In an acute-angled triangle $ABC$, the points $M$ and $N$ are the midpoints of the sides $AB$ and $AC$, respectively. For an arbitrary point $S$ lying on the side of $BC$ prove that the condition holds $(MB- MS)(NC-NS) \le 0$

Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.

Given a tetrahedron $ABCD$ with edges $|AD|=|BC|= a$, $|AC|=|BD|=b$, $AB=c$ and $|CD| = d$. Determine the smallest value of the sum $|AX|+|BX|+|CX|+|DX|$, where $X$ is any point in space.

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $E$ be the intersection of $BH$ and $AC$ and let $M$ and $N$ be the midpoints of $HB$ and $HO$, respectively. Let $I$ be the incenter of $AEM$ and $J$ be the intersection of $ME$ and $AI$. If $AO=20$, $AN=17$, and $\angle{ANM}=90^{\circ}$, then $\frac{AI}{AJ}=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Tristan Shin[/i]

Two circles $K_{1}$ and $K_{2}$, centered at $O_{1}$ and $O_{2}$ with radii $1$ and $\sqrt{2}$ respectively, intersect at $A$ and $B$. Let $C$ be a point on $K_{2}$ such that the midpoint of $AC$ lies on $K_{1}$. Find the length of the segment $AC$ if $O_{1}O_{2}=2$

$(HUN 3)$ In the plane $4000$ points are given such that each line passes through at most $2$ of these points. Prove that there exist $1000$ disjoint quadrilaterals in the plane with vertices at these points.

Let there be an acute triangle $ABC$, such that $\angle ABC < \angle ACB$. Let the perpendicular from $A$ to $BC$ hit the circumcircle of $ABC$ at $D$, and let $M$ be the midpoint of $AD$. The tangent to the circumcircle of $ABC$ at $A$ hits the perpendicular bisector of $AD$ at $E$, and the circumcircle of $MDE$ hits the circumcircle of $ABC$ at $F$. Let $G$ be the foot of the perpendicular from $A$ to $BD$, and $N$ be the midpoint of $AG$. Prove that $B, N, F$ are collinear.

Let $a_1 = 1$ and $a_{n+1} = a_n \cdot p_n$ for $n \geq 1$ where $p_n$ is the $n$th prime number, starting with $p_1 = 2$. Let $\tau(x)$ be equal to the number of divisors of $x$. Find the remainder when $$\sum_{n=1}^{2020} \sum_{d \mid a_n} \tau (d)$$ is divided by 91 for positive integers $d$. Recall that $d|a_n$ denotes that $d$ divides $a_n$. [i]Proposed by Minseok Eli Park (wolfpack)[/i]

Find the sum of all integers $x$, $x \ge 3$, such that $201020112012_x$ (that is, $201020112012$ interpreted as a base $x$ number) is divisible by $x-1$

Prove that there exists such a number $A$ that you can inscribe $1978$ different size squares in the plot of the function $y = A sin(x)$. (The square is inscribed if all its vertices belong to the plot.)

Show that the following implication holds for any two complex numbers $x$ and $y$: if $x+y$, $x^2+y^2$, $x^3+y^3$, $x^4+y^4\in\mathbb Z$, then $x^n+y^n\in\mathbb Z$ for all natural n.

Parallelogram $JHMT$ satisfies $JH=11$ and $HM=6,$ and point $P$ lies on $\overline{MT}$ such that $JP$ is an altitude of $JHMT.$ The circumcircles of $\triangle{HMP}$ and $\triangle{JMT}$ intersect at the point $Q\neq M.$ Let $A$ be the point lying on $\overline{JH}$ and the circumcircle of $\triangle{JMT}.$ If $MQ=10,$ then the perimeter of $\triangle{JAM}$ can be expressed in the form $\sqrt{a}+\tfrac{b}{c},$ where $a, \ b,$ and $c$ are positive integers, $a$ is not divisible by the square of any prime, and $b$ and $c$ are relatively prime. Find $a+b+c.$

Suppose that we have two natural numbers $x , y \le 100!$ with undetermined values. Prove that there exist natural numbers $m , n$ such that values of $x , y$ get uniquely determined according to value of $\varphi(d(my))+d(\varphi(nx))$. ( for each natural number $n$ , $d(n)$ is number of its positive divisors and $\varphi(n)$ is the number of the numbers less that $n$ which are relatively prime to $n$. ) [i]Proposed by Mehran Talaei[/i]

Attached to each of a number of objects is a tag which states the correct mass of the object. The tags have fallen off and have been replaced on the objects at random. We wish to determine if by chance all tags are in fact correct. We may use exactly once a horizontal lever which is supported at its middle. The objects can be hung from the lever at any point on either side of the support. The lever either stays horizontal or tilts to one side. Is this task always possible?

What's the area defined by equation $|x-1|+|y-1|=1$? $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}\pi\qquad\text{(D)}4$

Suppose that $V$ is a finite dimensional vector space over the real numbers equipped with an inner product and $S:V\times V \longrightarrow \mathbb R$ is a skew symmetric function that is linear for each variable when others are kept fixed. Prove there exists a linear transformation $T:V \longrightarrow V$ such that $\forall u,v \in V: S(u,v)=<u,T(v)>$. We know that there always exists $v\in V$ such that $W=<v,T(v)>$ is invariant under $T$. (it means $T(W)\subseteq W$). Prove that if $W$ is invariant under $T$ then the following subspace is also invariant under $T$: $W^{\perp}=\{v\in V:\forall u\in W <v,u>=0\}$. Prove that if dimension of $V$ is more than $3$, then there exist a two dimensional subspace $W$ of $V$ such that the volume defined on it by function $S$ is zero!!!! (This is the way that we can define a two dimensional volume for each subspace $V$. This can be done for volumes of higher dimensions.)

Find all positive integers $ a$ and $ b$ such that $ \frac{a^{4}\plus{}a^{3}\plus{}1}{a^{2}b^{2}\plus{}ab^{2}\plus{}1}$ is an integer.

There are $2018$ boxes $C_1$, $C_2$, $C_3$,..,$C_{2018}$. The $n$-th box $C_n$ contains $n$ balls. A move consists of the following steps: a) Choose an integer $k$ greater than $1$ and choose $m$ a multiple of $k$. b) Take a ball from each of the consecutive boxes $C_{m-1}$, $C_m$, $C_{m+1}$ and move the $3$ balls to the box $C_{m+k}$. With these movements, what is the largest number of balls we can get in the box $2018$?

Let be a function satisfying [url=http://mathworld.wolfram.com/CauchyFunctionalEquation.html]Cauchy's functional equation,[/url] and having the property that it's monotonic on a real interval. Prove that this function is globally monotonic. [i]Florian Dumitrel[/i]

On day $1$ of the new year, John Adams and Samuel Adams each drink one gallon of tea. For each positive integer $n$, on the $n$th day of the year, John drinks $n$ gallons of tea and Samuel drinks $n^2$ gallons of tea. After how many days does the combined tea intake of John and Samuel that year first exceed $900$ gallons? [i]Proposed by Aidan Duncan[/i] [hide=Solution] [i]Solution. [/i] $\boxed{13}$ The total amount that John and Samuel have drank by day $n$ is $$\dfrac{n(n+1)(2n+1)}{6}+\dfrac{n(n+1)}{2}=\dfrac{n(n+1)(n+2)}{3}.$$ Now, note that ourdesired number of days should be a bit below $\sqrt[3]{2700}$. Testing a few values gives $\boxed{13}$ as our answer. [/hide]

Two circles $c_1$ and $c_2$ with centres $O_1$ and $O_2$, respectively, are touching externally at $P$. On their common tangent at $P$, point $A$ is chosen, rays drawn from which touch the circles $c_1$ and $c_2$ at points $P_1$ and $P_2$ both different from $P$. It is known that $\angle P_1AP_2 = 120^o$ and angles $P_1AP$ and $P_2AP$ are both acute. Rays $AP_1$ and $AP_2$ intersect line $O_1O_2$ at points $G_1$ and $G_2$, respectively. The second intersection between ray $AO_1$ and $c_1$ is $H_1$, the second intersection between ray $AO_2$ and $c_2$ is $H_2$. Lines $G_1H_1$ and $AP$ intersect at $K$. Prove that if $G_1K$ is a tangent to circle $c_1$, then line $G_2A$ is tangent to circle $c_2$ with tangency point $H_2$.

An ant starts at corner $A$ of a square room $ABCD$ with side length $2\sqrt{2}$. In the middle of the room, there is a circular pillar of radius $1$ centered at the center of $ABCD$. What is the minimum distance it has to travel to get to corner $C$?

Given a sequence $\{ a_{n}\}$ such that $a_{n+1}=a_{n}+\frac{1}{2006}a_{n}^{2}$ , $n \in N$, $a_{0}=\frac{1}{2}$. Prove that $1-\frac{1}{2008}< a_{2006}< 1$.

For integers $a$ and $b$, $a + b$ is a root of $x^2 + ax + b = 0$. Compute the smallest possible value of $ab$.

1) Let $a,b,c\ge0$ and $ab+bc+ca=2.$ Prove that \[\frac{ab}{c+1}+\frac{bc}{a+1}+\frac{ca}{b+1}+2(a+b+c)\ge6.\] 2) Let $a,b,c\ge0$ and $ab+bc+ca=3.$ Prove that \[\frac{ab}{c+1}+\frac{bc}{a+1}+\frac{ca}{b+1}\ge\frac{3}{2}.\]