Find all natural numbers which can be written in the form $\frac{(a+b+c)^2}{abc}$ , where $a,b,c \in N$.

In the game of Yahtzee , players have to achieve various combinations of values with $5$ dice. In a round, a player can roll the dice three times. At the second and third rolls, he can choose which dice to re-roll and which to keep. What is the probability that a player achieves at least four $6$’s in a round, given that he plays with the optimal strategy to maximise this probability? Writing the answer as $p/q$ where $p$ and $q$ are coprime, you should submit the sum of all prime factors of $p$, counted with multiplicity. So for example if you obtained $\frac{p}{q} = \frac{3^4 \cdot 11}{ 2^5 \cdot 5}$ then the submitted answer should be $4 \cdot 3 + 11 = 23$.

Given triangle $ABC$ and its incircle $\omega$ prove you can use just a ruler and drawing at most 8 lines to construct points$A',B',C'$ on $\omega$ such that $A,B',C'$ and $B,C',A'$ and $C,A',B'$ are collinear.

For a 3-digit-number $n=\overline{abc}$, if $a,b,c$ can be three sides of an isosceles triangle (regular triangle included), then the number of such numbers is $\text{(A)}45\qquad\text{(B)}81\qquad\text{(C)}165\qquad\text{(D)}216$

Let $X = \{(x,y) \in \mathbb{R}^2 | y \geq 0, x^2+y^2 = 1\} \cup \{(x,0),-1\leq x\leq 1\} $ be the edge of the closed semicircle with radius 1. a) Let $n>1$ be an integer and $P_1,P_2,\dots,P_n \in X$. Show that there exists a permutation $\sigma \colon \{1,2,\dots,n\}\to \{1,2,\dots,n\}$ such that \[\sum_{j=1}^{n}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2\leq 8\]. Where $\sigma(n+1) = \sigma(1)$. b) Find all sets $\{P_1,P_2,\dots,P_n \} \subset X$ such that for any permutation $\sigma \colon \{1,2,\dots,n\}\to \{1,2,\dots,n\}$, \[\sum_{j=1}^{n}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2 \geq 8\]. Where $\sigma(n+1) = \sigma(1)$.

Let $ABCDE$ be a convex pentagon such that: $\angle ABC=90,\angle BCD=135,\angle DEA=60$ and $AB=BC=CD=DE$. Find angle $\angle DAE$.

In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.

(a) Prove that for any triangle $H_1$, there exists a triangle $H_2$ whose side lengths are equal to the lengths of the medians of $H_1$. (b) If $H_3$ is the triangle whose side lengths are equal to the lengths of the medians of $H_2$, prove that $H_1$ and $H_3$ are similar.

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.

Prove that there are only two sets of consecutive positive integers that satisfy that the sum of its elements is equal to $100$.

Gleb picked positive integers $N$ and $a$ ($a < N$). He wrote the number $a$ on a blackboard. Then each turn he did the following: he took the last number on the blackboard, divided the number $N$ by this last number with remainder and wrote the remainder onto the board. When he wrote the number $0$ onto the board, he stopped. Could he pick $N$ and $a$ such that the sum of the numbers on the blackboard would become greater than $100N$ ? Ivan Mitrofanov

Let be given a triangle $ABC$. Equilateral triangles $BCA_1$ and $BCA_2$ are drawn so that $A$ and $A_1$ are on one side of $BC$, whereas $A_2$ is on the other side. Points $B_1,B_2,C_1,C_2$ are analogously defined. Prove that \[S_{ABC} + S_{A_1B_1C_1} = S_{A_2B_2C_2}.\]

Show that a given natural number $A$ is the square of a natural number if and only if for any natural number $n$, at least one of the differences \[(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A, \cdots , (A + n)^2 - A\] is divisible by $n$.

In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$

Let $n \geq 2$ be a positive integer. Determine the number of $n$-tuples $(x_1, x_2, \ldots, x_n)$ such that $x_k \in \{0, 1, 2\}$ for $1 \leq k \leq n$ and $\sum_{k = 1}^n x_k - \prod_{k = 1}^n x_k$ is divisible by $3$.

Let $ABC$ be an acute triangle and let $X, Y , Z$ denote the midpoints of the shorter arcs $BC, CA, AB$ of its circumcircle, respectively. Let $M$ be an arbitrary point on side $BC$. The line through $M$, parallel to the inner angular bisector of $\angle CBA$ meets the outer angular bisector of $\angle BCA$ at point $N$. The line through $M$, parallel to the inner angular bisector of $\angle BCA$ meets the outer angular bisector of $\angle CBA$ at point $P$. Prove that lines $XM, Y N, ZP$ pass through a single point.

$A$ is a set satisfying the following the condition. Show that $2001+\sqrt{2001}$ is an element of $A$. [b]Condition[/b] (1) $1 \in A$ (2) If $x \in A$, then $x^2 \in A$. (3) If $(x-3)^2 \in A$, then $x \in A$.

Let $n$ be a positive integer. $S$ is a set of points such that the points in $S$ are arranged in a regular $2016$-simplex grid, with an edge of the simplex having $n$ points in $S$. (For example, the $2$-dimensional analog would have $\dfrac{n(n+1)}{2}$ points arranged in an equilateral triangle grid). Each point in $S$ is labeled with a real number such that the following conditions hold: (a) Not all the points in $S$ are labeled with $0$. (b) If $\ell$ is a line that is parallel to an edge of the simplex and that passes through at least one point in $S$, then the labels of all the points in $S$ that are on $\ell$ add to $0$. (c) The labels of the points in $S$ are symmetric along any such line $\ell$. Find the smallest positive integer $n$ such that this is possible. Note: A regular $2016$-simplex has $2017$ vertices in $2016$-dimensional space such that the distances between every pair of vertices are equal. [i]Proposed by James Lin[/i]

The number $2020$ has three different prime factors. What is their sum?

In an acute triangle $ABC$ points $D,E,F$ are located on the sides $BC,CA, AB$ such that \[ \frac{CD}{CE} = \frac{CA}{CB} , \frac{AE}{AF} = \frac{AB}{AC} , \frac{BF}{FD} = \frac{BC}{BA} \] Prove that $AD,BE,CF$ are altitudes of triangle $ABC$.

Points $A$, $B$, $C$, and $D$ lie on a circle. Let $AC$ and $BD$ intersect at point $E$ inside the circle. If $[ABE]\cdot[CDE]=36$, what is the value of $[ADE]\cdot[BCE]$? (Given a triangle $\triangle ABC$, $[ABC]$ denotes its area.)

[b]Q.[/b] Show that for any given positive integers $k, l$, there exists infinitely many positive integers $m$, such that $i) m \geqslant k$ $ii) \text{gcd}\left(\binom{m}{k}, l\right)=1$ [i]Suggested by pigeon_in_a_hole[/i]

$3$ green, $4$ yellow, and $5$ red balls are placed in a bag. (Large piles of balls of each colour are outside the bag.) Two balls of different colours are selected at random, and replaced by two balls of the third colour. If, at some point, there are $5$ green balls left in the bag, and there are at least as many yellow balls as red balls left in the bag, how many balls of each colour are left in the bag? Write your answer in the form $(g,y, r)$, where $g$ is the number of green balls and so on.

Let $p$ and $q$ be two positive integers such that $1 \le q \le p$. Also let $a = \left( p +\sqrt{p^2 + q} \right)^2$. a) Prove that the number $a$ is irrational. b) Show that $\{a\} > 0.75$.