Seven students in a class compare their marks in 12 subjects studied and observe that no two of the students have identical marks in all 12 subjects. Prove that we can choose 6 subjects such that any two of the students have different marks in at least one of these subjects.

A natural number $k$ is said $n$-squared if by colouring the squares of a $2n \times k$ chessboard, in any manner, with $n$ different colours, we can find $4$ separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of $n$, the smallest natural $k$ that is $n$-squared.

There is a chess tournament with $2n$ players ($n > 1$). There is at most one match between each pair of players. If it is not possible to find three players who all play each other, show that there are at most $n^2$ matches. Conversely, show that if there are at most $n^2$ matches, then it is possible to arrange them so that we cannot find three players who all play each other.

[asy] size(5cm); pen p=linewidth(3), dark_grey=gray(0.25), ll_grey=gray(0.90), light_grey=gray(0.75); transform dishift(real x) { return shift(x,x); } // Draw the table of latch of table path ell = ((0,0)--(0,-1)--(-0.1,-1)--(-0.1,-0.1)--(-1,-0.1)--(-1,0)--cycle); // the ell shape path corner = dishift(-0.85)*ell; // define the path path table = dishift(-1)*scale(5)*ell; // define the table by scaling the pulley filldraw(corner, ll_grey, light_grey+p); // base of pulley filldraw(table, ll_grey, grey+p); // table real block_size = 1.6; // template for block path block = unitsquare; pair block_center = (0.5,0.5); /* Resting block */ transform rest = shift(-5, -0.9) * scale(block_size); // transformation for resting block filldraw(rest * block, ll_grey, light_grey+p); // draw block draw(rest*(1,0.5)--dir(110), light_grey+p); // rope fr0m midpoint of right block to wheel label("$m$", rest * block_center, fontsize(16)); // label block /* Hanging block */ transform hang = shift(0.2,-4.1) * scale(block_size); // transformation for hanging block draw((1,0)--(1,-2.5), light_grey+p); // string of pulley filldraw(hang * block, ll_grey, light_grey+p); // fill it label("$M$",hang * block_center,fontsize(16)); // label the small m // Draws the actual pulley filldraw(unitcircle, grey, p); // outer boundary of pulley wheel filldraw(scale(0.4)*unitcircle, light_grey, p); // inner boundary of pulley wheel path pulley_body=arc((0,0),0.3,-40,130)--arc((-1,-1),0.5,130,320)--cycle; // defines "arm" of pulley filldraw(pulley_body, ll_grey, dark_grey+p); // draws the arm filldraw(scale(0.18)*unitcircle, ll_grey, dark_grey+p); // inner circle of pulley [/asy] A pulley system of two blocks, shown above, is released from rest. The block on the table, which has mass $m=1.0 \, \text{kg}$ slides after the time of release and hits the pulley to come to a dead stop. There was originally a distance of $ 1.0 \, \text{m} $ between the block and the pulley, which the block fully covers during the slide. From the time of release to the time of hitting the pulley, the angle that the rope above the table makes with the horizontal axis is a, nearly constant, $10.0^\circ$. The hanging block has mass $ M = 2.0 \, \text{kg} $. The table has a coefficient of friction of $0.50$ with the block that sits on it. The pulley is frictionless. Also, assume that, during the entire slide, the block never leaves the ground. Let $t$ be the number of seconds in takes for the $1.0\text{-m}$ slide. Find $100t$, rounded to two significant figures. [i](Ahaan Rungta, 4 points)[/i]

Let $n > 1$ be a given integer. An $n \times n \times n$ cube is composed of $n^3$ unit cubes. Each unit cube is painted with one colour. For each $n \times n \times 1$ box consisting of $n^2$ unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is listed only once). This way, we get $3n$ sets of colours, split into three groups according to the orientation. It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of $n$, the maximal possible number of colours that are present.

More than five competitors participated in a chess tournament. Each competitor played exactly once against each of the other competitors. Five of the competitors they each lost exactly two games. All other competitors each won exactly three games. There were no draws in the tournament. Determine how many competitors there were and show a tournament that verifies all conditions.

Cut a right triangle with an angle of $30^o$ into three isosceles non-acute triangles, among which there are no congruent ones. (Maria Rozhkova)

Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.

Let $f:(1,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying $f(x) \le x^2 \log(x)$ and $f'(x)>0$ for every $x \in (1,\infty)$. Prove that \[\int_1^{\infty} \frac{1}{f'(x)} dx=\infty.\]

If $x \in R-\{-7\}$, determine the smallest value of the expression $$\frac{2x^2 + 98}{(x + 7)^2}$$

Let $f(x) = x^4 + 2x^3 - x - 1$. (a) Prove that $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients. (b) Find the exact values of the 4 roots of $f(x)$.

Prove that every integer can be written as sum of $5$ third powers of integers.

$ABC$ is a triangle with $\angle A = 90^o$. Take $E$ such that the triangle $AEC$ is outside $ABC$ and $AE = CE$ and $\angle AEC = 90^o$. Similarly, take $D$ so that $ADB$ is outside $ABC$ and similar to $AEC$. $O$ is the midpoint of $BC$. Let the lines $OD$ and $EC$ meet at $D'$, and the lines $OE$ and $BD$ meet at $E'$. Find area $DED'E'$ in terms of the sides of $ABC$.

Finitely many cells of an infinite square board are colored black. Prove that one can choose finitely many squares in the plane of the board so that the following conditions are satisfied: (i) The interiors of any two different squares are disjoint; (ii) Each black cell lies in one of these squares; (iii) In each of these squares, the black cells cover at least $\frac15$ and at most $\frac45$ of the area of that square.

Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.

Suppose that $(i - 1)^{11}$ is a root of the quadratic $x^2 + Ax + B$ for integers $A$ and $B$, where $i =\sqrt{-1}$. Compute the value of $A + B$.