Equilateral triangles $AMB,BNC,CKA$ are constructed on the exterior of a triangle $ABC$. The perpendiculars from the midpoints of $MN, NK, KM$ to the respective lines $CA, AB, BC$ are constructed. Prove that these three perpendiculars pass through a single point.

Let $ C$ be a circle with center $ M$ in a plane $ V$, and $ P$ be a point not on the circle $ C$. $ (a)$ If $ P$ is fixed, prove that $ AP^2\plus{}BP^2$ is a constant for every diameter $ AB$ of the circle $ C$. $ (b)$ Let $ AB$ be a fixed diameter of $ C$ and $ P$ a point on a fixed sphere $ S$ not intersecting $ V$. Determine the points $ P$ on $ S$ that minimize $ AP^2\plus{}BP^2$.

A spaceborne energy storage device consists of two equal masses connected by a tether and rotating about their center of mass. Additional energy is stored by reeling in the tether; no external forces are applied. Initially the device has kinetic energy $E$ and rotates at angular velocity $\omega$. Energy is added until the device rotates at angular velocity $2\omega$. What is the new kinetic energy of the device? (a) $\sqrt{2}E$ (b) $2E$ (c) $2\sqrt{2}E$ (d) $4E$ (e) $8E$

Given an integer $n\ge 2$, show that there exist two integers $a,b$ which satisfy the following. For all integer $m$, $m^3+am+b$ is not a multiple of $n$.

A right circular cone pointing downward forms an angle of $60^\circ$ at its vertex. Sphere $S$ with radius $1$ is set into the cone so that it is tangent to the side of the cone. Three congruent spheres are placed in the cone on top of S so that they are all tangent to each other, to sphere $S$, and to the side of the cone. The radius of these congruent spheres can be written as $\tfrac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime. Find $a + b + c$. [asy] size(150); real t=0.12; void ball(pair x, real r, real h, bool ww=true) { pair xx=yscale(t)*x+(0,h); path P=circle(xx,r); unfill(P); draw(P); if(ww) draw(ellipse(xx-(0,r/2),0.85*r,t*r)); } pair X=(0,0); real H=17, h=5, R=h/2; draw(H*dir(120)--(0,0)--H*dir(60)); draw(ellipse((0,0.87*H),H/2,t*H/2)); pair Y=(R,h+2*R),C=(0,h); real r; for(int k=0;k<20;++k) { r=-(dir(30)*Y).x; Y-=(sqrt(3)/2*Y.x-r,abs(Y-C)-R-r)/3; } ball(Y.x*dir(90),r,Y.y,false); ball(X,R,h); ball(Y.x*dir(-30),r,Y.y); ball(Y.x*dir(210),r,Y.y);[/asy]

Let $D$ be an interior point of the triangle $ABC$. $CD$ and $AB$ intersect at $D_{c}$, $BD$ and $AC$ intersect at $D_{b}$, $AD$ and $BC$ intersect at $D_{a}$. Prove that there exists a triangle $KLM$ with orthocenter $H$ and the feet of altitudes $H_{k}\in LM, H_{l}\in KM, H_{m}\in KL$, so that $(AD_{c}D) = (KH_{m}H)$ $(BD_{c}D) = (LH_{m}H)$ $(BD_{a}D) = (LH_{k}H)$ $(CD_{a}D) = (MH_{k}H)$ $(CD_{b}D) = (MH_{l}H)$ $(AD_{b}D) = (KH_{l}H)$ where $(PQR)$ denotes the area of the triangle $PQR$

Show that the number $ 7^{100}-3^{100} $ has $ 85 $ digits and find its last $ 4 $ ones.

The inscribed sphere of a given tetrahedron touches all four faces of the tetrahedron at their respective centroids. Prove that the tetrahedron is regular.

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: [list] [*]In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin. [*]In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the first line. [*]In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line. [/list] Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

Let $A_1B_1C_1D_1A_2B_2C_2D_2$ be a unit cube, with $A_1B_1C_1D_1$ and $A_2B_2C_2D_2$ opposite square faces, and let $M$ be the center of face $A_2 B_2 C_2 D_2$. Rectangular pyramid $MA_1B_1C_1D_1$ is cut out of the cube. If the surface area of the remaining solid can be expressed in the form $a + \sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime, find $a+b$. [i]Author: Alex Zhu[/i]

Let $n$ be an integer greater than or equal to $2$. Consider real numbers $a_1, a_2, \dots, a_{2n}$ satisfying the condition \[ |a_k - a_{n+k}| \geqslant 1 \quad \text{for all } 1 \leqslant k \leqslant n. \] Determine the minimum possible value of \[ (a_1 - a_2)^2 + (a_2 - a_3)^2 + \dots + (a_{2n-1} - a_{2n})^2 + (a_{2n} - a_1)^2. \]

On a \(4 \times 4\) board, each cell is colored either black or white such that each row and each column have an even number of black cells. How many ways can the board be colored?

If you pick a random $3$-digit number, what is the probability that its hundreds digit is triple the ones digit?

A set of lines in the plan is called [i]nice [/i]i f every line in the set intersects an odd number of other lines in the set. Determine the smallest integer $k \ge 0$ having the following property: for each $2018$ distinct lines $\ell_1, \ell_2, ..., \ell_{2018}$ in the plane, there exist lines $\ell_{2018+1},\ell_{2018+2}, . . . , \ell_{2018+k}$ such that the lines $\ell_1, \ell_2, ..., \ell_{2018+k}$ are distinct and form a [i]nice [/i] set.

The circle $\omega$ with center at point $I$ inscribed in an triangle $ABC$ ($AB\neq AC$) touches the sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$, respectively. The circumcircles of triangles $ABC$ and $AEF$ intersect secondary at point $K.$ The lines $EF$ and $AK$ intersect at point $X$ and intersects the line $BC$ at points $Y$ and $Z$, respectively. The tangent lines to $\omega$, other than $BC$, passing through points $Y$ and $Z$ touch $\omega$ at points $P$ and $Q$, respectively. Let the lines $AP$ and $KQ$ intersect at the point $R$. Prove that if $M$ is a midpoint of segment $YZ,$ then $IR\perp XM$.

Let $D$ and $E$ be the midpoints of sides $[AB]$ and $[AC]$ of the triangle $ABC$. The circle of diameter $[AB]$ intersects the line $DE$ on the opposite side of $AB$ than $C$, in $X$. The circle of diameter $[AC]$ intersects $DE$ on the opposite side of $AC$ than $B$ in $Y$ . Let $T$ be the intersection of $BX$ and $CY$. Prove that the orthocenter of triangle $XY T$ lies on $BC$.

Let $P(x)$ be a quartic polynomial with integer coefficients and leading coefficient $1$ such that $P(\sqrt 2+\sqrt 3+\sqrt 6)=0$. Find $P(1)$.

The $64$ squares of an $8 \times 8$ chessboard are filled with positive integers in such a way that each integer is the average of the integers on the neighbouring squares. Show that in fact all the $64$ entries are equal.

Each number of the set $\{1,2, 3,4,5,6, 7,8\}$ is colored red or blue, following the following rules: (a) Number $4$ is colored red, and there is at least one blue number, (b) if two numbers $x,y$ have different colors and $x + y \le 8$, so the number $x + y$ is colored blue, (c) if two numbers $x,y$ have different colors and $x \cdot y \le 8$, then the number $x \cdot y$ is colored red. Find all the possible ways to color this set.

Find all solutions to $3^x-9^{x-1} = 2.$

If $p,q,$ and $r$ are primes with $pqr=7(p+q+r)$, find $p+q+r$.

Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

Three $n\times n$ squares form the figure $\Phi$ on the checkered plane as shown on the picture. (Neighboring squares are tpuching along the segment of length $n-1$.) Find all $n > 1$ for which the figure $\Phi$ can be covered with tiles $1\times 3$ and $3\times 1$ without overlapping.[img]https://pp.userapi.com/c850332/v850332712/115884/DKxvALE-sAc.jpg[/img]

Let $\triangle A_1B_1C_1$ be an equilateral triangle of area $60$. Chloe constructs a new triangle $\triangle A_2B_2C_2$ as follows. First, she flips a coin. If it comes up heads, she constructs point $A_2$ such that $B_1$ is the midpoint of $\overline{A_2C_1}$. If it comes up tails, she instead constructs $A_2$ such that $C_1$ is the midpoint of $\overline{A_2B_1}$. She performs analogous operations on $B_2$ and $C_2$. What is the expected value of the area of $\triangle A_2B_2C_2$?