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$.

Incircle of triangle $ABC$ is tangent to sides $BC,CA,AB$ at $D,E,F$,respectively.Points $P,Q$ are inside angle $BAC$ such that $FP=FB,FP||AC$ and $EQ=EC,EQ||AB$.Prove that $P,Q,D$ are collinear.

In the beginnig, all nine squares of $3\times 3$ chessboard contain $0$. At each step, we choose two squares sharing a common edge, then we add $1$ to them or $-1$ to them. Show that it is not possible to make all squares $2$, after a finite number of steps.

Sebastian has a certain number of rectangles with areas that sum up to 3 and with side lengths all less than or equal to $1$. Demonstrate that with each of these rectangles it is possible to cover a square with side $1$ in such a way that the sides of the rectangles are parallel to the sides of the square. [b]Note:[/b] The rectangles can overlap and they can protrude over the sides of the square.

Can the distances from a certain point on the plane to the vertices of a certain square be equal to $1, 4, 7$, and $8$ ?

Find the smallest number among the following numbers: $ \textbf{(A) }\sqrt{55}-\sqrt{52}\qquad\textbf{(B) }\sqrt{56}-\sqrt{53}\qquad\textbf{(C) }\sqrt{77}-\sqrt{74}\qquad\textbf{(D) }\sqrt{88}-\sqrt{85}\qquad\textbf{(E) }\sqrt{70}-\sqrt{67} $

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. Rays $\displaystyle \overrightarrow{OB}$ and $\displaystyle \overrightarrow{DC}$ intersect at $E$, and rays $\displaystyle \overrightarrow{OC}$ and $\displaystyle \overrightarrow{AB}$ intersect at $F$. Suppose that $AE = EC = CF = 4$, and the circumcircle of $ODE$ bisects $\overline{BF}$. Find the area of triangle $ADF$. [i]Proposed by Howard Halim[/i]

Let $A_{1}A_{2}\ldots A_{9}$ be a regular $9-$gon. Let $\{A_{1},A_{2},\ldots,A_{9}\}=S_{1}\cup S_{2}\cup S_{3}$ such that $|S_{1}|=|S_{2}|=|S_{3}|=3$. Prove that there exists $A,B\in S_{1}$, $C,D\in S_{2}$, $E,F\in S_{3}$ such that $AB=CD=EF$ and $A \neq B$, $C\neq D$, $E\neq F$.

Prove that in every convex quadrilateral of area $1$, the sum of the lengths of the sides and diagonals is not smaller than $2(2+\sqrt2)$.

Let $n$ be a positive integer such that there exists a positive integer that is less than $\sqrt{n}$ and does not divide $n$. Let $(a_1, . . . , a_n)$ be an arbitrary permutation of $1, . . . , n$. Let $a_{i1} < . . . < a_{ik}$ be its maximal increasing subsequence and let $a_{j1} > . . . > a_{jl}$ be its maximal decreasing subsequence. Prove that tuples $(a_{i1}, . . . , a_{ik})$ and $(a_{j1}, . . . , a_{jl} )$ altogether contain at least one number that does not divide $n$.

Determine the values that \(n\) can take so that the equation in \( x \) $$ x^4-(3n+2)x^2+n^2=0$$ has four different real roots \( x_1\), \(x_2\), \(x_3\) and \(x_4\) in arithmetic progression. That is, they satisfy that $$x_4-x_3=x_3-x_2=x_2-x_1$$

Let $S$ be the domain in the coordinate plane determined by two inequalities: \[y\geq \frac 12x^2,\ \ \frac{x^2}{4}+4y^2\leq \frac 18.\] Denote by $V_1$ the volume of the solid by a rotation of $S$ about the $x$-axis and by $V_2$, by a rotation of $S$ about the $y$-axis. (1) Find the values of $V_1,\ V_2$. (2) Compare the size of the value of $\frac{V_2}{V_1}$ and 1.

Given a set $S=\{1,2,3,\ldots,3n\},(n\in N^*)$, let $T$ be a subset of $S$, such that for any $x, y, z\in T$ (not necessarily distinct) we have $x+y+z\not \in T$. Find the maximum number of elements $T$ can have.

Given an integer $n$ greater than $1$, suppose $x_1,x_2,\ldots,x_n$ are integers such that none of them is divisible by $n$, and neither their sum. Prove that there exists atleast $n-1$ non-empty subsets $\mathcal I\subseteq \{1,\ldots, n\}$ such that $\sum_{i\in\mathcal I}x_i$ is divisible by $n$

It is given a right-angled triangle $ABC$ and its circumcircle $k$. (a) prove that the radii of the circle $k_1$ tangent to the cathets of the triangle and to the circle $k$ is equal to the diameter of the incircle of the triangle ABC. (b) on the circle $k$ there may be found a point $M$ for which the sum $MA+MB+MC$ is as large as possible.