Each of the $8$ vertices of a given cube is given a value $1$ or $-1$. Each of the $6$ faces is given the value of product of its four vertices. Let $A$ be the sum of all the $14$ values. Which are the possible values of $A$?

For positive integers $n \ge 3$ and $r \ge 1$, define $$P(n, r) = (n - 2)\frac{r^2}{2} - (n - 4) \frac{r}{2}$$ We call a triple $(a, b, c)$ of natural numbers, with $a \le b \le c$, an $n$-gonal Pythagorean triple if $P(n, a)+P(n, b) = P(n, c)$. (For $n = 4$, we get the usual Pythagorean triple.) (a) Find an $n$-gonal Pythagorean triple for each $n \ge 3$. (b) Consider all triangles $ABC$ whose sides are $n$-gonal Pythagorean triples for some $n \ge 3$. Find the maximum and the minimum possible values of angle $C$.

Given a positive integer $m,$ define the polynomial $$P_m(z) = z^4-\frac{2m^2}{m^2+1} z^3+\frac{3m^2-2}{m^2+1}z^2-\frac{2m^2}{m^2+1}z+1.$$ Let $S$ be the set of roots of the polynomial $P_5(z)\cdot P_7(z)\cdot P_8(z) \cdot P_{18}(z).$ Let $w$ be the point in the complex plane which minimizes $\sum_{z \in S} |z-w|.$ The value of $\sum_{z \in S} |z-w|^2$ equals $\tfrac{a}{b}$ for relatively prime positive integers $a$ and $b.$ Compute $a+b.$

For a positive integer $n$, denote by $\tau(n)$ the number of positive integer divisors of $n$, and denote by $\phi(n)$ the number of positive integers that are less than or equal to $n$ and relatively prime to $n$. Call a positive integer $n$ $\emph{good}$ if $\varphi (n) + 4\tau (n) = n$. For example, the number $44$ is good because $\varphi (44) + 4\tau (44)= 44$. Find the sum of all good positive integers $n$.

Suppose $(a,b)$ is an ordered pair of integers such that the three numbers $a$, $b$, and $ab$ form an arithmetic progression, in that order. Find the sum of all possible values of $a$. [i]Proposed by Nathan Xiong[/i]

Let $ABC$ be a given triangle. Find the locus $\mathbf M$ of all vertices $Z$ such that triangle $XYZ$ is equilateral where $X$ is any point of segment $AB$ and $Y\neq X$ lies on ray $AC.$

Let $G$ be an arbitrarily chosen finite simple graph. We write non-negative integers on the vertices of the graph such that for each vertex $v$ in $G,$ the number written on $v$ is equal to the number of vertices adjacent to $v$ where an even number is written. Prove that the number of ways to achieve this is a power of $2$.

In an isosceles triangle $ABC~(AC=BC)$, let $O$ be its circumcenter, $D$ the midpoint of $AC$ and $E$ the centroid of $DBC$. Show that $OE$ is perpendicular to $BD$.

Let $ABC$ be a triangle, and let $D$ be the point where the incircle meets side $BC$. Let $J_b$ and $J_c$ be the incentres of the triangles $ABD$ and $ACD$, respectively. Prove that the circumcentre of the triangle $AJ_bJ_c$ lies on the angle bisector of $\angle BAC$.

Determine all three integers $(x, y, z)$ that are solutions of the system $$x + y -z = 6$$ $$x^3 + y^3 -z^3 = 414$$

Determine all integer $n \ge 2$ such that it is possible to construct an $n * n$ array where each entry is either $-1, 0, 1$ so that the sums of elements in every row and every column are distinct

Show that there exists a positive integer $ k$ such that $ k \cdot 2^{n} \plus{} 1$ is composite for all $ n \in \mathbb{N}_{0}$.

Prove that \[\frac{1}{3}n^2 + \frac{1}{2}n + \frac{1}{6} \geq (n!)^{\frac{2}{n}},\] and let $n \geq 1$ be an integer. Prove that this inequality is only possible in the case $n = 1.$

Let $\alpha + \beta +\gamma = \pi$. Prove that $\sum_{cyc}{\sin 2\alpha} = 2\cdot \left(\sum_{cyc}{\sin \alpha}\right)\cdot\left(\sum_{cyc}{\cos \alpha}\right)- 2\sum_{cyc}{\sin \alpha}$.

A circle can be drawn around the quadrilateral $ABCD$. Let straight lines $AB$ and $CD$ intersect at point $M$, and straight lines $BC$ and $AD$ intersect at point $K$. (Points $B$ and $P$ lie on segments $AM$ and $AK$, respectively.) Let $P$ be the projection of point $M$ onto straight line $AK$, $L$ be the projection of point $K$ on the straight line $AM$. Prove that the straight line $LP$ divides the diagonal $BD$ in half.

A social network has $2019$ users, some pairs of whom are friends. Whenever user $A$ is friends with user $B$, user $B$ is also friends with user $A$. Events of the following kind may happen repeatedly, one at a time: [list] [*] Three users $A$, $B$, and $C$ such that $A$ is friends with both $B$ and $C$, but $B$ and $C$ are not friends, change their friendship statuses such that $B$ and $C$ are now friends, but $A$ is no longer friends with $B$, and no longer friends with $C$. All other friendship statuses are unchanged. [/list] Initially, $1010$ users have $1009$ friends each, and $1009$ users have $1010$ friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user. [i]Proposed by Adrian Beker, Croatia[/i]

Find all odd postive integers less than $2007$ such that the sum of all of its positive divisors is odd.

Let $m$ be a positive integer. Show that there exists a positive integer $n$ such that each of the $2m+1$ integers $$ 2^{n}-m,2^{n}-(m-1),\ldots,2^{n}+(m-1),2^{n}+m$$ is positive and composite.

Let $a, b$ be two real numbers such that $$\sqrt[3]{a}- \sqrt[3]{b} = 10, ,\,\,\,\,\,\, ab = \left( \frac{8 - a - b}{6}\right)^3$$ Find $a - b$.

Consider $70$-digit numbers with the property that each of the digits $1,2,3,...,7$ appear $10$ times in the decimal expansion of $n$ (and $8,9,0$ do not appear). Show that no number of this form can divide another number of this form.

On a set $G$ we are given an operation $*: G \times G \to G$, that for every pair $(x,y)$ of elements of $G$ gives back $x*y \in G$, and for every elements $x,y,z \in G$ the equation $(x*y)*z=x*(y*z)$ holds. $G$ is partitioned into three non-empty sets $A,B$ and $C$. Can it be that for every three elements $a \in A, b \in B, c \in C$ we have $a*b \in C, b*c \in A, c*a \in B$

We have $1000$ balls in $40$ different colours, $25$ balls of each colour. Determine the smallest $n$ for which the following holds: if you place the $1000$ balls in a circle, in any arbitrary way, then there are always $n$ adjacent balls which have at least $20$ different colours.

If $f(n)=\tfrac{1}{3}n(n1)(n+2)$, then $f(r)-f(r-1)$ equals: $\textbf{(A)}\ r(r+1) \qquad \textbf{(B)}\ (r+1)(r+2) \qquad \textbf{(C)}\ \tfrac{1}{3}r(r+1) \qquad\\ \textbf{(D)}\ \tfrac{1}{3}(r+1)(r+2) \qquad \textbf{(E)}\ \tfrac{1}{3}r(r+1)(r+2) $

Given the sequence $(a_n) $ satisfies $1=a_1< a_2 < a_3< \cdots<a_n $ and there exist real number $m$ such that $$\displaystyle\sum_{i=1}^{n-1} \sqrt[3]{\frac{a_{i+1}-a_i}{(2+a_i)^4}}\leq m $$ for any positive integer $ n $ not less than 2 . Find the minimum of $m.$

Rectangle $ PQRS$ lies in a plane with $ PQ = RS = 2$ and $ QR = SP = 6$. The rectangle is rotated $ 90^\circ$ clockwise about $ R$, then rotated $ 90^\circ$ clockwise about the point that $ S$ moved to after the first rotation. What is the length of the path traveled by point $ P$? ${ \textbf{(A)}\ (2\sqrt3 + \sqrt5})\pi \qquad \textbf{(B)}\ 6\pi \qquad \textbf{(C)}\ (3 + \sqrt {10})\pi \qquad \textbf{(D)}\ (\sqrt3 + 2\sqrt5)\pi \\ \textbf{(E)}\ 2\sqrt {10}\pi$