Suppose that $S=\{a_{1}, \cdots, a_{r}\}$ is a set of positive integers, and let $S_{k}$ denote the set of subsets of $S$ with $k$ elements. Show that \[\text{lcm}(a_{1}, \cdots, a_{r})=\prod_{i=1}^{r}\prod_{s\in S_{i}}\gcd(s)^{\left((-1)^{i}\right)}.\]

Prove that if the sides $a, b, c$ of a non-equilateral triangle satisfy $a + b = 2c$, then the line passing through the incenter and centroid is parallel to one of the sides of the triangle.

Let $ n \in \mathbb{Z}^\plus{}$ and let $ a, b \in \mathbb{R}.$ Determine the range of $ x_0$ for which \[ \sum^n_{i\equal{}0} x_i \equal{} a \text{ and } \sum^n_{i\equal{}0} x^2_i \equal{} b,\] where $ x_0, x_1, \ldots , x_n$ are real variables.

Prove that there are infinitely many primes representable in the form $m^2+mn+n^2$ for some integers $m,n$ .

Emily’s broken clock runs backwards at five times the speed of a regular clock. Right now, it is displaying the wrong time. How many times will it display the correct time in the next 24 hours? It is an analog clock (i.e. a clock with hands), so it only displays the numerical time, not AM or PM. Emily’s clock also does not tick, but rather updates continuously.

Let $ \mathcal{M} $ a set of $ 3n\ge 3 $ planar points such that the maximum distance between two of these points is $ 1 $. Prove that: [b]a)[/b] among any four points,there are two aparted by a distance at most $ \frac{1}{\sqrt{2}} . $ [b]b)[/b] for $ n=2 $ and any $ \epsilon >0, $ it is possible that $ 12 $ or $ 15 $ of the distances between points from $ \mathcal{M} $ lie in the interval $ (1-\epsilon , 1]; $ but any $ 13 $ of the distances can´t be found all in the interval $ \left(\frac{1}{\sqrt 2} ,1\right]. $ [b]c)[/b] there exists a circle of diameter $ \sqrt{6} $ that contains $ \mathcal{M} . $ [b]d)[/b] some two points of $ \mathcal{M} $ are on a distance not exceeding $ \frac{4}{3\sqrt n-\sqrt 3} . $

In the parallelogram $ABCD$, rays are released from its vertices towards its interior. The rays coming out of the vertices $A{}$ and $D{}$ intersect at $E{}$ and the rays coming out of the vertices $B{}$ and $C{}$ at point $F{}$. It is known that $\angle BAE=\angle BCF$ and $\angle CDE = \angle CBF$. Prove that $AB \parallel EF$. [i]Proposed by V. Eisenstadt[/i]

Is there an eight-digit number without zero digits, which when divided by the first digit gives the remainder $1$, when divided by the second digit will give the remainder $2$, ..., when divided by the eighth digit will give the remainder $8$?

Given an integer $n>1$ and a $n\times n$ grid $ABCD$ containing $n^2$ unit squares, each unit square is colored by one of three colors: Black, white and gray. A coloring is called [i]symmetry[/i] if each unit square has center on diagonal $AC$ is colored by gray and every couple of unit squares which are symmetry by $AC$ should be both colred by black or white. In each gray square, they label a number $0$, in a white square, they will label a positive integer and in a black square, a negative integer. A label will be called $k$-[i]balance[/i] (with $k\in\mathbb{Z}^+$) if it satisfies the following requirements: i) Each pair of unit squares which are symmetry by $AC$ are labelled with the same integer from the closed interval $[-k,k]$ ii) If a row and a column intersectes at a square that is colored by black, then the set of positive integers on that row and the set of positive integers on that column are distinct.If a row and a column intersectes at a square that is colored by white, then the set of negative integers on that row and the set of negative integers on that column are distinct. a) For $n=5$, find the minimum value of $k$ such that there is a $k$-balance label for the following grid [asy] size(4cm); pair o = (0,0); pair y = (0,5); pair z = (5,5); pair t = (5,0); dot("$A$", y, dir(180)); dot("$B$", z); dot("$C$", t); dot("$D$", o, dir(180)); fill((0,5)--(1,5)--(1,4)--(0,4)--cycle,gray); fill((1,4)--(2,4)--(2,3)--(1,3)--cycle,gray); fill((2,3)--(3,3)--(3,2)--(2,2)--cycle,gray); fill((3,2)--(4,2)--(4,1)--(3,1)--cycle,gray); fill((4,1)--(5,1)--(5,0)--(4,0)--cycle,gray); fill((0,3)--(1,3)--(1,1)--(0,1)--cycle,black); fill((2,5)--(4,5)--(4,4)--(2,4)--cycle,black); fill((2,1)--(3,1)--(3,0)--(2,0)--cycle,black); fill((2,1)--(3,1)--(3,0)--(2,0)--cycle,black); fill((4,3)--(5,3)--(5,2)--(4,2)--cycle,black); for (int i=0; i<=5; ++i) { draw((0,i)--(5,i)^^(i,0)--(i,5)); } [/asy] b) Let $n=2017$. Find the least value of $k$ such that there is always a $k$-balance label for a symmetry coloring.

The roots of the equation $x^2+5x-7=0$ are $x_1$ and $x_2$. What is the value of $x_1^3+5x_1^2-4x_1+x_1^2x_2-4x_2$ ? $\textbf{(A)}\ -15 \qquad\textbf{(B)}\ 175+25\sqrt{53} \qquad\textbf{(C)}\ -50 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ \text{None}$

On the planet Automory, there are infinitely many inhabitants. Every Automorian loves exactly one Automorian and honours exactly one Automorian. Additionally, the following can be noticed: $\bullet$ each Automorian is loved by some Automorian; $\bullet$ if Automorian $A$ loves Automorian $B$, then also all Automorians honouring $A$ love $B$, $\bullet$if Automorian $A$ honours Automorian $B$, then also all Automorians loving $A$ honour $B$. Is it correct to claim that every Automorian honours and loves the same Automorian?

$ \forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $ a_n, b_n, c_n \in \mathbb{Z}$ such \[ \left(1 \plus{} 4 \cdot \sqrt[3]{2} \minus{} 4 \cdot \sqrt[3]{4} \right)^n \equal{} a_n \plus{} b_n \cdot \sqrt[3]{2} \plus{} c_n \cdot \sqrt[3]{4}.\] Prove that $ c_n \equal{} 0$ implies $ n \equal{} 0.$

Write out the positive integers consisting of only $1$s, $6$s, and $9$s in ascending order as in: $1,6,9,11,16,\dots$. a. Find the order of $1996$ in the sequence. b. Find the $1996$th term in the sequence.

$\log_8 2 = 0.2525$ in base $8$ (to $4$ places of decimals). Find $\log_8 4$ in base $8$ (to $4$ places of decimals).

Let $ABC$ be a triangle with $\angle C=90^\circ$, and $D$ be a point outside $ABC$, such that $\angle ADC=\angle BAC$. The segments $CD$ and $AB$ meet at point $E$. It is known that the distance from $E$ to $AC$ is equal to the circumradius of triangle $ADE$. Find the angles of triangle $ABC$.

In $\Delta ABC$, $AD \perp BC$ at $D$. $E,F$ lie on line $AB$, such that $BD=BE=BF$. Let $I,J$ be the incenter and $A$-excenter. Prove that there exist two points $P,Q$ on the circumcircle of $\Delta ABC$ , such that $PB=QC$, and $\Delta PEI \sim \Delta QFJ$ .

For Kelvin the Frog's birthday, Alex the Kat gives him one brick weighing $x$ pounds, two bricks weighing $y$ pounds, and three bricks weighing $z$ pounds, where $x,y,z$ are positive integers of Kelvin the Frog's choice. Kelvin the Frog has a balance scale. By placing some combination of bricks on the scale (possibly on both sides), he wants to be able to balance any item of weight $1,2,\ldots,N$ pounds. What is the largest $N$ for which Kelvin the Frog can succeed?

Let $ABCD$ be a quadrilateral with sides $\left|\overline{AB}\right|=2$, $\left|\overline{BC}\right|=\left|\overline{CD}\right|=4$ and $\left|\overline{DA}\right|=5$. The opposite angles, $\angle A$ and $\angle C$ are equal. The length of diagonal $BD$ equals $\textbf{(A)}~2\sqrt6$ $\textbf{(B)}~3\sqrt3$ $\textbf{(C)}~3\sqrt6$ $\textbf{(D)}~2\sqrt3$

Six segments are such that any three can form a triangle. Is it true that these segments can be used to form a tetrahedron? (S. Markelov)

Find the number of different positive integer triples $(x, y,z)$ satisfying the equations $x^2 + y -z = 100$ and $x + y^2 - z = 124$:

Let $\omega_1$ and $\omega_2$ be two circles with centers $O_1$ and $O_2$. The two circles intersect at $A$ and $B$. $\ell$ is the circles' common external tangent that is closer to $B$, and it meets $\omega_1$ at $T_1$ and $\omega_2$ at $T_2$. Let $C$ be the point on line $AB$ not equal to $A$ that is the same distance from $\ell$ as $A$ is. Given that $O_1O_2=15$, $AT_1=5$ and $AT_2=12$, find $AC^2+{T_1T_2}^2$. [i]Proposed by Zachary Perry[/i]

Given a triangle $OAB$ with the vetices $O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0)$ in the $xyz$ space. Let $V$ be the cone obtained by rotating the triangle around the $x$-axis. Find the volume of the solid obtained by rotating the cone $V$ around the $y$-axis.

In an acute triangle $ABC$ with $|AB| \not= |AC|$, let $I$ be the incenter and $O$ the circumcenter. The incircle is tangent to $\overline{BC}, \overline{CA}$ and $\overline{AB}$ in $D,E$ and $F$ respectively. Prove that if the line parallel to $EF$ passing through $I$, the line parallel to $AO$ passing through $D$ and the altitude from $A$ are concurrent, then the point of concurrence is the orthocenter of the triangle $ABC$. [i]Proposed by Petar Nizié-Nikolac[/i]

How many ordered four-tuples of integers $(a,b,c,d)$ with $0 < a < b < c < d < 500$ satisfy $a + d = b + c$ and $bc - ad = 93$?

Let $ABCD$ be a convex quadrilateral with $\angle DAB=\angle BCD=90^{\circ}$ and $\angle ABC> \angle CDA$. Let $Q$ and $R$ be points on segments $BC$ and $CD$, respectively, such that line $QR$ intersects lines $AB$ and $AD$ at points $P$ and $S$, respectively. It is given that $PQ=RS$.Let the midpoint of $BD$ be $M$ and the midpoint of $QR$ be $N$.Prove that the points $M,N,A$ and $C$ lie on a circle.