Two circles are given in the plane, $\Omega$ and inside it $\omega$. The center of $\omega$ is $I$. $P$ is a point moving on $\Omega$. The second intersection of the tangents from $P$ to $\omega$ and circle $\Omega$ are $Q$ and $R.$ The second intersection of circle $IQR$ and lines $PI$, $PQ$ and $PR$ are $J$, $S$ and $T,$ respectively. The reflection of point $J$ across line $ST$ is $K.$ Prove that lines $PK$ are concurrent.

Let $ A,B$ denote two distinct fixed points in space. Let $ X, P$ denote variable points (in space), while $ K,N, n$ denote positive integers. Call $ (X,K,N,P)$ admissible if \[ (N \minus{} K) \cdot PA \plus{} K \cdot PB \geq N \cdot PX.\] Call $ (X,K,N)$ admissible if $ (X,K,N,P)$ is admissible for all choices of $ P.$ Call $ (X,N)$ admissible if $ (X,K,N)$ is admissible for some choice of $ K$ in the interval $ 0 < K < N.$ Finally, call $ X$ admissible if $ (X,N)$ is admissible for some choice of $ N, (N > 1).$ Determine: [b](a)[/b] the set of admissible $ X;$ [b](b)[/b] the set of $ X$ for which $ (X, 1989)$ is admissible but not $ (X, n), n < 1989.$

Determine the remainder when $$\sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor$$ is divided by 100, where $\lfloor x \rfloor$ denotes the largest integer not greater than $x$.

If $3x^3-9x^2+kx-12$ is divisible by $x-3$, then it is also divisible by: ${{ \textbf{(A)}\ 3x^2-x+4 \qquad\textbf{(B)}\ 3x^2-4 \qquad\textbf{(C)}\ 3x^2+4 \qquad\textbf{(D)}\ 3x-4 }\qquad\textbf{(E)}\ 3x+4 } $

(a) On each square of a squared sheet of paper of size $20 \times 20$ there is a soldier. Vanya chooses a number $d$ and Petya moves the soldiers to new squares in such a way that each soldier is moved through a distance of at least $d$ (the distance being measured between the centres of the initial and the new squares) and each square is occupied by exactly one soldier. For which $d$ is this possible? (Give the maximum possible $d$, prove that it is possible to move the soldiers through distances not less than $d$ and prove that there is no greater $d$ for which this procedure may be carried out.) (b) Answer the same question as (a), but with a sheet of size $21 \times 21$. (SS Krotov, Moscow)

If $ x$, $ y$, $ z$, and $ n$ are natural numbers, and $ n\geq z$ then prove that the relation $ x^n \plus{} y^n \equal{} z^n$ does not hold.

A positive integer is called [i]digit-reduced[/i] if at most nine different digits occur in its decimal representation (leading $0$s are omitted.) Let $M$ be a finite set of [i]digit-reduced[/i] numbers. Show that the sum of the reciprocals of the elements in $M$ is less than $180$.

Let $C_1$ and $C_2$ be two concentric circles with $C_1$ inside $C_2$. Let $P_1$ and $P_2$ be two points on $C_1$ that are not diametrically opposite. Extend the segment $P_1P_2$ past $P_2$ until it meets the circle $C_2$ in $Q_2$. The tangent to $C_2$ at $Q_2$ and the tangent to $C_1$ at $P_1$ meet in a point $X$. Draw from X the second tangent to $C_2$ which meets $C_2$ at the point $Q_1$. Show that $P_1X$ bisects angle $Q_1P_1Q_2$.

Let $o$, $r$, $g$, $t$, $n$, $i$, $z$, $e$, and $d$ be positive reals. Show that \[ \sqrt{(d+o+t+t+e+d)(o+r+z+i+n+g)} > \sqrt{ti} + \sqrt{go} + \sqrt[6]{orz}. \] when $d^2e \geq \tfrac{2}{1434}$. Proposed by [i]David Fox[/i]

Determine the number of ten-digit positive integers with the following properties: $\bullet$ Each of the digits $0, 1, 2, . . . , 8$ and $9$ is contained exactly once. $\bullet$ Each digit, except $9$, has a neighbouring digit that is larger than it. (Note. For example, in the number $1230$, the digits $1$ and $3$ are the neighbouring digits of $2$ while $2$ and $0$ are the neighbouring digits of $3$. The digits $1$ and $0$ have only one neighbouring digit.) [i](Karl Czakler)[/i]

Let $r > n$ be positive integers. A "good word" is an $n$-tuple $\langle a_1,\dots, a_n \rangle$ of distinct positive integers between 1 and $r$. A "play" consist of changing a integer $a_i$ of a good word, in such a way that the resulting word is still a good word. The distance between two good words $A= \langle a_1,\dots, a_n \rangle$ and $B = \langle b_1,\dots, b_n \rangle$ is the minimun number of plays needed to obtain B from A. Find the maximun posible distance between two good words.

Let $a, b, c$ and $d$ be positive real numbers. Prove that $$\frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a} +\frac{d - a}{a + b } \ge 0 $$

A positive integer $m$ has the property that $m^2$ is expressible in the form $4n^2-5n+16$ where $n$ is an integer (of any sign). Find the maximum value of $|m-n|.$

Find the number of ordered pairs $(a, b)$ of positive integers that are solutions of the following equation: \[a^2 + b^2 = ab(a+b).\]

Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.

A positive integer $n$ satisfies the equation $(n+1)! + (n+2)! = n! \cdot 440$. What is the sum of the digits of $n$? $\textbf{(A) }2\qquad\textbf{(B) }5\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }15$

The range of function $y=x+\sqrt{x^2-3x+2}(x\in\mathbb{R})$ is________.

In a mathematical competition, in which $6$ problems were posed to the participants, every two of these problems were solved by more than $\frac 25$ of the contestants. Moreover, no contestant solved all the $6$ problems. Show that there are at least $2$ contestants who solved exactly $5$ problems each. [i]Radu Gologan and Dan Schwartz[/i]

Prove that for each positive integer n,the equation $x^{2}+15y^{2}=4^{n}$ has at least $n$ integer solution $(x,y)$

In a football league, whenever a player is transferred from a team $X$ with $x$ players to a team $Y$ with $y$ players, the federation is paid $y-x$ billions liras by $Y$ if $y \geq x$, while the federation pays $x-y$ billions liras to $X$ if $x > y$. A player is allowed to change as many teams as he wishes during a season. Suppose that a season started with $18$ teams of $20$ players each. At the end of the season, $12$ of the teams turn out to have again $20$ players, while the remaining $6$ teams end up with $16,16, 21, 22, 22, 23$ players, respectively. What is the maximal amount the federation may have won during the season?

$a_1,a_2,...,a_9$ are nonnegative reals with sum $1$. Define $S$ and $T$ as below: $$S=\min\{a_1,a_2\}+2\min\{a_2,a_3\}+...+9\min\{a_9,a_1\}$$ $$T=\max\{a_1,a_2\}+2\max\{a_2,a_3\}+...+9\max\{a_9,a_1\}$$ When $S$ reaches its maximum, find all possible values of $T$.

For $\triangle ABC$ and its circumcircle $\omega$, draw the tangents at $B,C$ to $\omega$ meeting at $D$. Let the line $AD$ meet the circle with center $D$ and radius $DB$ at $E$ inside $\triangle ABC$. Let $F$ be the point on the extension of $EB$ and $G$ be the point on the segment $EC$ such that $\angle AFB=\angle AGE=\angle A$. Prove that the tangent at $A$ to the circumcircle of $\triangle AFG$ is parallel to $BC$. [i]Proposed by 61plus[/i]

A triangle has no angle greater than $90^o$. Show that the sum of the medians is greater than four times the circumradius.

Given a complex number $z$ satisfies $\operatorname{Im}(z)=z^2-z$, find all possible values of $|z|$.

Given five $n$-digit binary numbers. For each two numbers their digits coincide exactly on $m$ places. There is no place with the common digit for all the five numbers. Prove that $$2/5 \le m/n \le 3/5$$