We know that there exists a positive integer with $7$ distinct digits which is multiple of each of them. What are its digits? (Paolo Leonetti)

Find all quadrilaterals which can be covered (without overlappings) with squares with side $ 1$ and equilateral triangles with side $ 1$. (Emanuele Tron)

Do there exist (not necessarily distinct) primes $p_1,..., p_k$ and $q_1,...,q_n$ such that $$p_1! \cdot \cdot \cdot p_k! \cdot 2017 = q_1! \cdot \cdot \cdot q_n! \cdot 2016 \,\,?$$ (Paolo Leonetti)

Let $p_n$ be the $n$-th prime, so that $p_1 = 2, p_2 = 3,...$ and define $$X_n = \{0\} \cup \{p_1,...,p_n\}$$for each positive integer $n$. Find all $n$ for which there exist $A,B \subseteq N$ such that$ |A|,|B| \ge 2$ and $$X_n = A + B$$, where $A + B :=\{a + b : a \in A; b \in B \}$ and $N := \{0,1, 2,...\}$. (Salvatore Tringali)

Find the smallest integer $n > 3$ such that, for each partition of $\{3, 4,..., n\}$ in two sets, at least one of these sets contains three (not necessarily distinct) numbers $ a, b, c$ for which $ab = c$. (Alberto Alfarano)

Fix reals $x, y,z > 0$ such that $x + y + z = \sqrt[5]{x} + \sqrt[5]{y} +\sqrt[5]{z}$ . Prove that $x^x y^y z^z \ge 1$. (Paolo Leonetti)

Fix $2n$ distinct reals $x_1,y_1,...,x_n,y_n$ and dene the $n\times n$ matrix where its $(i, j)$-th element is $x_i + y_j$ for all $i, j = 1,..., n$. Show that if the products of the numbers in each column is always the same, then also the products of the numbers in each row is always the same. ( Alberto Alfarano)

Fix $a_1, . . . , a_n \in (0, 1)$ and define $$f(I) = \prod_{i \in I} a_i \cdot \prod_{j \notin I} (1 - a_j)$$for each $I \subseteq \{1, . . . , n\}$. Assuming that $$\sum_{I\subseteq \{1,...,n\}, |I| odd} {f(I)} = \frac12,$$show that at least one $a_i$ has to be equal to $\frac12$. (Paolo Leonetti)

Given a triangle $ABC$, let $ P$ be the point which minimizes the sum of squares of distances from the sides of the triangle. Let $D, E, F$ the projections of $ P$ on the sides of the triangle $ABC$. Show that $P$ is the barycenter of $DEF$. (Jack D’Aurizio)

Let $(x_n)_{n\in Z}$ and $(y_n)_{n\in Z}$ be two sequences of integers such that $|x_{n+2} - x_n| \le 2$ and $x_n + x_m = y_{n^2+m^2}$ for all $n, m \in Z$. Show that the sequence of $x_n$s takes at most $6$ distinct values. (Paolo Leonetti)

Let $p$ be a sufficiently large prime. Show that the number of distinct residues taken by the set $$\{1 + \frac12 + ... + \frac{1}{n}: n = 1, 2,..., p - 1\}$$modulo $p$ has at least $\sqrt[4]{p}$ elements. (Carlo Sanna)

1 and only round Hello guys, I am back with, I hope, a good new. There will be in nearly one month the 5th edition of the "Oliforum contest". It is an individual and telematic contest, with some training problems at national/imo level. Maybe it can be useful to some forumers around here. Some details below: 1- The contest is made by a unique round, starting in the early afternoon of Tuesday 31 January and ending at 23:59 of Sunday 05 February (Rome meridian +1GTM) (in practice, 5.5 days). 2- Every problem will have some points, from 0 to 7. One additional point could be given for each clear, correct, and original solution. 3- There are no age contraints. 4- Non-olympic arguments could be used, even if it is preferred to not do it. 5- It is enough to send the solutions by mail, at the address that you can find below. Enrollment is not necessary. 6- Time of incoming solutions will be considered only for ex-aequo positions. 7- How to send solutions. - You need to send solutions to the following mail: $\text{ leonetti.paolo (AT) gmail.com }$. - Solutions need to be attached in a unique attached .pdf file with a reasonable size. - The .pdf file must be written with $\LaTeX$ or in a way that can be easily understood. - You have to rename the .pdf file with the nickname that you have here on MathLinks. - Try to be clear everytime. Previous editions: { Ed. 2008, round 1, round 2, round 3 } , { Ed. 2009, round 1, round 2 }, { Ed. 2012 round 1 }, { Ed. 2013 round 1 }. I will add below the list of problems on the 7th February, together with the list with people who sent correctly their solutions. Comments and suggestions are welcome. Cheers, Paolo Leonetti Italian version