Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

Compare the magnitude of the following three numbers. $$ \sqrt[3]{\frac{25}{3}} ,\, \sqrt[3]{\frac{1148}{135}} ,\, \frac{\sqrt[3]{25}}{3} + \sqrt[3]{\frac{6}{5}} $$

A positive integer is [i]detestable[/i] if the sum of its digits is a multiple of $11$. How many positive integers below $10000$ are detestable? [i]Proposed by Giacomo Rizzo[/i]

For three real numbers $ a,b,c>1, $ prove the inequality: $ \log_{a^2b} a +\log_{b^2c} b +\log_{c^2a} c\le 1. $

What is the product of the real roots of the equation \[x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}\,\,?\]

In the plane seven different points $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3$ are given. The points $P_1, P_2, P_3, P_4$ are on the straight line $p$, the points $Q_1, Q_2, Q_3$ are not on $p$. By each of the three points $Q_1, Q_2, Q_3$ the perpendiculars are drawn on the straight lines connecting points different of them. Prove that the maximum's number of the possibles intersections of all perpendiculars is to 286, if the points $Q_1, Q_2, Q_3$ are taken in account as intersections.

Solve the system $$\begin{cases} x+\log\left(x+\sqrt{x^2+1}\right)=y \\ y+\log\left(y+\sqrt{y^2+1}\right)=z \\ z+\log\left(z+\sqrt{z^2+1}\right)=x \end{cases}$$

A circle is inscribed in an angle with vertex $O$, touching its sides at points $M$ and $N$. On an arc $MN$ nearest to point $O$, an arbitrary point $P$ is selected. At point $P$, a tangent is drawn to the circle $P$, intersecting the sides of the angle at points $A$ and $B$. Prove that that the length of the segment $AB$ is the smallest when $P$ is its midpoint.

Find all prime $p$, for each of which there are such natural $ x$ and $y$ such that $p^x = y^3 + 1$.

A sequence of integers $(a_0,...,a_k)$ is said to be [i]medaled[/i] if, for each $i = 0,...,k$, there are exactly $a_i$ elements of the sequence equal to $i$. For example, $(1,2,1,0)$ is a [i]medaled [/i] seqence. Indicates all [i]medaled [/i] sequences $(a_0,...,a_{2015})$.

Let positive real numbers $ a,b$ satisfy $ b \minus{} a > 2.$ Prove that for any two distinct integers $ m,n$ belonging to $ [a,b),$ there always exists non-empty set $ S$ consisting of certain integers belonging to $ [ab,(a \plus{} 1)(b \plus{} 1))$ such that $ \frac {\displaystyle\prod_{x\in S}}{mn}$ is square of a rational number.

Let $ABC$ be a triangle such that $|AC|>|AB|$. Let $X$ be on line $AB$ (closer to $A$) such that $|BX|=|AC|$ and let $Y$ be on the segment $AC$ such that $|CY|=|AB|$. Intersection of lines $XY$ and bisector of $BC$ is point $P$. Prove that $\angle BPC+\angle BAC = 180^\circ$.

Find all positive integers $n$ for which there do not exist $n$ consecutive composite positive integers less than $n!$.

We will say that an octagon is integral if its is equiangular, its vertices are lattice points (i.e., points with integer coordinates), and its area is an integer. For example, the figure on the right shows an integral octagon of area $21$. Determine, with proof, the smallest positive integer $K$ so that for every positive integer $k\geq K$, there is an integral octagon of area $k$. [asy] size(200); defaultpen(linewidth(0.8)); draw((-1/2,0)--(17/2,0)^^(0,-1/2)--(0,15/2)); for(int i=1;i<=6;++i){ draw((0,i)--(17/2,i),linetype("4 4")); } for(int i=1;i<=8;++i){ draw((i,0)--(i,15/2),linetype("4 4")); } draw((2,1)--(1,2)--(1,3)--(4,6)--(5,6)--(7,4)--(7,3)--(5,1)--cycle,linewidth(1)); label("$1$",(1,0),S); label("$2$",(2,0),S); label("$x$",(17/2,0),SE); label("$1$",(0,1),W); label("$2$",(0,2),W); label("$y$",(0,15/2),NW); [/asy]

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

Don Miguel places a token in one of the $(n+1)^2$ vertices determined by an $n \times n$ board. A [i]move[/i] consists of moving the token from the vertex on which it is placed to an adjacent vertex which is at most $\sqrt2$ away, as long as it stays on the board. A [i]path[/i] is a sequence of moves such that the token was in each one of the $(n+1)^2$ vertices exactly once. What is the maximum number of diagonal moves (those of length $\sqrt2$) that a path can have in total?

A circle inscribed in a square, Has two chords as shown in a pair. It has radius $2$, And $P$ bisects $TU$. The chords' intersection is where? Answer the question by giving the distance of the point of intersection from the center of the circle. [asy] size(100); defaultpen(linewidth(0.8)); draw(unitcircle); draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle); label("$A$",(-1,1),SE); label("$B$",(1,1),SE); label("$C$",(1,-1),SE); label("$D$",(-1,-1),SE); pair M=(1,0),N=(0,-1),T=(-1,0),U=(0,1),P=dir(135); draw(P--M^^(-1,-1)--(1,1)); label("$M$",M,SE); label("$N$",N,SE); label("$T$",T,SE); label("$U$",U,SE); label("$P$",P,dir(270)); dot(origin^^(-1,1)^^(-1,-1)^^(1,-1)^^(1,1)^^M^^N^^T^^U^^P); [/asy]

For a positive integer $n$, let $A(n)$ be the equal to the remainder when $n$ is divided by $11$ and let $T(n)=A(1)+A(2)+A(3)+ \dots + A(n)$. Find the value of $$A(T(2021))$$

Suppose we have an (infinite) cone $ \mathcal C$ with apex $ A$ and a plane $ \pi$. The intersection of $ \pi$ and $ \mathcal C$ is an ellipse $ \mathcal E$ with major axis $ BC$, such that $ B$ is closer to $ A$ than $ C$, and $ BC \equal{} 4$, $ AC \equal{} 5$, $ AB \equal{} 3$. Suppose we inscribe a sphere in each part of $ \mathcal C$ cut up by $ \mathcal E$ with both spheres tangent to $ \mathcal E$. What is the ratio of the radii of the spheres (smaller to larger)?

The total number of languages used in KAUST is $n$. For each positive integer $k \le n$, let $A_k$ be the set of all those people in KAUST who can speak at least $k$ languages; and let $B_k$ be the set of all people $P$ in KAUST with the property that, for any $k$ pairwise different languages (used in KAUST), $P$ can speak at least one of these $k$ languages. Prove that (a) If $2k \ge n + 1$ then $A_k \subseteq B_k$ (b) If $2k \le n + 1$ then $A_k \supseteq B_k.$ Nguyễn Duy Thái Sơn

Determine the largest number of steps for $\gcd(k,76)$ to terminate over all choices of $0 < k < 76$, using the following algorithm for gcd. Give your answer in the form $(n,k)$ where $n$ is the maximal number of steps and $k$ is the $k$ which achieves this. If multiple $k$ work, submit the smallest one. \begin{tabular}{l} 1: \textbf{FUNCTION} $\text{gcd}(a,b)$: \\ 2: $\qquad$ \textbf{IF} $a = 0$ \textbf{RETURN} $b$ \\ 3: $\qquad$ \textbf{ELSE RETURN} $\text{gcd}(b \bmod a,a)$ \end{tabular}

Let $\Gamma$ be the circumscribed circle of a triangle $ABC$ and let $D$ be a point at line segment $BC$. The circle passing through $B$ and $D$ tangent to $\Gamma$ and the circle passing through $C $and $D$ tangent to $\Gamma$ intersect at a point $E \ne D$. The line $DE$ intersects $\Gamma$ at two points $X$ and $Y$ . Prove that $|EX| = |EY|$.

Three circles are pairwise tangent, with none of them lying inside another. The centres of the circles are the corners of a triangle with circumference $1$. What is the smallest possible value for the sum of the areas of the circles?

Find all integers$ n$ such that $n^2 + 24n + 35$ is a square.

Calculate the volume of the tetrahedron $ ABCD $ given the edges $ AB = b $, $ AC = c $, $ AD = d $ and the angles $ \measuredangle CAD = \beta $, $ \measuredangle DAB = \gamma $ and $ \measuredangle BAC = \delta$.