Let $a < b < c < d < e$ be real numbers. We calculate all possible sums in pairs of these 5 numbers. Of these 10 sums, the three smaller ones are 32, 36, 37, while the two larger ones are 48 and 51. Determine all possible values ​​that $e$ can take.

Let $k\geq 1$ be an integer. In a group of $2k+1$ people, some are sincere (they always tell the truth) and the rest are unpredictable (sometimes they tell the truth and sometimes they lie). It is known that the unpredictable ones are at most $k$. Someone outside the group must determine who is sincere and who is unpredictable through a sequence of steps. In each step he chooses two people $A$ and $B$ from the group and asks $A$ is $B$ sincere? Show that after $3k$ steps the stranger will be able to classify with certainty the $2k+1$ people in the group. (Before asking each question, the answers to the previous questions are known.) Clarification: Each of the $2k+1$ people in the group knows which ones are sincere and which ones are unpredictable.

Three circles, with radii of $1, 1$, and $2$, are externally tangent to each other. The minimum possible area of a quadrilateral that contains and is tangent to all three circles can be written as $a + b\sqrt{c}$ where $c$ is not divisible by any perfect square larger than $1$. Find $a + b + c$

Find the largest possible integer $k$, such that the following statement is true: Let $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain \[ \left. \begin{array}{rcl} & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\ & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\ \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\ \end{array}\right.\] Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_j$, $r_j$, $w_j$. [i]Proposed by Michal Rolinek, Czech Republic[/i]

Let $O$ be the circumcenter of triangle $ABC$. Define $O_{A0} = O_{B0} = O_{C0} = O$. Recursively, define $O_{An}$ to be the circumcenter of $\vartriangle BO_{A(n-1)}C$. Similarly define $O_{Bn}, O_{Cn}$. Find all $n \ge 1$ so that for any triangle $ABC$ such that $O_{An}, O_{Bn}, O_{Cn}$ all exist, it is true that $AO_{An}, BO_{Bn}, CO_{Cn}$ are concurrent. (Li4)

Does there exist a convex pentagon such that for any of its inner angles, the angle bisector contains one of the diagonals?

What is the value of the product$$\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)?$$ $\textbf{(A) }\frac{7}{6}\qquad\textbf{(B) }\frac{4}{3}\qquad\textbf{(C) }\frac{7}{2}\qquad\textbf{(D) }7\qquad\textbf{(E) }8$

The sequences $(a_n)$ and $(b_n)$ are defined by $a_1 = b_1 = 1$ and $a_{n+1} = a_n +b_n, b_{n+1} = a_nb_n$ for $n = 1,2,...$ Show that every two distinct terms of the sequence $(a_n)$ are coprime

For what values of $k{}$ can a regular octagon with side-length $k{}$ be cut into $1 \times 2{}$ dominoes and rhombuses with side-length 1 and a $45^\circ{}$ angle?

Let $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$, and $DE=12$. Denote $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$.

There are $n \ge 7$ points in the plane, no $3$ of which are collinear. At least $7$ pairs of points are joined by line segments. For every aforementioned line segment $s$, let $t(s)$ be the number of triangles for which the segment $s$ is a side. Prove that there exist different line segments $s_1, s_2, s_3,$ and $s_4$ such that \[t(s_1) = t(s_2) = t(s_3) = t(s_4)\] holds. Proposed by [i]Viktor Simjanoski[/i]

A sequence of polynomial $f_n(x)\ (n=0,1,2,\cdots)$ satisfies $f_0(x)=2,f_1(x)=x$, \[f_n(x)=xf_{n-1}(x)-f_{n-2}(x),\ (n=2,3,4,\cdots)\] Let $x_n\ (n\geqq 2)$ be the maximum real root of the equation $f_n(x)=0\ (|x|\leqq 2)$ Evaluate \[\lim_{n\to\infty} n^2 \int_{x_n}^2 f_n(x)dx\]

A positive integer $ K$ is given. Define the sequence $ (a_n)$ by $ a_1\equal{}1$ and $ a_n$ is the $ n$-th natural number greater than $ a_{n\minus{}1}$ which is congruent to $ n$ modulo $ K$. $ (a)$ Find an explicit formula for $ a_n$. $ (b)$ What is the result if $ K\equal{}2?$

Let $a, b, c$ be non-negative numbers with $a+b+c = 3$. Prove the inequality \[\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{a^2+1} \geq \frac 32.\]

Demonstrate that if $ z_1,z_2\in\mathbb{C}^* $ satisfy the relation: $$ z_1\cdot 2^{\big| z_1\big|} +z_2\cdot 2^{\big| z_2\big|} =\left( z_1+z_2\right)\cdot 2^{\big| z_1 +z_2\big|} , $$ then $ z_1^6=z_2^6 $

Height of a circular truncated cone is $8$. Center of sphere $O_1$ with a radius of $2$ is on the axis of the circular truncated cone. Sphere $O_1$ is tangent to the top surface and the flank. We can put another sphere $O_2$, satisfying that sphere $O_2$ with a radius of $3$ have only one common point with sphere $O_1$, bottom surface and the flank. Besides $O_2$, how many spheres can we put inside the circular truncated cone? $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4$

The point $O$ inside the convex polygon makes isosceles triangle with all the pairs of its vertices. Prove that $O$ is the centre of the circumscribed circle. [u]other formulation:[/u] $P$ is a convex polygon and $X$ is an interior point such that for every pair of vertices $A, B$, the triangle $XAB$ is isosceles. Prove that all the vertices of $P$ lie on a circle with center $X$.

(a) Find all strictly monotone functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x+f(y))=f(x)+y\quad\text{for all real}\ x,y. \] (b) If $n>1$ is an integer, prove that there is no strictly monotone function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[ f(x+f(y))=f(x)+y^n\quad \text{for all real}\ x, y.\]

Determine the number of $2021$-tuples of positive integers such that the number $3$ is an element of the tuple and consecutive elements of the tuple differ by at most $1$. [i]Walther Janous (Austria)[/i]

(a) Show that, for each positive integer $n$, the number of monic polynomials of degree $n$ with integer coefficients having all its roots on the unit circle is finite. (b) Let $P(x)$ be a monic polynomial with integer coefficients having all its roots on the unit circle. Show that there exists a positive integer $m$ such that $y^m=1$ for each root $y$ of $P(x)$.

Numbers $1,\ldots,2025$ are written in a circle in increasing order. For every three consecutive numbers $i,j,k$ we consider the polynomial $(x-i)(x-j)(x-k)$. Let $s(x)$ be the sum of all $2025$ these polynomials. Prove that $s(x)$ has an integral root. [i]A. Voidelevich[/i]

A square, a regular pentagon, and a regular hexagon are all inscribed in the same circle. The $15$ vertices of these polygons divide the circle into at most $15$ arcs. Let $M$ be the degree measure of the longest of these arcs. Find the minimum possible value for $M$.

Let $X=\{1,2,3,...,12\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{2,3,5,7,8\}$.

Suppose that $G$ is a finite group generated by the two elements $g$ and $h,$ where the order of $g$ is odd. Show that every element of $G$ can be written in the form \[g^{m_1}h^{n_1}g^{m_2}h^{n_2}\cdots g^{m_r}h^{n_r}\] with $1\le r\le |G|$ and $m_n,n_1,m_2,n_2,\dots,m_r,n_r\in\{1,-1\}.$ (Here $|G|$ is the number of elements of $G.$)

$\textbf{(Lucas Numbers)}$ The Lucas numbers are defined by $L_0 = 2$, $L_1 = 1$, and $L_{n+2} = L_{n+1} + L_n$ for every $n \ge 0$. There are $N$ integers $1 \le n \le 2016$ such that $L_n$ contains the digit $1$. Estimate $N$. An estimate of $E$ earns $\left\lfloor 20 - 2|N-E| \right\rfloor$ or $0$ points, whichever is greater.