Point $A,B,C,D,E$ lie on a line in this order, such that $BC=CD=\sqrt{AB\cdot DE},$ $P$ doesn't lie on the line, and satisfys that $PB=PD.$ Point $K,L$ lie on the segment $PB,PD,$ respectively, such that $KC$ bisects $\angle BKE,$ and $LC$ bisects $\angle ALD.$ Prove that $A,K,L,E$ are concyclic.

Let $A',B',C'$ be the midpoints of the sides $BC, CA, AB$, respectively, of an acute non-isosceles triangle $ABC$, and let $D,E,F$ be the feet of the altitudes through the vertices $A,B,C$ on these sides respectively. Consider the arc $DA'$ of the nine point circle of triangle $ABC$ lying outside the triangle. Let the point of trisection of this arc closer to $A'$ be $A''$. Define analogously the points $B''$ (on arc $EB'$) and $C''$(on arc $FC'$). Show that triangle $A''B''C''$ is equilateral.

On a table, there are $2025$ empty boxes numbered $1,2,\dots ,2025$, and $2025$ balls with weights $1,2,\dots ,2025$. Starting with Vadim, Vadim and Marian take turns selecting a ball from the table and placing it into an empty box. After all $2025$ turns, there is exactly one ball in each box. Denote the weight of the ball in box $i$ by $w_i$. Marian wins if $$\sum_{i=1}^{2025}i\cdot w_i\equiv 0 \pmod{23}.$$ If both players play optimally, can Marian guarantee a win? [i]Proposed by Pavel Ciurea[/i]

Prove that for every $n \in \mathbb{N}$ the following proposition holds: $7|3^n +n^3$ if and only if $7|3^{n} n^3 +1$.

A certain Oxford professor, assigned to espionage cryptography services British, role played by Dirk Bogarde in a film, recruits his proposing small attention exercises, such as mentally reading a word the other way around. Frequently he does it with his own name: $SEBASTIAN$, what will there be to read $NAITSABES$. He wonders if there is any movement of the plane or of space that transforms one of these words in the other, just as they appear written. And if it had been called $AVITO$, like a certain Unamuno character? Give a reasoned explanation for each answer.

Find all pairs of positive prime numbers $(p, q)$ such that $$p^5 + p^3 + 2 = q^2 - q.$$

An integer $1$ is written on the blackboard. We are allowed to perform the following operations:to change the number $x$ to $3x+1$ of to $[\frac{x}{2}]$. Prove that we can get all positive integers using this operations.

The product of two of the four roots of the quartic equation $x^4 - 18x^3 + kx^2+200x-1984=0$ is $-32$. Determine the value of $k$.

Find positive real numbers $x,y,z$ that are solutions of the system $x+y+z=xy+yz+zx$ and $xyz=1$ , and have the smallest possible sum.

For the real numbers $a_1,...,a_n, b_1,...,b_m$ , the following relations hold: 1) $|a_i|= |b_j|=1$, $i=1,...,n$ ,$j=1,...,m$ 2) $a_1\sqrt{2+a_2\sqrt{2+...+a_n\sqrt2}}=b_1\sqrt{2+b_2\sqrt{2+...+b_m\sqrt2}}$ Prove that $n = m$ and $a_i=b_i$ , $i=1,...,n$

Solve in real numbers the equation $$\log_7 (6^x+1)=\log_6(7^x-1).$$ [i]Mathematical Gazette[/i]

A circle with integer radius $r$ is centered at $(r, r)$. Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \le i \le 14$ and are tangent to the circle, where $a_i$, $b_i$, and $c_i$ are all positive integers and $c_1 \le c_2 \le \cdots \le c_{14}$. What is the ratio $\frac{c_{14}}{c_1}$ for the least possible value of $r$? $\textbf{(A)} ~\frac{21}{5} \qquad\textbf{(B)} ~\frac{85}{13} \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~\frac{39}{5} \qquad\textbf{(E)} ~17 $

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

A notch is cut in a cylindrical vertical tree trunk. The notch penetrates to the axis of the cylinder and is bounded by two half-planes. Each half-plane is bounded by a horizontal line passing through the axis of the cylinder. The angle between the two half-planes is $\theta$. Prove that the volume of the notch is minimized (for given tree and $\theta$) by taking the bounding planes at equal angles to the horizontal plane.

Call a triple of numbers [b]Nice[/b] if one of them is the average of the other two. Assume that we have $2k+1$ distinct real numbers with $k^2$ [b] Nice[/b] triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios Proposed by [i]Morteza Saghafian[/i]

A tetrahedron is to be cut by three planes which form a parallelepiped whose three faces and all vertices lie on the surface of the tetrahedron. (a) Can this be done so that the volume of the parallelepiped is at least $ \frac{9}{40}$ of the volume of the tetrahedron? (b) Determine the common point of the three planes if the volume of the parallelepiped is $ \frac{11}{50}$ of the volume of the tetrahedron.

Let $\mathcal{S}$ be a sphere with radius $2.$ There are $8$ congruent spheres whose centers are at the vertices of a cube, each has radius $x,$ each is externally tangent to $3$ of the other $7$ spheres with radius $x,$ and each is internally tangent to $\mathcal{S}.$ There is a sphere with radius $y$ that is the smallest sphere internally tangent to $\mathcal{S}$ and externally tangent to $4$ spheres with radius $x.$ There is a sphere with radius $z$ centered at the center of $\mathcal{S}$ that is externally tangent to all $8$ of the spheres with radius $x.$ Find $18x + 5y + 4z.$

Rectangle $ABCD$ has side lengths $AB=84$ and $AD=42$. Point $M$ is the midpoint of $\overline{AD}$, point $N$ is the trisection point of $\overline{AB}$ closer to $A$, and point $O$ is the intersection of $\overline{CM}$ and $\overline{DN}$. Point $P$ lies on the quadrilateral $BCON$, and $\overline{BP}$ bisects the area of $BCON$. Find the area of $\triangle{CDP}$.

In a lottery, a person must select six distinct numbers from $1, 2, 3,\dots, 36$ to put on a ticket. The lottery commitee will then draw six distinct numbers randomly from $1, 2, 3, \ldots, 36$. Any ticket with numbers not containing any of these $6$ numbers is a winning ticket. Show that there is a scheme of buying $9$ tickets guaranteeing at least one winning ticket, but $8$ tickets are not enough to guarantee a winning ticket in general.

Let $N$ be the number of non-empty subsets $T$ of $S = \{1, 2, 3, 4, . . . , 2020\}$ satisfying $\max (T) >1000$. Compute the largest integer $k$ such that $3^k$ divides $N$.

Let $\mathbb{Z^+}$ denote the set of positive integers. Find all functions $f:\mathbb{Z^+} \to \mathbb{Z^+}$ satisfying the condition $$ f(a) + f(b) \mid (a + b)^2$$ for all $a,b \in \mathbb{Z^+}$

Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below. \[ \begin{array}{l} 2x_1+x_2+x_3+x_4+x_5=6 \\ x_1+2x_2+x_3+x_4+x_5=12 \\ x_1+x_2+2x_3+x_4+x_5=24 \\ x_1+x_2+x_3+2x_4+x_5=48 \\ x_1+x_2+x_3+x_4+2x_5=96 \\ \end{array} \]

For a positive integer k, let $f_{1}(k)$ be the square of the sum of the digits of k. (For example $f_{1}(123)=(1+2+3)^{2}=36$.) Let $f_{n+1}(k)=f_{1}(f_{n}(k))$. Determine the value of the $f_{2007}(2^{2006})$. Justify your claim.

The orthogonal projection of a tetrahedron onto a plane containing one of its faces is a trapezoid of area $1$, which has only one pair of parallel sides. a) Is it possible that the orthogonal projection of this tetrahedron onto a plane containing another its face is a square of area $1$? b) The same question for a square of area $1/2019$. (Mikhail Evdokimov)

1. Find $N$, the smallest positive multiple of $45$ such that all of its digits are either $7$ or $0$. 2. Find $M$, the smallest positive multiple of $32$ such that all of its digits are either $6$ or $1$. 3. How many elements of the set $\{1,2,3,...,1441\}$ have a positive multiple such that all of its digits are either $5$ or $2$?