Let $n\geq 3$ be a positive integer. Consider an $n\times n$ square. In each cell of the square, one of the numbers from the set $M=\{1,2,\ldots,2n-1\}$ is to be written. One such filling is called [i]good[/i] if, for every index $1\leq i\leq n,$ row no. $i$ and column no. $i,$ together, contain all the elements of $M$. [list=a] [*]Prove that there exists $n\geq 3$ for which a good filling exists. [*]Prove that for $n=2017$ there is no good filling of the $n\times n$ square. [/list]

$F(0)=3$ and $F(n)=F(n-1)+4$ when $n$ is positive. Find $F(F(F(5)))$.

In the triangle $ABC$, the lengths of the sides $BC, CA$ and $AB$ are $a,b$ and $c{}$ respectively. Several segments are drawn from the vertex $C{}$, which cut the triangle $ABC$ into several triangles. Find the smallest number $M{}$ for which, with each such cut, the sum of the radii of the circles inscribed in triangles does not exceed $M{}$. [i]Porposed by O. Titov[/i]

Find all values of $k$ by which it is possible to divide any triangular region in $k$ quadrilaterals of equal area.

Let the nonnegative numbers $a_1, a_2,... a_9$, where $a_1 = a_9 = 0$ and let at least one of the numbers is nonzero. Denote the sentence $(P)$: '' For $2 \le i \le 8$ there is a number $a_i$, such that $a_{i - 1} + a_{i + 1} <ka_i $”. a) Show that the sentence $(P)$ is true for $k = 2$. b) Determine whether is the sentence $(P)$ true for $k = \frac{19}{10}$

Let $x, y, z$ be positive numbers such that $xyz \le 2$ and $\frac{1}{x^2}+ \frac{1}{y^2}+ \frac{1}{z^2}< k$, for some real $k \ge 2$. Find all values of $k$ such that the conditions above imply that there exist a triangle having the side-lengths $x, y, z$.

Find all triples $ (x,y,z)$ of natural numbers satisfying $ 2xz\equal{}y^2$ and $ x\plus{}z\equal{}1987$.

The number of solutions to $\{1,~2\}\subseteq~X~\subseteq~\{1,~2,~3,~4,~5\}$, where $X$ is a subset of $\{1,~2,~3,~4,~5\}$ is $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad \textbf{(E) }\text{None of these}$

Investigate, in any way which yields significant results, the existence, in the plane, of the configuration consisting of an ellipse simultaneously tangent to four distinct concentric circles.

Suppose a grid with $2m$ rows and $2n$ columns is given, where $m$ and $n$ are positive integers. You may place one pawn on any square of this grid, except the bottom left one or the top right one. After placing the pawn, a snail wants to undertake a journey on the grid. Starting from the bottom left square, it wants to visit every square exactly once, except the one with the pawn on it, which the snail wants to avoid. Moreover, it wants to finish in the top right square. It can only move horizontally or vertically on the grid. On which squares can you put the pawn for the snail to be able to finish its journey?

For any two points $A (x_1 , y_1)$ and $B (x_2, y_2)$, the distance $r (A, B)$ between them is determined by the equality $r(A, B) = max\{| x_1- x_2 | , | y_1 - y_2 |\}$. Prove that the triangle inequality $r(A, C) + r(C, B) \ge r(A, B)$. holds for the distance introduced in this way . Let $A$ and $B$ be two points of the plane . Find the locus of points $C$ for which a) $r(A, C) + r(C, B) = r(A, B)$ b) $r(A, C) = r(C, B).$

The area of a square is $144$. An equilateral triangle has the same perimeter as the square. The area of a regular hexagon is $6$ times the area of the equilateral triangle. What is the perimeter of the hexagon?

Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles were displayed on the odometer, where $abc$ is a 3-digit number with $a \ge 1$ and $a+b+c \le 7$. At the end of the trip, where the odometer showed $cba$ miles. What is $a^2+b^2+c^2$? $ \textbf{(A) } 26 \qquad\textbf{(B) }27\qquad\textbf{(C) }36\qquad\textbf{(D) }37\qquad\textbf{(E) }41\qquad $

Let $ABC$ be a triangle and let $D$ be a point on the segment $BC$ such that $AD=BC$. \\ Suppose $\angle{CAD}=x^{\circ}, \angle{ABC}=y^{\circ}$ and $\angle{ACB}=z^{\circ}$ and $x,y,z$ are in an arithmetic progression in that order where the first term and the common difference are positive integers. Find the largest possible value of $\angle{ABC}$ in degrees.

Determine all possible values of $x+\frac{1}{x}$ , where the real number $x$ satisfies the equation $$x^4+5x^3-4x^2+5x+1=0$$ and solve this equation.

Is it possible to partition a triangle, with line segments, into exactly five isosceles triangles? All the triangles in concern are assumed to be nondegenerated triangles.

In the Cartesian plane, a line segment is called [i]tame[/i] if it lies parallel to one of the coordinate axes and its distance to this axis is an integer. Otherwise it is called [i]wild[/i]. Let $m$ and $n$ be odd positive integers. The rectangle with vertices $(0,0),(m,0),(m,n)$ and $(0,n)$ is partitioned into finitely many triangles. Let $M$ be the set of these triangles. Assume that (1) Each triangle from $M$ has at least one tame side. (2) For each tame side of a triangle from $M$, the corresponding altitude has length $1$. (3) Each wild side of a triangle from $M$ is a common side of exactly two triangles from $M$. Show that at least two triangles from $M$ have two tame sides each.

Let $k$ be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay $1$ coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.

There are two $3$-digit numbers which end in $99$. These two numbers are also the product of two integers which differ by $2$. What is the sum of these two numbers?

Points $E$ and $C$ are chosen on a semicircle with diameter $AB$ and center $O$ such that $OE \perp AB$ and the intersection point $D$ of $AC$ and $OE$ is inside the semicircle. Find all values of $\angle{CAB}$ for which the quadrilateral $OBCD$ is tangent.

Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$. [i]Proposed by Charles Leytem, Luxembourg[/i]

How many noncongruent integer-sided triangles with positive area and perimeter less than $15$ are neither equilateral, isosceles, nor right triangles? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$

Points $A$ and $B$ are chosen on a circle $k$. Let AP and $BQ$ be segments of the same length tangent to $k$, drawn on different sides of line $AB$. Prove that the line $AB$ bisects the segment $PQ$.

Let $S_n$ be the number of subsets of the first $n$ positive integers that have the same number of even values and odd values; the empty set counts as one of these subsets. Compute the smallest positive integer $n$ such that $S_n$ is a multiple of $2020$.

Do there exist functions $f : R \to R$ and $g : R \to R$, $g$ being periodic, such that $$x^3= f(\lfloor x \rfloor ) + g(x)$$ for all real $x$ ?