Let $a$ and $b$ positive real numbers such that $(65a^2 + 2ab + b^2)(a^2 + 8ab + 65b^2) = (8a^2 + 39ab + 7b^2)^2$. Then one possible value of $\frac{a}{b}$ satises $2 \left(\frac{a}{b}\right) = m +\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.

Given positive integers $m$ and $n$ so there is a chessboard with $mn$ $1 \times 1$ grids. Colour the grids into red and blue (Grids that have a common side are not the same colour and the grid in the left corner at the bottom is red). Now the diagnol that goes from the left corner at the bottom to the top right corner is coloured into red and blue segments (Every segment has the same colour with the grid that contains it). Find the sum of the length of all the red segments.

$A$ is the set of points of a convex $n$-gon on a plane. The distinct pairwise distances between any $2$ points in $A$ arranged in descending order is $d_1>d_2>...>d_m>0$. Let the number of unordered pairs of points in $A$ such that their distance is $d_i$ be exactly $\mu _i$, for $i=1, 2,..., m$. Prove: For any positive integer $k\leq m$, $\mu _1+\mu _2+...+\mu _k\leq (3k-1)n$.

In triangle $ABC$, $M$ is midpoint of $AC$, and $D$ is a point on $BC$ such that $DB=DM$. We know that $2BC^{2}-AC^{2}=AB.AC$. Prove that \[BD.DC=\frac{AC^{2}.AB}{2(AB+AC)}\]

Let $\mathcal P$ be the set of all points in $\mathbb R^n$ with rational coordinates. For the points $A,B \in \mathcal l{P}$, one can move from $A$ to $B$ if the distance $AB$ is $1$. Prove that every point in $\mathcal l{ P}$ can be reached from any other point in $\mathcal{P}$ by a finite sequence of moves if and only if $n \geq 5$.

Let $d$ denote the greatest common divisor of the natural numbers $a$ and $b$, and let $d'$ denote the greatest common divisor of the natural numbers $a'$ and $b'$. Prove that $dd'$ is the greatest common divisor of the four numbers $$ aa' , \ \ ab' , \ \ ba' , \ \ bb' .$$

Let a regular $ (2n \plus{}1)\minus{}$gon be inscribed in a circle of radius $ r.$ We consider all the triangles whose vertices are from those of the regular $ (2n \plus{} 1)\minus{}$gon. [b](a)[/b] How many triangles among them contain the center of the circle in their interior? [b](b)[/b] Find the sum of the areas of all those triangles that contain the center of the circle in their interior.

We consider three distinct half-lines $Ox, Oy, Oz$ in a plane. Prove the existence and uniqueness of three points $A \in Ox, B \in Oy, C \in Oz$ such that the perimeters of the triangles $OAB,OBC,OCA$ are all equal to a given number $2p > 0.$

The smallest positive integer which can be expressed as sum of positive perfect cubes (possibly with repetition and/or with a single element sum) in at least two different ways in $$(A) 8$$ $$(B) 1729$$ $$(C) 2023$$ $$(D) 2024$$

Consider the sequence of integers $0, 1, 2, 4, 6, 9, 12,...$ obtained by starting with zero, adding $1$, then adding $1$ again, then adding $2$, and adding $2$ again, then adding $3$, and adding $3$ again, and so on. If we call the subsequent terms of this sequence $a_0, a_1, a_2, ...$, then we have $a_0 = 0$, and $a_{2n-1} = a_{2n-2} + n$ , $a_{2n} = a_{2n-1} + n$ for all integers $n \ge 1$. Find all integers $k \ge 0$ for which $a_k$ is the square of an integer.

A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not inclued the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed $1000x.$

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?

In an isosceles triangle with base $a$, lateral side $b$, and height to the base $v$, it holds that $\frac a2+v\ge b\sqrt2$. Find the angles of the triangle. Compute its area if $b=8\sqrt2$.

There are several white and black points. Every white point is connected with every black point by a segment. Each segment is equipped with a positive integer. For any closed circuit the product of the integers on the segments passed in the direction from white to black point is equal to the product of the integers on the segments passed in the opposite direction. Can one always place the integer at each point so that the integer on each segment is the product of the integers at its ends?

Let $\mathbf M\subseteq\mathbb R^2$ be a set with the following properties: 1) there is a pair $(a,b)\in\mathbf M$ such that $ab(a-b)\neq0,$ 2) if $\left(x_1,y_1\right),\left(x_2,y_2\right)\in\mathbf M$ and $c\in\mathbb R$ then also \[\left(cx_1,cy_1\right),\left(x_1+x_2,y_1+y_2\right),\left(x_1x_2,y_1y_2\right)\in\mathbf M.\] Show that in fact \[\mathbf M=\mathbb R^2.\]

Given a convex quadrilateral $ABCD$. The medians of the triangle $ABC$ intersect at point $M$, and the medians of the triangle $ACD$ at point$ N$. The circle, circumscibed around the triangle $ACM$, intersects the segment $BD$ at the point $K$ lying inside the triangle $AMB$ . It is known that $\angle MAN = \angle ANC = 90^o$. Prove that $\angle AKD = \angle MKC$.

Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins.

Three players Alexander, Beverley and Catherine participate in a tournament (all of them play the same number of games with each other). Is it possible that Alexander gets more points than the others, Catherine gets less points than the others, but Alexander has a smaller number of wins than the others and Catherine has a greater number of wins than the others? (A win scores $1$ point, a draw scores $\frac12$.) (A Rubin,)

Is it possible to put $1987$ points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine a non-degenerate triangle with rational area? [i](IMO Problem 5)[/i] [i]Proposed by Germany, DR[/i]

Find the smallest prime number $p$ for which the number $p^3+2p^2+p$ has exactly $42$ divisors.

If $a$ and $b$ are integers such that $x^2 - x - 1$ is a factor of $ax^3 + bx^2 + 1$, then $b$ is $ \textbf{(A)}\ -2\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2 $

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

A positive integer is randomly selected from among the first $2011$ primes. What is the probability that it is even?

A hilly island has $2023$ lookouts. It is known that each of them is in line of sight with at least $42$ of the other lookouts. For any two distinct lookouts $X$ and $Y$ there is a positive integer $n$ and lookouts $A_1,A_2,\dots,A_{n+1}$ such that $A_1=X$ and $A_{n+1}=Y$ and $A_1$ is in line of sight with $A_2$, $A_2$ with $A_3$, $\dots$ and $A_n$ with $A_{n+1}$. The smallest such number $n$ is called the [i]viewing distance[/i] of $X$ and $Y$. Determine the largest possible viewing distance that can exist between two lookouts under these conditions.

Alexis and Joshua are walking along the beach when they decide to draw symbols in the sand. Alex draws only stars and only draws them in pairs while Joshua draws only squares in trios. "Let's see how many rows of $15$ adjacent symbols we can make this way," suggests Josh. Alexis is game for the task and the two get busy drawing. Some of their rows look like \[\begin{array}{ccccccccccccccc} \vspace{10pt}*&*&*&*&*&*&*&*&*&*&*&*&\blacksquare&\blacksquare&\blacksquare\\\vspace{10pt}\blacksquare&\blacksquare&\blacksquare&*&*&*&*&*&*&*&*&*&*&*&*\\\vspace{10pt}\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&*&*&*&*&*&*&\blacksquare&\blacksquare&\blacksquare \\\vspace{10pt}\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare\\\vspace{10pt} *&*&*&*&*&*&\blacksquare&\blacksquare&\blacksquare&*&*&*&*&*&*\end{array}\] The twins decide to count each of the first two rows above as distinct rows, even though one is the mirror image of the other. But the last of the rows above is its own mirror image, so they count it only once. Around an hour later, the twins realized that they had drawn every possible row exactly once using their rules of stars in pairs and squares in trips. How many rows did they draw in the sand?