Let $O$ be the circumcenter and $I$ be the incenter of $\vartriangle$, $P$ is the reflection from $I$ through $O$, the foot of perpendicular from $P$ to $BC,CA,AB$ is $X,Y,Z$, respectively. Prove that $AP^2+PX^2=BP^2+PY^2=CP^2+PZ^2$.

Let $(x_j,y_j)$, $1\le j\le 2n$, be $2n$ points on the half-circle in the upper half-plane. Suppose $\sum_{j=1}^{2n}x_j$ is an odd integer. Prove that $\displaystyle{\sum_{j=1}^{2n}y_j \ge 1}$.

Let $ABC$ be a triangle with $AC = 28$, $BC = 33$, and $\angle ABC = 2\angle ACB$. Compute the length of side $AB$. [i]2016 CCA Math Bonanza #10[/i]

Given two distinct points $A,B$ in the plane, for each point $C$ not on the line $AB$, we denote by $G$ and $I$ the centroid and incenter of the triangle $ABC$, respectively. (a) For $0<\alpha<\pi$, let $\Gamma$ be the set of points $C$ in the plane such that $\angle\left(\overrightarrow{CA},\overrightarrow{CB}\right)=\alpha+2k\pi$ as an oriented angle, where $k\in\mathbb Z$. If $C$ describes $\Gamma$, show that points $G$ and $I$ also descibre arcs of circles, and determine these circles. (b) Suppose that in addition $\frac\pi3<\alpha<\pi$. For which positions of $C$ in $\Gamma$ is $GI$ minimal? (c) Let $f(\alpha)$ denote the minimal $GI$ from the part (b). Give $f(\alpha)$ explicitly in terms of $a=AB$ and $\alpha$. Find the minimum value of $f(\alpha)$ for $\alpha\in\left(\frac\pi3,\pi\right)$.

$240$ students are participating a big performance show. They stand in a row and face to their coach. The coach askes them to count numbers from left to right, starting from $1$. (Of course their counts be like $1,2,3,...$)The coach askes them to remember their number and do the following action: First, if your number is divisible by $3$ then turn around. Then, if your number is divisible by $5$ then turn around. Finally, if your number is divisible by $7$ then turn around. (a) How many students are face to coach now? (b) What is the number of the $66^{\text{th}}$ student counting from left who is face to coach?

Let $ f(x)$ be a nonnegative, integrable function on $ (0,2\pi)$ whose Fourier series is $ f(x)\equal{}a_0\plus{}\sum_{k\equal{}1}^{\infty} a_k \cos (n_k x)$, where none of the positive integers $ n_k$ divides another. Prove that $ |a_k| \leq a_0$. [i]G. Halasz[/i]

A candy company makes $5$ colors of jellybeans, which come in equal proportions. If I grab a random sample of $5$ jellybeans, what is the probability that I get exactly $2$ distinct colors?

Let $[z]$ denote the greatest integer not exceeding $z$. Let $x$ and $y$ satisfy the simultaneous equations \[ \begin{array}{c} y=2[x]+3, \\ y=3[x-2]+5. \end{array} \]If $x$ is not an integer, then $x+y$ is $\textbf {(A) } \text{an integer} \qquad \textbf {(B) } \text{between 4 and 5} \qquad \textbf {(C) } \text{between -4 and 4} \qquad \textbf {(D) } \text{between 15 and 16} \qquad \textbf {(E) } 16.5$

A cube is cut by a plane such that the cross section is a pentagon. Show there is a side of the pentagon of length $\ell$ such that the inequality holds: \[ |\ell-1|>\frac{1}{5} \]

Eric has assembled a convex polygon $P$ from finitely many centrally symmetric (not necessarily congruent or convex) polygonal tiles. Prove that $P$ is centrally symmetric. [i]Proposed by Josef Tkadlec - Czech Republic[/i]

Exactly $4n$ numbers in set $A= \{ 1,2,3,...,6n \} $ of natural numbers painted in red, all other in blue. Proved that exist $3n$ consecutive natural numbers from $A$, exactly $2n$ of which numbers is red.

Let $P$ and $P^{\prime }$ be two parallelograms with equal area, and let their sidelengths be $a,$ $b$ and $a^{\prime },$ $b^{\prime }.$ Assume that $a^{\prime }\leq a\leq b\leq b^{\prime },$ and moreover, it is possible to place the segment $b^{\prime }$ such that it completely lies in the interior of the parallelogram $P.$ Show that the parallelogram $P$ can be partitioned into four polygons such that these four polygons can be composed again to form the parallelogram $% P^{\prime }.$

What is the smallest positive integer having $24$ positive divisors?

Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ the reflection of $B$ across $\ell$. Compute the expected value of $OP^2$. [i]Proposed by Lewis Chen[/i]

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuously differentiable,strictly convex function.Let $H$ be a Hilbert space and $A,B$ be bounded,self adjoint linear operators on $H$.Prove that,if $f(A)-f(B)=f'(B)(A-B)$ then $A=B$.

If $x,y\in(-2,2),xy=-1$, then the minumum value of $u=\frac{4}{4-x^2}+\frac{9}{9-y^2}$ is $\text{(A)}\frac{8}{5}\qquad\text{(B)}\frac{24}{11}\qquad\text{(C)}\frac{12}{7}\qquad\text{(D)}\frac{12}{5}\qquad$

A serie of positive integers $a_{1}$,$a_{2}$,. . . ,$a_{n}$ is $auto-delimited$ if for every index $i$ that holds $1\leq i\leq n$, there exist at least $a_{i}$ terms of the serie such that they are all less or equal to $i$. Find the maximum value of the sum $a_{1}+a_{2}+\cdot \cdot \cdot+a_{n}$, where $a_{1}$,$a_{2}$,. . . ,$a_{n}$ is an $auto-delimited$ serie.

Find the number of unordered pairs $\{ A,B \}$ of subsets of an n-element set $X$ that satisfies the following: (a) $A \not= B$ (b) $A \cup B = X$

In the fraction below and its decimal notation (with period of length $ 4$) every letter represents a digit, and different letters denote different digits. The numerator and denominator are coprime. Determine the value of the fraction: $ \frac{ADA}{KOK}\equal{}0.SNELSNELSNELSNEL...$ $ Note.$ Ada Kok is a famous dutch swimmer, and "snel" is Dutch for "fast".

Bolek draw a trapezoid $ABCD$ trapezoid ($AB // CD$) on the board, with its midsegment line $EF$ in it. Point intersection of his diagonal $AC, BD$ denote by $P,$ and his rectangular projection on line $AB$ denote by $Q$. Lolek, wanting to tease Bolek, blotted from the board everything except segments $EF$ and $PQ$. When Bolek saw it, wanted to complete the drawing and draw the original trapezoid, but did not know how to do it. Can you help Bolek?

If $f$ is a polynomial sends $\mathbb Z$ to $\mathbb Z$ and for $n\in\mathbb N_{\ge2}$, there exists $x\in\mathbb Z$ so that $n\nmid f(x)$, show that for every $k\in\mathbb Z$, there is a non-negative integer $t$ and $a_1,\ldots,a_t\in\{-1,1\}$ such that $$a_1f(1)+\ldots+a_tf(t)=k.$$

Prove that $\mathbb R^{2}$ has a dense subset such that has no three collinear points.

Find all functions $f : R\rightarrow R$ such that $f ( f (x)+y) = f (x^2 -y)+4 f (x)y$ for all $x,y \in R$ .

Given that $20^{22}+1$ has exactly $4$ prime divisors $p_1<p_2<p_3<p_4$, determine $p_1+p_2$.

Consider a white 100×100 square. Several cells (not necessarily neighbouring) were painted black. In each row or column that contains some black cells their number is odd. Hence we may consider the middle black cell for this row or column (with equal numbers of black cells in both opposite directions). It so happened that all the middle black cells of such rows lie in different columns and all the middle black cells of the columns lie in different rows. a) Prove that there exists a cell that is both the middle black cell of its row and the middle black cell of its column. b) Is it true that any middle black cell of a row is also a middle black cell of its column?