Let $ n\geqslant 3$ and $ P$ a convex $ n$-gon. Show that $ P$ can be, by $ n \minus{} 3$ non-intersecting diagonals, partitioned in triangles such that the circumcircle of each triangle contains the whole area of $ P$. Under which conditions is there exactly one such triangulation?

Find the smallest possible value of $n$ such that $n+2$ people can stand inside or on the border of a regular $n$-gon with side length $6$ feet where each pair of people are at least $6$ feet apart. [i]Proposed by Jeff Lin[/i]

Let $G$ be the set of points $(x, y)$ such that $x$ and $y$ are positive integers less than or equal to 6. A [i]magic grid[/i] is an assignment of an integer to each point in $G$ such that, for every square with horizontal and vertical sides and all four vertices in $G$, the sum of the integers assigned to the four vertices is the same as the corresponding sum for any other such square. A magic grid is formed so that the product of all 36 integers is the smallest possible value greater than 1. What is this product?

A square $ABCD$ with side length $1$ is inscribed in a circle. A smaller square lies in the circle with two vertices lying on segment $AB$ and the other two vertices lying on minor arc $AB$. Compute the area of the smaller square.

Let $U$ be the incenter of a triangle $ABC$ and $O_{1}, O_{2}, O_{3}$ be the circumcenters of the triangles $BCU, CAU, ABU$ , respectively. Prove that the circumcircles of the triangles $ABC$ and $O_{1}O_{2}O_{3}$ have the same center.

Let $ABC$ be an acute-angled triangle and let $AN$, $BM$ and $CP$ the altitudes with respect to the sides $BC$, $CA$ and $AB$, respectively. Let $R$, $S$ be the pojections of $N$ on the sides $AB$, $CA$, respectively, and let $Q$, $W$ be the projections of $N$ on the altitudes $BM$ and $CP$, respectively. (a) Show that $R$, $Q$, $W$, $S$ are collinear. (b) Show that $MP=RS-QW$.

Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$. (Here we always assume that an angle bisector is a ray.) [i]Proposed by Sergey Berlov, Russia[/i]

Given three planes $\alpha,\beta,\gamma$. Intersection angle between any two planes are all $\theta$.$\alpha\cap\beta=a,\beta\cap\gamma=b,\gamma\cap\alpha=c$. Given two conditions: A: $\theta>\frac{\pi}{3}$ B: $a,b,c$ share one point. $(\text{A})$A is sufficient but unnecessary condition of B. $(\text{B})$A is necessary but insufficient condition of B. $(\text{C})$A is sufficient and necessary condition of B. $(\text{D})$None above

Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?

Each of the numbers in the set $N = \{1, 2, 3, \cdots, n - 1\}$, where $n \geq 3$, is colored with one of two colors, say red or black, so that: [i](i)[/i] $i$ and $n - i$ always receive the same color, and [i](ii)[/i] for some $j \in N$, relatively prime to $n$, $i$ and $|j - i|$ receive the same color for all $i \in N, i \neq j.$ Prove that all numbers in $N$ must receive the same color.

Let $M$ be a point inside a triangle $ABC$ . The lines $AM , BM , CM$ intersect $BC, CA, AB$ at $A_{1}, B_{1}, C_{1}$, respectively. Assume that $S_{MAC_{1}}+S_{MBA_{1}}+S_{MCB_{1}}= S_{MA_{1}C}+S_{MB_{1}A}+S_{MC_{1}B}$ . Prove that one of the lines $AA_{1}, BB_{1}, CC_{1}$ is a median of the triangle $ABC.$

We call two circles in the space fighting if they are intersected or they are clipsed. Find a good necessary and sufficient condition for four distinct points $A,B,A',B'$ such that each circle passing through $A,B$ and each circle passing through $A',B'$ are fighting circles. [i]proposed by Ali Khezeli[/i]

Let $ABC$ be an acute triangle, and let $X$ be a variable interior point on the minor arc $BC$ of its circumcircle. Let $P$ and $Q$ be the feet of the perpendiculars from $X$ to lines $CA$ and $CB$, respectively. Let $R$ be the intersection of line $PQ$ and the perpendicular from $B$ to $AC$. Let $\ell$ be the line through $P$ parallel to $XR$. Prove that as $X$ varies along minor arc $BC$, the line $\ell$ always passes through a fixed point. (Specifically: prove that there is a point $F$, determined by triangle $ABC$, such that no matter where $X$ is on arc $BC$, line $\ell$ passes through $F$.) [i]Robert Simson et al.[/i]

Parallel lines $\ell_1$, $\ell_2$, $\ell_3$, $\ell_4$ are evenly spaced in the plane, in that order. Square $ABCD$ has the property that $A$ lies on $\ell_1$ and $C$ lies on $\ell_4$. Let $P$ be a uniformly random point in the interior of $ABCD$ and let $Q$ be a uniformly random point on the perimeter of $ABCD$. Given that the probability that $P$ lies between $\ell_2$ and $\ell_3$ is $\frac{53}{100}$ , the probability that $Q$ lies between $\ell_2$ and $\ell_3$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

Let $z_1,\ldots,z_n$ and $w_1,\ldots,w_n$ $(n\in\mathbb N)$ be complex numbers such that $$|\epsilon_1z_1+\ldots+\epsilon_nz_n|\le|\epsilon_1w_1+\ldots+\epsilon_nw_n|$$holds for every choice of $\epsilon_1,\ldots,\epsilon_n\in\{-1,1\}$. Prove that $$|z_1|^2+\ldots+|z_n|^2\le|w_1|^2+\ldots+|w_n|^2.$$

x,y,z positive real numbers such that $x^2+y^2+z^2=25$ Find the min price of $A=\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}$

For non-empty number sets $S, T$, define the sets $S+T=\{s+t\mid s\in S, t\in T\}$ and $2S=\{2s\mid s\in S\}$. Let $n$ be a positive integer, and $A, B$ be two non-empty subsets of $\{1,2\ldots,n\}$. Show that there exists a subset $D$ of $A+B$ such that 1) $D+D\subseteq 2(A+B)$, 2) $|D|\geq\frac{|A|\cdot|B|}{2n}$, where $|X|$ is the number of elements of the finite set $X$.

The student wrote on the board three natural numbers that are consecutive members of one arithmetic progression. Then he erased the commas separating the numbers, resulting in a seven-digit number. What is the largest number that could result?

A point $F$ inside a triangle $ABC$ is chosen so that $\angle AFB = \angle BFC = \angle CFA$. The line passing through $F$ and perpendicular to $BC$ meets the median from $A$ at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the points $A_1, B_1$, and $C_1$ are three vertices of some regular hexagon, and that the three remaining vertices of that hexagon lie on the sidelines of $ABC$.

In a triangle $ABC$ with $AB<AC<BC$, the perpendicular bisectors of $AC$ and $BC$ intersect $BC$ and $AC$ at $K$ and $L$, respectively. Let $O$, $O_1$, and $O_2$ be the circumcentres of triangles $ABC$, $CKL$, and $OAB$, respectively. Prove that $OCO_1O_2$ is a parallelogram.

Find all real polynomials $ g(x)$ of degree at most $ n \minus{} 3$, $ n\geq 3$, knowing that all the roots of the polynomial $ f(x) \equal{} x^n \plus{} nx^{n \minus{} 1} \plus{} \frac {n(n \minus{} 1)}2 x^{n \minus{} 2} \plus{} g(x)$ are real.

Circles $\omega_{1}$ and $\omega_{2}$ intersect at points $A$ and $B$. Let $t_{1}$ and $t_{2}$ be the tangents to $\omega_{1}$ and $\omega_{2}$, respectively, at point $A$. Let the second intersection of $\omega_{1}$ and $t_{2}$ be $C$, and let the second intersection of $\omega_{2}$ and $t_{1}$ be $D$. Points $P$ and $E$ lie on the ray $AB$, such that $B$ lies between $A$ and $P$, $P$ lies between $A$ and $E$, and $AE = 2 \cdot AP$. The circumcircle to $\bigtriangleup BCE$ intersects $t_{2}$ again at point $Q$, whereas the circumcircle to $\bigtriangleup BDE$ intersects $t_{1}$ again at point $R$. Prove that points $P$, $Q$, and $R$ are collinear.

On a non-rhombus parallelogram $ ABCD, $ the vertex $ B $ is projected on $ AC $ in the point $ E. $ The perpendicular on $ BD $ thru $ E $ intersects the lines $ BC $ and $ AB $ in $ F $ and $ G, $ respectively. Show that $ EF=EG $ if and only if $ \angle ABC=90^{\circ } . $ [i]Mircea Becheanu[/i]

a) The product of $n$ integers equals $n$, and their sum is zero. Prove that $n$ is divisible by $4$. b) Let $n$ is divisible by $4$. Prove that there exist $n$ integers such, that their product equals $n$, and their sum is zero.

Each square in a $5 \times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules: [list] [*] Any filled square with two or three filled neighbors remains filled. [*] Any empty square with exactly three filled neighbors becomes a filled square. [*] All other squares remain empty or become empty. [/list] A sample transformation is shown in the figure below. [asy] import geometry; unitsize(0.6cm); void ds(pair x) { filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,gray+opacity(0.5),invisible); } ds((1,1)); ds((2,1)); ds((3,1)); ds((1,3)); for (int i = 0; i <= 5; ++i) { draw((0,i)--(5,i)); draw((i,0)--(i,5)); } label("Initial", (2.5,-1)); draw((6,2.5)--(8,2.5),Arrow); ds((10,2)); ds((11,1)); ds((11,0)); for (int i = 0; i <= 5; ++i) { draw((9,i)--(14,i)); draw((i+9,0)--(i+9,5)); } label("Transformed", (11.5,-1)); [/asy] Suppose the $5 \times 5$ grid has a border of empty squares surrounding a $3 \times 3$ subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.) [asy] import geometry; unitsize(0.6cm); void ds(pair x) { filldraw(x -- (1,0) + x -- (1,1) + x -- (0,1)+x -- cycle,gray+opacity(0.5),invisible); } for (int i = 1; i < 4; ++ i) { for (int j = 1; j < 4; ++j) { label("?",(i + 0.5, j + 0.5)); } } for (int i = 0; i <= 5; ++i) { draw((0,i)--(5,i)); draw((i,0)--(i,5)); } label("Initial", (2.5,-1)); draw((6,2.5)--(8,2.5),Arrow); ds((11,2)); for (int i = 0; i <= 5; ++i) { draw((9,i)--(14,i)); draw((i+9,0)--(i+9,5)); } label("Transformed", (11.5,-1)); [/asy] $$\textbf{(A) 14}~\textbf{(B) 18}~\textbf{(C) 22}~\textbf{(D) 26}~\textbf{(E) 30}$$