Define a sequence $\{a_n\}_{n\geq 1}$ of real numbers by \[a_1=2,\qquad a_{n+1} = \frac{a_n^2+1}{2}, \text{ for } n\geq 1.\] Prove that \[\sum_{j=1}^{N} \frac{1}{a_j + 1} < 1\] for every natural number $N$.

Assuming $a\neq3$, $b\neq4$, and $c\neq5$, what is the value in simplest form of the following expression? $$\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}$$ $\textbf{(A) } -1 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } \frac{abc}{60} \qquad \textbf{(D) } \frac{1}{abc} - \frac{1}{60} \qquad \textbf{(E) } \frac{1}{60} - \frac{1}{abc}$

We consider dissections of regular $n$-gons into $n - 2$ triangles by $n - 3$ diagonals which do not intersect inside the $n$-gon. A [i]bicoloured triangulation[/i] is such a dissection of an $n$-gon in which each triangle is coloured black or white and any two triangles which share an edge have different colours. We call a positive integer $n \ge 4$ [i]triangulable[/i] if every regular $n$-gon has a bicoloured triangulation such that for each vertex $A$ of the $n$-gon the number of black triangles of which $A$ is a vertex is greater than the number of white triangles of which $A$ is a vertex. Find all triangulable numbers.

Quadrilateral $ABCD$ is inscribed in a circle, $M$ is the intersection point of the lines $AB$ and $CD$ and $N$ is the intersection point of the lines $BC$ and $AD$. It is known that $BM = DN$. Prove that $CM = CN$. (F Nazarov)

The train schedule in Hummut is hopelessly unreliable. Train $A$ will enter Intersection $X$ from the west at a random time between $9:00$ am and $2:30$ pm; each moment in that interval is equally likely. Train $B$ will enter the same intersection from the north at a random time between $9:30$ am and $12:30$ pm, independent of Train $A$; again, each moment in the interval is equally likely. If each train takes $45$ minutes to clear the intersection, what is the probability of a collision today?

Given a triangle $ABC$, points $D,E,F$ lie on sides $BC,CA,AB$ respectively. Moreover, the radii of incircles of $\triangle AEF, \triangle BFD, \triangle CDE$ are equal to $r$. Denote by $r_0$ and $R$ the radii of incircles of $\triangle DEF$ and $\triangle ABC$ respectively. Prove that $r+r_0=R$.

Does there exists two functions $f,g :\mathbb{R}\rightarrow \mathbb{R}$ such that: $\forall x\not =y : |f(x)-f(y)|+|g(x)-g(y)|>1$ Time allowed for this problem was 75 minutes.

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 $2018$?