Find all positive integer number $n$ such that $[\sqrt{n}]-2$ divides $n-4$ and $[\sqrt{n}]+2$ divides $n+4$. Note: $[r]$ denotes the integer part of $r$.

Let $ABC$ be a triangle with circumcenter $O$. Point $X$ is the intersection of the parallel line from $O$ to $AB$ with the perpendicular line to $AC$ from $C$. Let $Y$ be the point where the external bisector of $\angle BXC$ intersects with $AC$. Let $K$ be the projection of $X$ onto $BY$. Prove that the lines $AK, XO, BC$ have a common point.

We call a set of cells on a checkered plane [i]rook-connected[/i] if from any of its cells one can get to any other by moving along the cells of this set by moving the rook (the rook is allowed to fly through fields that do not belong to our set). Prove that a [i]rook-connected[/i] set of $100$ cells can be divided into pairs of cells, lying in one row or in one column.

Let $n$ be a positive integer and set $S=\{n,n+1,n+2,...,5n\}$ $a)$ If set $S$ is divided into two disjoint sets , prove that there exist three numbers $x$, $y$ and $z$(possibly equal) which belong to same subset of $S$ and $x+y=z$ $b)$ Does $a)$ hold for set $S=\{n,n+1,n+2,...,5n-1\}$

If $ p $ is an odd prime, then the following characterization holds. $$ 2^{p-1}\equiv 1\pmod{p^2}\iff \sum_{2=q}^{(p-1)/2} q^{p-2}\equiv -1\pmod p $$ [i]Marius Cavachi[/i]

Determine the number of integers $n \ge 2$ for which the congruence \[x^{25}\equiv x \; \pmod{n}\] is true for all integers $x$.

Anya, Borya, and Vasya listed words that could be formed from a given set of letters. They each listed a different number of words : Anya listed the most, Vasya the least . They were awarded points as follows. Each word listed by only one of them scored $2$ points for this child. Each word listed by two of them scored $1$ point for each of these two children. Words listed by all three of them scored $0$ points. Is it possible that Vasya got the highest score, and Anya the lowest? (A Shapovalov)

A hundred of silver coins are laid down in a line. A wizard can convert silver coin into golden one in $3$ seconds. Each golden coin, which is near the coin being converted, reduces this time by $1$ second. What minimal time is required for the wizard to convert all coins to gold?

The incircle of the acute-angled triangle $ABC$ is tangent to the sides $AB, BC, CA$ at points $C_1, A_1, B_1,$ respectively, and $I$ is the incenter. Let the midpoint of side $BC$ be $M.$ Let $J$ be the foot of the altitude drawn from $M$ to $C_1B_1.$ The tangent drawn from $B$ to the circumcircle of $\triangle BIC$ intersects $IJ$ at $X.$ If the circumcircle of $\triangle AXI$ intersects $AB$ at $Y,$ prove that $BY = BM.$

Let $ABCD$ be a convex cyclic quadrilateral. Prove that: $a)$ the number of points on the circumcircle of $ABCD$, like $M$, such that $\frac{MA}{MB}=\frac{MD}{MC}$ is $4$. $b)$ The diagonals of the quadrilateral which is made with these points are perpendicular to each other.

$A$ circle $\gamma$ is drawn and let $AB$ be a diameter. The point $C$ on $\gamma$ is the midpoint of the line segment $BD$. The line segments $AC$ and $DO$, where $O$ is the center of $\gamma$, intersect at $P$. Prove that there is a point $E$ on $AB$ such that $P$ is on the circle with diameter $AE.$

Prove the following assertion: The four altitudes of a tetrahedron $ABCD$ intersect in a point if and only if \[AB^2 + CD^2 = BC^2 + AD^2 = CA^2 + BD^2.\]

A frog is positioned at the origin in the coordinate plane. From the point $(x,y)$, the frog can jump to any of the points $(x+1, y), (x+2, y), (x, y+1),$ or $(x, y+2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0,0)$ and ends at $(4,4)$.

Find the number of even permutations of $ \{1,2,\ldots,n\}$ with no fixed points.

A function $f$ from the positive integers to the positive integers is called [i]Canadian[/i] if it satisfies $$\gcd\left(f(f(x)), f(x+y)\right)=\gcd(x, y)$$ for all pairs of positive integers $x$ and $y$. Find all positive integers $m$ such that $f(m)=m$ for all Canadian functions $f$.

On the plane are given $n > 2$ points, no three of which are collinear. Prove that among all closed polygonal lines passing through these points, any one with the minimum length is non-selfintersecting.

For every $n \in N$, is it possible to make a figure consisting of $n+1$ points, where $n$ points lie on one line and one point is not on that line, so that each pair of those points is an integer distance from each other?

For $x>0$, let $f(x)=x^x$. Find all values of $x$ for which $f(x)=f'(x)$.

The set $M = \{1, 2, . . . , 2n\}$ is partitioned into $k$ nonintersecting subsets $M_1,M_2, \dots, M_k,$ where $n \ge k^3 + k.$ Prove that there exist even numbers $2j_1, 2j_2, \dots, 2j_{k+1}$ in $M$ that are in one and the same subset $M_i$ $(1 \le i \le k)$ such that the numbers $2j_1 - 1, 2j_2 - 1, \dots, 2j_{k+1} - 1$ are also in one and the same subset $M_j (1 \le j \le k).$

For each positive integer $ n$, define $ f(n)$ as the number of digits $ 0$ in its decimal representation. For example, $ f(2)\equal{}0$, $ f(2009)\equal{}2$, etc. Please, calculate \[ S\equal{}\sum_{k\equal{}1}^{n}2^{f(k)},\] for $ n\equal{}9,999,999,999$. [i]Yudi Satria, Jakarta[/i]

An acute triangle has area $84$ and perimeter $42$, with each side being at least $10$ units long. Let $S$ be the set of points that are within $5$ units of some vertex of the triangle. What fraction of the area of $S$ lies outside the triangle? [i]Proposed by Nathan Ramesh

Find all real numbers $x, y,z,t \in [0, \infty)$ so that $$x + y + z \le t, \,\,\, x^2 + y^2 + z^2 \ge t \,\,\, and \,\,\,x^3 + y^3 + z^3 \le t.$$

Intersections between a plane and 12 edges of a cube are all $\alpha$, then $\sin\alpha=$________.

If a polygon has both an inscribed circle and a circumscribed circle, then define the [i]halo[/i] of that polygon to be the region inside the circumcircle but outside the incircle. In particular, all regular polygons and all triangles have halos. (a) What is the area of the halo of a square with side length 2? (b) What is the area of the halo of a 3-4-5 right triangle? (c) What is the area of the halo of a regular 2016-gon with side length 2?

A [i]regular octahedron[/i] has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of $Q$? [asy] // Note: This diagram was not made by me. import graph; // The Solid // To save processing time, do not use three (dimensions) // Project (roughly) to two size(15cm); pair Fr, Lf, Rt, Tp, Bt, Bk; Lf=(0,0); Rt=(12,1); Fr=(7,-1); Bk=(5,2); Tp=(6,6.7); Bt=(6,-5.2); draw(Lf--Fr--Rt); draw(Lf--Tp--Rt); draw(Lf--Bt--Rt); draw(Tp--Fr--Bt); draw(Lf--Bk--Rt,dashed); draw(Tp--Bk--Bt,dashed); label(rotate(-8.13010235)*slant(0.1)*"$Q$", (4.2,1.6)); label(rotate(21.8014095)*slant(-0.2)*"$?$", (8.5,2.05)); pair g = (-8,0); // Define Gap transform real a = 8; draw(g+(-a/2,1)--g+(a/2,1), Arrow()); // Make arrow // Time for the NET pair DA,DB,DC,CD,O; DA = (6.92820323028,0); DB = (3.46410161514,6); DC = (DA+DB)/3; CD = conj(DC); O=(0,0); transform trf=shift(3g+(0,3)); path NET = O--(-2*DA)--(-2DB)--(-DB)--(2DA-DB)--DB--O--DA--(DA-DB)--O--(-DB)--(-DA)--(-DA-DB)--(-DB); draw(trf*NET); label("$7$",trf*DC); label("$Q$",trf*DC+DA-DB); label("$5$",trf*DC-DB); label("$3$",trf*DC-DA-DB); label("$6$",trf*CD); label("$4$",trf*CD-DA); label("$2$",trf*CD-DA-DB); label("$1$",trf*CD-2DA); [/asy] $\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}~5\qquad$