On a highway, a pedestrian and a cyclist were going in the same direction, while a cart and a car were coming from the opposite direction. All were travelling at different constant speeds. The cyclist caught up with the pedestrian at $10$ o'clock. After a time interval, she met the cart, and after another time interval equal to the first, she met the car. After a third time interval, the car met the pedestrian, and after another time interval equal to the third, the car caught up with the cart. If the pedestrian met the car at $11$ o'clock, when did he meet the cart?

Let $a$ be a positive number. Assume that the parameterized curve $C: \ x=t+e^{at},\ y=-t+e^{at}\ (-\infty <t< \infty)$ is touched to $x$ axis. (1) Find the value of $a.$ (2) Find the area of the part which is surrounded by two straight lines $y=0, y=x$ and the curve $C.$

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.