In isosceles right-angled triangle $ABC$, $CA = CB = 1$. $P$ is an arbitrary point on the sides of $ABC$. Find the maximum of $PA \cdot PB \cdot PC$.

Let $K$ be a finite field and $f:K\to K^*$. Prove that there is a reducible polynomial $P\in K[X]$ s.t. $P(x)=f(x),\forall x\in K$. [i]Marian Andronache[/i]

Austen has a regular icosahedron ($20$-sided polyhedron with all triangular faces). He randomly chooses $3$ distinct points among the vertices and constructs the circle through these three points. The expected value of the total number of the icosahedron’s vertices that lie on this circle can be written as $m/n$ for relatively prime positive integers $m$ and $n.$ Find $m + n.$

Let $\triangle ABC$ be a triangle with $\angle A \neq 60^\circ$. Let $I_B, I_C$ be the $B, C$-excenters of triangle $ABC$, let $B^\prime$ be the reflection of $B$ with respect to $AC$, and let $C^\prime$ be the reflection of $C$ with respect to $AB$. Let $P$ be the intersection of $I_C B^\prime$ and $I_B C^\prime$. Denote by $P_A, P_B, P_C$ the reflections of the point $P$ with respect to $BC, CA, AB$. Show that the three lines $A P_A, B P_B, C P_C$ meet at a single point.

Show that, for every integer $r \ge 2$, there exists an $r$-chromatic simple graph (no loops, nor multiple edges) which has no cycle of less than $6$ edges

Let $TABCD$ be a pyramid with top vertex $T$, such that its base $ABCD$ is a square of side length 4. It is given that, among the triangles $TAB$, $TBC$, $TCD$ and $TDA$, one can find an isosceles triangle and a right-angled triangle. Find all possible values for the volume of the pyramid.

In $\triangle AEF$, let $B$ and $D$ be on segments $AE$ and $AF$ respectively, and let $ED$ and $FB$ intersect at $C$. Define $K,L,M,N$ on segments $AB,BC,CD,DA$ such that $\frac{AK}{KB}=\frac{AD}{BC}$ and its cyclic equivalents. Let the incircle of $\triangle AEF$ touch $AE,AF$ at $S,T$ respectively; let the incircle of $\triangle CEF$ touch $CE,CF$ at $U,V$ respectively. Prove that $K,L,M,N$ concyclic implies $S,T,U,V$ concyclic.

Let $a, b, c$ be the lengths of sides of a triangle. Prove the inequality $$(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.$$

Let $n>1$ be an integer. There are $n$ orangutoads, conveniently numbered $1,2,\dots{},n$, each sitting at an integer position on the number line. They take turns moving in the order $1,2,\dots{},n$, and then going back to $1$ to start the process over; they stop if any orangutoad is ever unable to move. To move, an orangutoad chooses another orangutoad who is at least $2$ units away from her towards them by a a distance of $1$ unit. (Multiple orangutoads can be at the same position.) Show that eventually some orangutoad will be unable to move.

Let $c$ be a real number. If the inequality $$f(c)\cdot f(-c)\ge f(a)$$ holds for all $f(x)=x^2-2ax+b$ where $a$ and $b$ are arbitrary real numbers, find all possible values of $c$.

Let $O$ and $\Omega$ denote the circumcenter and circumcircle, respectively, of scalene triangle $\triangle ABC$. Furthermore, let $M$ be the midpoint of side $BC$. The tangent to $\Omega$ at $A$ intersects $BC$ and $OM$ at points $X$ and $Y$, respectively. If the circumcircle of triangle $\triangle OXY$ intersects $\Omega$ at two distinct points $P$ and $Q$, prove that $PQ$ bisects $\overline{AM}$. [i]Proposed by Andrew Wen[/i]

Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.

Find all integer pairs $(a,b)$ such that $a\cdot b$ divides $a^2+b^2+3$.

Let $n$ be a positive integer such that $2n+1$ and $3n+1$ are both perfect squares. Show that $5n+3$ is a composite number.

Find pairs of positive integers $(n, k)$ satisfying \[(n + 1)^k - 1 = n!\]

Let $f : Z_{\ge 0} \to Z_{\ge 0}$ be a function which satisfies for all integer $n \ge 0$: (a) $f(2n + 1)^2 - f(2n)^2 = 6f(n) + 1$, (b) $f(2n) \ge f(n)$ where $Z_{\ge 0}$ is the set of nonnegative integers. Solve the equation $f(n) = 1000$

There are two identical clocks on the wall, one showing the current Moscow time and the other showing current local time. The minimum distance between the ends of their hour hands equals $m$ and the maximum distance equals $M$. Find the distance between the centres of the clocks. (S Fomin, Leningrad)

Let $d$ be a positive divisor of the number $A = 1024^{1024}+5$ and suppose that $d$ can be expressed as $d = 2x^2+2xy+3y^2$ for some integers $x,y$. Which remainder we can have when divide $d$ by $20$ ?

We consider the division of a chess board $8 \times 8$ in p disjoint rectangles which satisfy the conditions: [b]a)[/b] every rectangle is formed from a number of full squares (not partial) from the 64 and the number of white squares is equal to the number of black squares. [b]b)[/b] the numbers $\ a_{1}, \ldots, a_{p}$ of white squares from $p$ rectangles satisfy $a_1, , \ldots, a_p.$ Find the greatest value of $p$ for which there exists such a division and then for that value of $p,$ all the sequences $a_{1}, \ldots, a_{p}$ for which we can have such a division. [color=#008000]Moderator says: see [url]https://artofproblemsolving.com/community/c6h58591[/url][/color]

In a nonisosceles triangle $ABC$, point $I$ is its incentre and $\Gamma$ is its circumcircle. Points $E$ and $D$ lie on $\Gamma$ and the circumcircle of triangle $BIC$ respectively such that $AE$ and $ID$ are both perpendicular to $BC$. Let $M$ be the midpoint of $BC$, $N$ be the midpoint of arc $BC$ on $\Gamma$ containing $A$, $F$ is the point of tangency of the $A-$excircle on $BC$, and $G$ is the intersection of line $DE$ with $\Gamma$. Prove that lines $GM$ and $NF$ intersect at a point located on $\Gamma$. (Possibly proposed by Farras Faddila)

Let $ABCD$ be a trapezoid, with $AD \parallel BC$, let $M$ be the midpoint of $AD$, and let $C_1$ be symmetric point to $C$ with respect to line $BD$. Segment $BM$ meets diagonal $AC$ at point $K$, and ray $C_1K$ meets line $BD$ at point $H$. Prove that $\angle{AHD}$ is a right angle. [i]Proposed by Giorgi Arabidze, Georgia[/i]

The triangle $ABC$ is right in $A$ and $R$ is the midpoint of the hypotenuse $BC$ . On the major leg $AB$ the point $P$ is marked such that $CP = BP$ and on the segment $BP$ the point $Q$ is marked such that the triangle $PQR$ is equilateral. If the area of triangle $ABC$ is $27$, calculate the area of triangle $PQR$ .

A sequence of integers $a_1,a_2,\ldots $ is such that $a_1=1,a_2=2$ and for $n\ge 1$, \[a_{n+2}=\left\{\begin{array}{cl}5a_{n+1}-3a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is even},\\ a_{n+1}-a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is odd},\end{array}\right. \] Prove that $a_n\not= 0$ for all $n$.

In a triangle $ABC$, let $a,b,c$ be its sides and $m,n,p$ be the corresponding medians. For every $\alpha>0$, let $\lambda(\alpha)$ be the real number such that $$a^\alpha+b^\alpha+c^\alpha=\lambda(\alpha)^\alpha\left(m^\alpha+n^\alpha+p^\alpha\right)^\alpha.$$ (a) Compute $\lambda(2)$. (b) Find the limit of $\lambda(\alpha)$ as $\alpha$ approaches $0$. (c) For which triangles $ABC$ is $\lambda(\alpha)$ independent of $\alpha$?

In a certain state there were 2004 cities connected by roads so that from any city one could get to any other. It is known that when it is prohibited to travel on any of the roads, the least of them any city could be reached to any other. The Minister of Transport and the Minister of Internal Affairs take turns introducing restrictions on the roads while there is possibility, one-way traffic (on one road per turn), and minister, after whose move it became impossible to leave any city to reach any other, immediately resigns. First the Minister of Transport walks. Can any of the ministers force the resignation of another, regardless of his performance? [hide=original wording]В некотором государстве было 2004 города, соединенных дорогами так, что из любого города можно было добраться до любого другого. Известно, что при запрещенном проезде по любой из дорог, по-прежнему из любого города можно было добраться до любого другого. Министр транспорта и министр внутренних дел по очереди вводят на дорогах, пока есть возможность, одностороннее движение (на одной дороге за ход), причем министр, после хода которого из какого-либо города стало невозможно добраться до какого-либо другого, немедленно уходит в отставку. Первым ходит министр транспорта. Может ли кто-либо из министров добиться отставки другого независимо от его игры?[/hide]