Let $P$ be a point in the interior of $\triangle ABC$. The lengths of the sides of $\triangle ABC$ is $a,b,c$, and the distance from $P$ to the sides of $\triangle ABC$ is $p,q,r$. Show that the circumradius $R$ of $\triangle ABC$ satisfies \[\displaystyle R\le \frac{a^2+b^2+c^2}{18\sqrt[3]{pqr}}.\] When does equality hold?

Find all primes $p$ such that the number of distinct positive factors of $p^2+2543$ is less than 16.

A club provides 30 snacks to 18 members, and each member orders 3 different snacks. It is known that every snack is ordered by at least one member, and that any two members order at most one same snack. Is it possible to find 12 snacks, such that the snacks ordered by any member is not completely in these 12 snacks?

It is known that $\triangle ABC$ is an acute triangle. Let $C'$ be the foott of the perpendicular from $C$ to $AB$, and $D$, $E$ two distinct points on $CC'$. The feet of the perpendiculars from $D$ to $AC$ and $BC$ are $F$ and $G$, respectively. Show that if $DGEF$ is a parallelogram then $ABC$ is isosceles.

Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying \[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\] for any two positive integers $ m$ and $ n$. Remark. The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers: $ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$). Proposed by Mohsen Jamali, Iran

Given a triangle $ABC$, $A_1$ divides the length of the path $CAB$ into two equal parts, and define $B_1$ and $C_1$ analogously. Let $l_A$, $l_B$, $l_C$ be the lines passing through $A_1$, $B_1$ and $C_1$ and being parallel to the bisectors of $\angle A$, $\angle B$, and $\angle C$. Show that $l_A$, $l_B$, $l_C$ are concurrent.

Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals. Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded? Proposed by Mihai Bălună, Romania

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

The set $\{1,2,\dots\>,n\}$ is called $P$. The function $f: P \to \{1,2,\dots\>,m\}$ satisfies \[f(A\cap B)=\min (f(A), f(B)).\] What is the relationship between the number of possible functions $f$ with the sum $\displaystyle \sum_{j=1}^m j^n$? There is a nice and easy solution to this. Too bad I did not think of it...

Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,A_2,\dots,A_n$ the sequence of arithmetic means defined by \[ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. \] Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality \[ n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 \] and establish the cases of equality. Proposed by Finbarr Holland, Ireland

Given a triangle $ABC$, we construct a circle $\Gamma$ through $B,C$ with center $O$. $\Gamma$ intersects $AC, AB$ at points $D$, $E$, respectively($D$, $E$ are distinct from $B$ and $C$). Let the intersection of $BD$ and $CE$ be $F$. Extend $OF$ so that it intersects the circumcircle of $\triangle ABC$ at $P$. Show that the incenters of triangles $PBD$ and $PCE$ coincide.

Given an integer ${n>1}$, denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$. For each ${n>1}$, find the remainder of $P_{n}$ on division by $n$. Proposed by John Murray, Ireland