In an acute triangle, the distance from the midpoint of any side to the opposite vertex is equal to the sum of the distances from it to sides of the triangle. Prove that this triangle is equilateral.

What is the area of the largest circle contained in an equilateral triangle of area $8\sqrt3$?

The sides of triangles $ABC$ and $ACD$ satisfy the following conditions: $AB = AD = 3$ cm, $BC = 7$ cm, $DC = 11$ cm. What values can the side length $AC$ take if it is an integer number of centimeters, is the average in $\Delta ACD$ and the largest in $\Delta ABC$?

Let $AB$ be a diameter of a circle; let $t_1$ and $t_2$ be the tangents at $A$ and $B$, respectively; let $C$ be any point other than $A$ on $t_1$; and let $D_1D_2. E_1E_2$ be arcs on the circle determined by two lines through $C$. Prove that the lines $AD_1$ and $AD_2$ determine a segment on $t_2$ equal in length to that of the segment on $t_2$ determined by $AE_1$ and $AE_2.$

In the convex quadrilateral $ABCD$ we have that $\angle BCD = \angle ADC \ge 90 ^o$. The bisectors of $\angle BAD$ and $\angle ABC$ intersect in $M$. Prove that if $M \in CD$, then $M$ is the middle of $CD$.

Let $x_0,x_1,x_2,\ldots$ be a sequence of rational numbers defined recursively as follows: $x_0$ can be any rational number and, for $n\ge 0$, \[ x_{n+1}=\begin{cases} \left|\frac{x_n}2-1\right| & \text{if the numerator of }x_n\text{ is even}, \\ \left|\frac1{x_n}-1\right| & \text{if the numerator of }x_n\text{ is odd},\end{cases} \] where by numerator of a rational number we mean the numerator of the fraction in its lowest terms. Prove that for any value of $x_0$: (a) the sequence contains only finitely many distinct terms; (b) the sequence contains exactly one of the numbers $0$ and $2/3$ (namely, either there exists an index $k$ such that $x_k=0$, or there exists an index $m$ such that $x_m=2/3$, but not both).

A given tetrahedron $ ABCD$ is isoceles, that is, $ AB\equal{}CD$, $ AC\equal{}BD$, $ AD\equal{}BC$. Show that the faces of the tetrahedron are acute-angled triangles.

Let $n$ be a positive integer. In how many ways can an $n \times n$ table be filled with integers from $0$ to $5$ such that a) the sum of each row is divisible by $2$ and the sum of each column is divisible by $3$ b) the sum of each row is divisible by $2$, the sum of each column is divisible by $3$ and the sum of each of the two diagonals is divisible by $6$?

a) Find a pair(s) of integers $(x,y)$ such that: $y^2=x^3+2017$ b) Prove that there isn't integers $x$ and $y$, with $y$ not divisible by $3$, such that: $y^2=x^3-2017$

Find the number of integers $n$ that satisfy $\textit{both}$ of the following conditions: [list=i] [*]$208<n<2008$, [*]$n$ has the same remainder when divided by $24$ or by $30$.[/list]

Let $ABC$ be an acute triangle. Determine the locus of points $M$ in the interior of the triangle such that $AB-FG=\frac{MF \cdot AG+MG \cdot BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to lines $BC$ and $AC$, respectively.

Let $SABC$ is pyramid, such that $SA\le 4$, $SB\ge 7$, $SC\ge 9$, $AB=5$, $BC\le 6$ and $AC\le 8$. Find max value capacity(volume) of the pyramid $SABC$.

A regular pentagon is constructed externally on each side of a regular pentagon of side $1$. The figure is then folded and the two edges of the external pentagons meeting at each vertex of the original pentagon are glued together. Find the volume of water that can be poured into the obtained container.

If for real numbers $x$ and $y$ holds $\left(x+\sqrt{1+y^2}\right)\left(y+\sqrt{1+x^2}\right)=1$ prove that $$\left(x+\sqrt{1+x^2}\right)\left(y+\sqrt{1+y^2}\right)=1$$

Let $N$ be the set of all non-negative integers and for each $n \in N$ denote $n'= n +1$. The function $A : N^3 \to N$ is defined as follows: (i) $A(0, m, n) = m'$ for all $m, n \in N$ (ii) $A(k', 0, n) =\left\{ \begin{array}{ll} n & if \, \, k = 0 \\ 0 & if \, \,k = 1, \\ 1 & if \, \, k > 1 \end{array} \right.$ for all $k, n \in N$ (iii) $A(k', m', n) = A(k, A(k',m,n), n)$ for all $k,m, n \in N$. Compute $A(5, 3, 2)$. (H. Nestra)

Jerry likes to play with numbers. One day, he wrote all the integers from $1$ to $2024$ on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them with either their sum or their product. (For example, Jerry's first step might have been to erase $1, 2, 3$, and $5$, and then write either $11$, their sum, or $30$, their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the board were odd. What is the maximum possible number of integers on the board at that time? $ \textbf{(A) }1010 \qquad \textbf{(B) }1011 \qquad \textbf{(C) }1012 \qquad \textbf{(D) }1013 \qquad \textbf{(E) }1014 \qquad $