Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.

Let $ABCD$ be a parallelogram and point $M$ be the midpoint of $[AB]$ so that the quadrilateral $MBCD$ is cyclic. If $N$ is the point of intersection of the lines $DM$ and $BC$, and $P \in BC$, then prove that the ray $(DP$ is the angle bisector of $\angle ADM$ if and only if $PC = 4BC$.

Incircle of $\triangle ABC$ has centre $I$ and touches sides $AC, AB$ at $E,F$, respectively. The perpendicular bisector of segment $AI$ intersects side $AC$ at $P$. On side $AB$ a point $Q$ is chosen so that $QI \perp FP$. Prove that $EQ \perp AB$.

Let \( I \) and \( M \) be the incenter and the centroid of a scalene triangle \( ABC \), respectively. A line passing through point \( I \) parallel to \( BC \) intersects \( AC \) and \( AB \) at points \( E \) and \( F \), respectively. Reconstruct triangle \( ABC \) given only the marked points \( E, F, I, \) and \( M \). [i]Proposed by Hryhorii Filippovskyi[/i]

Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$. Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.

The triangle $ABC$ is inscribed in the circle. Construct a point $P$ such that the points of intersection of lines $AP, BP$ and $CP$ with this circle are the vertices of an equilateral triangle. (A. Zaslavsky)

On grid paper, mark three nodes so that in the triangle they formed, the sum of the two smallest medians equals to half-perimeter.

There are $27$ unit cubes. We are marking one point on each of the two opposing faces, two points on each of the other two opposing faces, and three points on each of the remaining two opposing faces of each cube. We are constructing a $3\times 3 \times 3$ cube with these $27$ cubes. What is the least number of marked points on the faces of the new cube? $ \textbf{(A)}\ 54 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 72 \qquad\textbf{(D)}\ 90 \qquad\textbf{(E)}\ 96 $

Let $ ABC$ be an acute-angled triangle; $ AD$ be the bisector of $ \angle BAC$ with $ D$ on $ BC$; and $ BE$ be the altitude from $ B$ on $ AC$. Show that $ \angle CED > 45^\circ .$ [b][weightage 17/100][/b]

Two congruent squares $Q$ and $Q'$ are given in the plane. Show that they can be divided into parts $T_1, T_2, \ldots , T_n$ and $T'_1 , T'_2 , \ldots , T'_n$, respectively, such that $T'_i$ is the image of $T_i$ under a translation for $i=1,2, \ldots, n.$

Given a circle and a point $P$ inside it, different from the center. We consider pairs of circles tangent to the given internally and to each other at point $P$. Find the locus of the points of intersection of the common external tangents to these circles.

Prove that there exists a positive constant $c$ such that the following statement is true: Consider an integer $n > 1$, and a set $\mathcal S$ of $n$ points in the plane such that the distance between any two different points in $\mathcal S$ is at least 1. It follows that there is a line $\ell$ separating $\mathcal S$ such that the distance from any point of $\mathcal S$ to $\ell$ is at least $cn^{-1/3}$. (A line $\ell$ separates a set of points S if some segment joining two points in $\mathcal S$ crosses $\ell$.) [i]Note. Weaker results with $cn^{-1/3}$ replaced by $cn^{-\alpha}$ may be awarded points depending on the value of the constant $\alpha > 1/3$.[/i] [i]Proposed by Ting-Feng Lin and Hung-Hsun Hans Yu, Taiwan[/i]

Let $ACBD$ be a asquare and $K,L,M,N$ be points of $AB,BC,CD,DA$ respectively. If $O$ is the center of the square , prove that the expression $$ \overrightarrow{OK}\cdot \overrightarrow{OL}+\overrightarrow{OL}\cdot\overrightarrow{OM}+\overrightarrow{OM}\cdot\overrightarrow{ON}+\overrightarrow{ON}\cdot\overrightarrow{OK}$$ is independent of positions of $K,L,M,N$, (i.e. is constant )

Let $n\geq3$ be a positive integer. Let $C_1,C_2,C_3,\ldots,C_n$ be unit circles in the plane, with centres $O_1,O_2,O_3,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that \[ \sum\limits^{}_{1\leq i<j\leq n}{1\over O_iO_j}\leq{(n-1)\pi\over 4}. \]

Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$

Each of the $12$ edges of a cube is labeled $0$ or $1$. Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the $6$ faces of the cube equal to $2?$ $\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }16\qquad\textbf{(E) }20$

On the coordinate plane, at some points with integer coordinates, there is a pebble (a finite number of pebbles). It is allowed to make the following move: select a pair of pebbles, take some vector $\vec{a}$ with integer coordinates and then move one of the selected pebbles to vector $\vec{a}$, and the other to the opposite vector $-\vec{a}$; it is forbidden that there should be more than one pebble at one point. Is it always possible to achieve a situation in which all the pebbles lie on the same straight line in a few moves? [i] K. Ivanov [/i]

Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. Segment $PQ$ is tangent to $\omega_1$ at $P$ and to $\omega_2$ at $Q$, and $A$ is closer to $PQ$ than $B$. Point $X$ is on $\omega_1$ such that $PX\parallel QB$, and point $Y$ is on $\omega_2$ such that $QY\parallel PB$. Given that $\angle APQ=30^\circ$ and $\angle PQA=15^\circ$, find the ratio $AX/AY$.

We are given triangles $ABC$ and $DEF$ such that $D\in BC, E\in CA, F\in AB$, $AD\perp EF, BE\perp FD, CF\perp DE$. Let the circumcenter of $DEF$ be $O$, and let the circumcircle of $DEF$ intersect $BC,CA,AB$ again at $R,S,T$ respectively. Prove that the perpendiculars to $BC,CA,AB$ through $D,E,F$ respectively intersect at a point $X$, and the lines $AR,BS,CT$ intersect at a point $Y$, such that $O,X,Y$ are collinear. [i]Proposed by Sammy Luo[/i]

A square in the [i]xy[/i]-plane has area [i]A[/i], and three of its vertices have [i]x[/i]-coordinates $2,0,$ and $18$ in some order. Find the sum of all possible values of [i]A[/i].

Prove that the diagonals of a quadrilateral are perpendicular to each other if and only if the midpoints of it's sides lie on a circle.

Evaluate \[\int_0^a \sqrt{2ax-x^2}\ dx \ (a>0)\]

A)Let $x,y$ be two complex numbers on the unit circle so that: $\frac{\pi }{3} \le \arg (x)-\arg (y) \le \frac{5 \pi }{3}$ Prove that for any $z \in \mathbb{C}$ we have: $|z|+|z-x|+|z-y| \ge |zx-y|$ B)Let $x,y$ be two complex numbers so that: $\frac{\pi }{3} \le \arg (x)-\arg (y) \le \frac{2 \pi }{3}$ Prove that for any $z \in \mathbb{C}$ we have: $|z|+|z-y|+|z-x| \ge | \frac{\sqrt{3}}{2} x +(y-\frac{x}{2})i|$

Acute scalene triangle $ABC$ has $G$ as its centroid and $O$ as its circumcenter. Let $H_a,\, H_b,\, H_c$ be the projections of $A,\, B,\, C$ on respective opposite sides and $D,\, E,\, F$ be the midpoints of $BC,\, CA,\, AB$ in that order. $\overrightarrow{GH_a},\, \overrightarrow{GH_b},\, \overrightarrow{GH_c}$ intersect $(O)$ at $X,\,Y,\,Z$ respectively. a. Prove that the circle $(XCE)$ pass through the midpoint of $BH_a$ b. Let $M,\, N,\, P$ be the midpoints of $AX,\, BY,\, CZ$ respectively. Prove that $\overleftrightarrow{DM},\, \overleftrightarrow{EN},\,\overleftrightarrow{FP}$ are concurrent.

How many diagonals does $11$-sided convex polygon have?