For $m\in \mathbb{N}$ define $m?$ be the product of first $m$ primes. Determine if there exists positive integers $m,n$ with the following property : \[ m?=n(n+1)(n+2)(n+3) \] Proposed by Matko Ljulj

Let $P$ be a point inside a triangle $ABC$. A line through $P$ parallel to $AB$ meets $BC$ and $CA$ at points $L$ and $F$, respectively. A line through $P$ parallel to $BC$ meets $CA$ and $BA$ at points $M$ and $D$ respectively, and a line through $P$ parallel to $CA$ meets $AB$ and $BC$ at points $N$ and $E$ respectively. Prove $$ [PDBL] \cdot [PECM] \cdot [PFAN]=8\cdot [PFM] \cdot [PEL] \cdot [PDN] $$ Proposed by Steve Dinh

We are given a combination lock consisting of $6$ rotating discs. Each disc consists of digits $0, 1, 2,\ldots , 9$ in that order (after digit $9$ comes $0$). Lock is opened by exactly one combination. A move consists of turning one of the discs one digit in any direction and the lock opens instantly if the current combination is correct. Discs are initially put in the position $000000$, and we know that this combination is not correct. a) What is the least number of moves necessary to ensure that we have found the correct combination? b) What is the least number of moves necessary to ensure that we have found the correct combination, if we know that none of the combinations $000000, 111111, 222222, \ldots , 999999$ is correct? Proposed by Ognjen Stipetić and Grgur Valentić

Let $a,b,c$ be positive reals satisfying : \[ \frac{a}{1+b+c}+\frac{b}{1+c+a}+\frac{c}{1+a+b}\ge \frac{ab}{1+a+b}+\frac{bc}{1+b+c}+\frac{ca}{1+c+a} \] Then prove that : \[ \frac{a^2+b^2+c^2}{ab+bc+ca}+a+b+c+2\ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}) \] Proposed by Dimitar Trenevski

In each field of a table there is a real number. We call such $n \times n$ table silly if each entry equals the product of all the numbers in the neighbouring fields. a) Find all $2 \times 2$ silly tables. b) Find all $3 \times 3$ silly tables.

Palindrome is a sequence of digits which doesn't change if we reverse the order of its digits. Prove that a sequence $(x_n)^{\infty}_{n=0}$ defined as $x_n=2013+317n$ contains infinitely many numbers with their decimal expansions being palindromes.

We call a sequence of $n$ digits one or zero a code. Subsequence of a code is a palindrome if it is the same after we reverse the order of its digits. A palindrome is called nice if its digits occur consecutively in the code. (Code $(1101)$ contains $10$ palindromes, of which $6$ are nice.) a) What is the least number of palindromes in a code? b) What is the least number of nice palindromes in a code?

Given a triangle $ABC$ let $D$, $E$, $F$ be orthogonal projections from $A$, $B$, $C$ to the opposite sides respectively. Let $X$, $Y$, $Z$ denote midpoints of $AD$, $BE$, $CF$ respectively. Prove that perpendiculars from $D$ to $YZ$, from $E$ to $XZ$ and from $F$ to $XY$ are concurrent.