Let $ABC$ be a triangle and $I$ its incenter. Suppose $AI=\sqrt{2}$, $BI=\sqrt{5}$, $CI=\sqrt{10}$ and the inradius is $1$. Let $A'$ be the reflection of $I$ across $BC$, $B'$ the reflection across $AC$, and $C'$ the reflection across $AB$. Compute the area of triangle $A'B'C'$.

Determine all the sets of six consecutive positive integers such that the product of some two of them . added to the product of some other two of them is equal to the product of the remaining two numbers.

A point $A_0$ and two lines $d_1$ and $d_2$ are given in the space. For each nonnegative integer $n$ we denote by $B_n$ the projection of $A_n$ on $d_2,$ and by $A_{n+1}$ the projection of $B_n$ on $d_1.$ Prove that there exist two segments $[A'A''] \subset d_1$ and $[B'B''] \subset d_2$ of length $0.001$ and a nonnegative integer $N$ such that $A_n \in [A'A'']$ and $B_n \in [B'B'']$ for any $n \ge N.$

$$\int_{-\infty}^\infty\frac{x\arctan(x)}{x^4+1}\,dx$$ [i]Proposed by Connor Gordon[/i]

A $ 4\times 4$ block of calendar dates is shown. The order of the numbers in the second row is to be reversed. Then the order of the numbers in the fourth row is to be reversed. Finally, the numbers on each diagonal are to be added. What will be the positive difference between the two diagonal sums? \[ \setlength{\unitlength}{5mm} \begin{picture}(4,4)(0,0) \multiput(0,0)(0,1){5}{\line(1,0){4}} \multiput(0,0)(1,0){5}{\line(0,1){4}} \put(0,3){\makebox(1,1){\footnotesize{1}}} \put(1,3){\makebox(1,1){\footnotesize{2}}} \put(2,3){\makebox(1,1){\footnotesize{3}}} \put(3,3){\makebox(1,1){\footnotesize{4}}} \put(0,2){\makebox(1,1){\footnotesize{8}}} \put(1,2){\makebox(1,1){\footnotesize{9}}} \put(2,2){\makebox(1,1){\footnotesize{10}}} \put(3,2){\makebox(1,1){\footnotesize{11}}} \put(0,1){\makebox(1,1){\footnotesize{15}}} \put(1,1){\makebox(1,1){\footnotesize{16}}} \put(2,1){\makebox(1,1){\footnotesize{17}}} \put(3,1){\makebox(1,1){\footnotesize{18}}} \put(0,0){\makebox(1,1){\footnotesize{22}}} \put(1,0){\makebox(1,1){\footnotesize{23}}} \put(2,0){\makebox(1,1){\footnotesize{24}}} \put(3,0){\makebox(1,1){\footnotesize{25}}} \end{picture} \]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10$

In acute triangle $ABC$ ($\triangle ABC$ is not an isosceles triangle), $I$ is its incentre, and circle $ \omega$ is its inscribed circle. $\odot\omega$ touches $BC, CA, AB$ at $D, E, F$ respectively. $AD$ intersects with $\odot\omega$ at $J$ ($J\neq D$), and the circumcircle of $\triangle BCJ$ intersects $\odot\omega$ at $K$ ($K\neq J$). The circumcircle of $\triangle BFK$ and $\triangle CEK$ meet at $L$ ($L\neq K$). Let $M$ be the midpoint of the major arc $BAC$. Prove that $M, I, L$ are collinear.

A polynomial $P$ with real coefficients is called [i]great,[/i] if for some integer $a>1$ and for all integers $x$, there exists an integer $z$ such that $aP(x)=P(z)$. Find all [i]great[/i] polynomials. [i]Proposed by A. Golovanov[/i]

Compute the largest possible radius of a circle contained in the region defined by $|x+|y|| \le 1$ in the coordinate plane.

There is a ring outside of Saturn. In order to distinguish if the ring is actually a part of Saturn or is instead part of the satellites of Saturn, we need to know the relation between the velocity $v$ of each layer in the ring and the distance $R$ of the layer to the center of Saturn. Which of the following statements is correct? $\textbf{(A) }$ If $v \propto R$, then the layer is part of Saturn. $\textbf{(B) }$ If $v^2 \propto R$, then the layer is part of the satellites of Saturn. $\textbf{(C) }$ If $v \propto 1/R$, then the layer is part of Saturn. $\textbf{(D) }$ If $v^2 \propto 1/R$, then the layer is part of Saturn. $\textbf{(E) }$ If $v \propto R^2$, then the layer is part of the satellites of Saturn.

$ABCD$ is a rectangle. The segment $MA$ is perpendicular to plane $ABC$ . $MB= 15$ , $MC=24$ , $MD=20$. Find the length of $MA$ .

Let $ S(n)$ denote the sum of the digits of a natural number $ n$ (in base $ 10$). Prove that for every $ n$, $ S(2n) \le 2S(n) \le 10S(2n)$. Prove also that there is a positive integer $ n$ with $ S(n)\equal{}1996S(3n)$.

Three main diagonals of a hegular hexagon divide the hegular hexagon into six regular triangles. Note them $A,B,C,D,E,F$. In one part, grow one kind of plant. Also, adjacent parts must be grown with different plants. If we are given four kinds of plants, then the number of wanys to grow plants is________.

Any rational number admits a non-decimal representation unlimited decimal expansion. This development has the particularity of being periodic. Examples: $\frac{1}{7} = 0.142857142857…$ has a period $6$ while $\frac{1}{11}=0.0909090909 …$ $2$ periodic. What are the reciprocals of the prime integers with a period less than or equal to five?

The set $S\subseteq \mathbb{R}$ is given with the properties: $(a) \mathbb{Z}\subset S$, $(b) (\sqrt 2 +\sqrt 3)\in S$, $(c)$ If $x,y\in S$ then $x+y\in S$, and $(d)$ If $x,y\in S$ then $x\cdot y\in S$. Prove that $(\sqrt 2+\sqrt 3)^{-1}\in S$.

Cut an acute triangle, one of whose sides is equal to the altitude drawn, by two straight cuts, into four parts, from which you can fold a square.

Triangle $ABC$ has sides $AB = 14$, $BC = 13$, and $CA = 15$. It is inscribed in circle $\Gamma$, which has center $O$. Let $M$ be the midpoint of $AB$, let $B'$ be the point on $\Gamma$ diametrically opposite $B$, and let $X$ be the intersection of $AO$ and $MB'$. Find the length of $AX$.

The Fibonacci sequence is defined by $ F_0\equal{}0, F_1\equal{}1$ and $ F_{n\plus{}2}\equal{}F_n\plus{}F_{n\plus{}1}$ for $ n \ge 0$. Prove that: $ (a)$ The statement $ "F_{n\plus{}k}\minus{}F_n$ is divisible by $ 10$ for all $ n \in \mathbb{N}"$ is true if $ k\equal{}60$ but false for any positive integer $ k<60$. $ (b)$ The statement $ "F_{n\plus{}t}\minus{}F_n$ is divisible by $ 100$ for all $ n \in \mathbb{N}"$ is true if $ t\equal{}300$ but false for any positive integer $ t<300$.

Find all rational numbers $r$ such that the equation $rx^2 + (r + 1)x + r = 1$ has integer solutions.

Find all positive integers $k$, so that there exists a polynomial $f(x)$ with rational coefficients, such that for all sufficiently large $n$, $$f(n)=\text{lcm}(n+1, n+2, \ldots, n+k).$$

Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$.

$10$ cities are connected by one-way air routes in a way so that each city can be reached from any other by several connected flights. Let $n$ be the smallest number of flights needed for a tourist to visit every city and return to the starting city. Clearly $n$ depends on the flight schedule. Find the largest $n$ and the corresponding flight schedule.

Show that there exist infinitely many pairs $(a, b)$ of positive integers with the property that $a+b$ divides $ab+1$, $a-b$ divides $ab-1$, $b>1$ and $a>b\sqrt{3}-1$

On a connected graph $G$, one may perform the following operations: [list] [*]choose a vertice $v$, and add a vertice $v'$ such that $v'$ is connected to $v$ and all of its neighbours [*] choose a vertice $v$ with odd degree and delete it [/list] Show that for any connected graph $G$, we may perform a finite number of operations such that the resulting graph is a clique. Proposed by [i]idonthaveanaopsaccount[/i]

Let $S$ be the set of interior points of a sphere and $C$ be the set of interior points of a circle. Find, with proof, whether there exists a function $f:S\rightarrow C$ such that $d(A,B)\le d(f(A),f(B))$ for any two points $A,B\in S$ where $d(X,Y)$ denotes the distance between the points $X$ and $Y$. [i]Marius Cavachi[/i]

The excircle $\omega_B$ of triangle $ABC$ opposite $B$ touches side $AC$, rays $BA$ and $BC$ at $B_1, C_1$ and $A_1$, respectively. Point $D$ lies on major arc $A_1C_1$ of $\omega_B$. Rays $DA_1$ and $C_1B_1$ meet at $E$. Lines $AB_1$ and $BE$ meet at $F$. Prove that line $FD$ is tangent to $\omega_B$ (at $D$).