should the first sentence read: Suppose $l(M, XYZ)$ is a Simson line of the triangle $XYZ$ [u]with respect to[/u] $M$. ? Since it appears weird that a Simson line that passes a point is to be constructed. However, this is Unsolved after all, so I'm not sure.[/color]

In triangle $ABC$, $AB>AC$. The bisectors of $\angle{B},\angle{C}$ intersect the sides $AC,AB$ at $P,Q$, respectively. Let $I$ be the incenter of $\Delta ABC$. Suppose that $IP=IQ$. How much isthe value of $\angle A$?

[b]Q[/b]. $ABC$ is an acute - angled triangle with $\odot(ABC)$ and $\Omega$ as the circumcircle and incircle respectively. Let $D, E, F$ to be the respective intouch points on $\overline{BC}, \overline{CA}$ and $\overline{AB}$. Circle $\gamma_A$ is drawn internally tangent to sides $\overline{AC}, \overline{AB}$ and $\odot(ABC)$ at $X, Y$ and $Z$ respectively. Another circle $(\omega)$ is constructed tangent to $\overline{BC}$ at $\mathcal{T}_1$ and internally tangent to $\odot(ABC)$ at $\mathcal{T}_2$. A tangent is drawn from $A$ such that it touches $\omega$ at $W$ and meets $BC$ at $V$, with $V$ lying inside $\odot(ABC)$. Now if $\overline{EF}$ meets $\odot(BC)$ at $\mathcal{X}_1$ and $\mathcal{X}_2$, opposite to vertex $B$ and $C$ respectively, where $\odot(BC)$ denotes the circle with $BC$ as diameter, prove that the set of lines $\{\overline{B\mathcal{X}_1}, \overline{ZS}, \overline{C\mathcal{X}_2}, \overline{DU}, \overline{YX}, \overline{\mathcal{T}_1W} \}$ are concurrent where $S$ is the mid-point of $\widehat{BC}$ containing $A$ and $U$ is the anti-pode of $D$ with respect to $\Omega$. If the line joining that concurrency point and $A$ meets $\odot(ABC)$ at $N\not = A$ prove that $\overline{AD}, \overline{ZN}$ and $\gamma_A$ pass through a common point. [i] Proposed by srijonrick[/i]

In triangle $ABC$ the midpoint of $BC$ is called $M$. Let $P$ be a variable interior point of the triangle such that $\angle CPM=\angle PAB$. Let $\Gamma$ be the circumcircle of triangle $ABP$. The line $MP$ intersects $\Gamma$ a second time at $Q$. Define $R$ as the reflection of $P$ in the tangent to $\Gamma$ at $B$. Prove that the length $|QR|$ is independent of the position of $P$ inside the triangle.

All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products? $\textbf{(A)}\ 312 \qquad \textbf{(B)}\ 343 \qquad \textbf{(C)}\ 625 \qquad \textbf{(D)}\ 729 \qquad \textbf{(E)}\ 1680$

Let $P$ be a point inside the acute angle $\angle BAC$. Suppose that $\angle ABP = \angle ACP = 90^{\circ}$. The points $D$ and $E$ are on the segments $BA$ and $CA$, respectively, such that $BD = BP$ and $CP = CE$. The points $F$ and $G$ are on the segments $AC$ and $AB$, respectively, such that $DF$ is perpendicular to $AB$ and $EG$ is perpendicular to $AC$. Show that $PF = PG$.