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Find all $6$ -digit natural numbers which are both a perfect square and a cube.
Given a triangle $A B C$ with $\widehat{A}=30^{\circ}$ , $\widehat{B}=20^{\circ}$ . On the side $A B$ choose the point $D$ such that $A D=B C$ . Find the value of the angle $\widehat{B C D}$ .
Assume that $a, b \in \mathbb{R}$ and $a^{2}+b^{2}+16=8 a+6 b$ . Show that
a) $10 \leq 4 a+3 b \leq 40$ .
b) $7 b \leq 24 a$ .
Given a half circle with the center $O,$ the diameter $B C .$ Choose a point $G$ inside the half circle so that $\widehat{B G O}=135^{\circ}$ . The line which is perpendicular to $G B$ at $G$ intersects the half circle at $A$ . The incircle $I$ of $A B C$ is tangent to $B C$ , $C A$ respectively at $D$ and $E .$ Show that $G$ lies on $E D$ .
Suppose that $x, y, z$ are positive numbers satisfying $x+y \leq 2 z$ . Find the minimum value of the expression $P = x y + z + y x + z - x + y 2 z .$ $P=\frac{x}{y+z}+\frac{y}{x+z}-\frac{x+y}{2 z}.$
Show the inequality $(x + y x - y) 2020 + (y + z y - z) 2020 + (z + x z - x) 2020 > 2 1010 3 1009$ $\left(\frac{x+y}{x-y}\right)^{2020}+\left(\frac{y+z}{y-z}\right)^{2020}+\left(\frac{z+x}{z-x}\right)^{2020}>\frac{2^{1010}}{3^{1009}}$ where $x, y, z$ are different numbers.
Solve the system of equations $⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x 2 x 3 \dots x 2020 x 1 = x 31 - 3 x 1 = x 32 - 3 x 2 \dots = x 32019 - 3 x 2019 = x 32020 - 3 x 2020$ $\begin{cases}x_{2} &=x_{1}^{3}-3 x_{1} \\ x_{3} &=x_{2}^{3}-3 x_{2} \\ \ldots & \ldots \\ x_{2020} &=x_{2019}^{3}-3 x_{2019} \\ x_{1} &=x_{2020}^{3}-3 x_{2020}\end{cases}$
Given a right triangle $A B C$ with the right angle $A$ and the altitude $A H$ . On the line segment $A H$ choose a point $I$ , the line $C I$ intersects $A B$ at $E .$ On the side $A C$ choose the point $F$ such that $E F$ is parallel to $B C$ . The line which passes through $F$ and is perpendicular to $C E$ at $N$ intersects $B I$ at $M$ . Let $D$ be the intersection between $A N$ and $B C$ . Prove that four points $M$ , $N$ , $D$ , $C$ both lies on a circle.
Let $x$ , $y$ be real numbers. Find the minimum value of the expression $P = sin 4 x (sin 4 y + cos 4 y + 9 8 cos 2 x \cdot sin 2 2 y) + cos 4 x .$ $P=\sin ^{4} x\left(\sin ^{4} y+\cos ^{4} y+\frac{9}{8} \cos ^{2} x \cdot \sin ^{2} 2 y\right)+\cos ^{4} x.$
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $f (2 f (x) + 2 y) = x + f (2 f (y) + x), \forall x, y \in R .$ $f(2 f(x)+2 y)=x+f(2 f(y)+x),\, \forall x, y \in \mathbb{R}.$
There are $n$ $(n \geq 2)$ soccer teams attending a tournament. Each team will play with all other teams once. The winning team get 3 points, the losing team gets 0 point; and if the match ties, both teams get 1 point. After the tournament, we recognize that all teams got different total points. What is the possible minimal value for the difference between the team with the most points and the team with the least points?
Given a triangle $A B C$ with $I$ is the center of the excircle relative to the vertex $A$ . This circle is tangent to $B C$ , $C A$ , $A B$ respectively at $M, N, P$ . Let $E$ be the intersection between $M N$ and $B I$ , and $F$ be the intersection between $M P$ and $CI$ . The line $B C$ intersects $A E$ , $A F$ respectively at $G$ , $D$ . Show that $A I$ is parallel to the line passing through $M$ and the center of the Euler circle of AGD
Compare the value of the expression $A=\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+\ldots+\frac{9999}{10000}$ $A=\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+\ldots+\frac{9999}{10000}$ with the numbers $98$ and $99$ .
Let $A B C$ be a triangle with $\widehat{A B C}=30^{\circ}$ , $\widehat{A C B}=20^{\circ}$ . The perpendicular bisector of $A C$ cuts $B C$ at $E$ and cuts the ray $B A$ at $F$ . Prove that $A F=E F$ and $A C=B E$ .
Find the intergers $x, y$ satisfying $\frac{x+2 y}{x^{2}+y^{2}}=\frac{7}{20}.$ $\frac{x+2 y}{x^{2}+y^{2}}=\frac{7}{20}.$
Prove that $\frac{a+b}{2} \geq \sqrt{a b}+\frac{(a-b)^{2}(3 a+b)(a+3 b)}{8(a+b)\left(a^{2}+6 a b+b^{2}\right)}$ $\frac{a+b}{2} \geq \sqrt{a b}+\frac{(a-b)^{2}(3 a+b)(a+3 b)}{8(a+b)\left(a^{2}+6 a b+b^{2}\right)}$ for positive numbers $a, b$ .
Solve the inequation $(x-1) \sqrt{x^{2}-2 x+5}-4 x \sqrt{x^{2}+1} \geq 2(x+1)$ $(x-1) \sqrt{x^{2}-2 x+5}-4 x \sqrt{x^{2}+1} \geq 2(x+1)$
On a line, take three distinct points $A$ , $B$ , $C$ in this order. Draw the tangents $A D$ , $A E$ to the circle with diameter $B C$ ( $D$ and $E$ are the touching points). Draw $D H \perp C E$ at $H$ . Let $P$ be the midpoint of $D H$ . The line $C P$ cuts again the circle at $Q$ . Prove that the circle passing through the three points $A$ , $D$ , $Q$ is tangent to the line $A C$ .
$M$ is a point on the incircle of triangle $A B C$ . Let $K$ , $H$ , $J$ be respectively the orthogonal projections of $M$ on the lines $A B$ , $B C$ , $C A$ . Determine the position of $M$ so that the sum $M K+M H+M J$ .
a) attains its greatest value
b) attains its least value
Determine all sequences of positive integers $\left(x_{n}\right)$ $(n=1,2,3, \ldots)$ satisfying $x_{1}=1,\, x_{2}>1,\quad x_{n+2}=\frac{1+x_{n+1}^{4}}{x_{n}},\,\forall n=1,2,3, \ldots$ $x_{1}=1,\, x_{2}>1,\quad x_{n+2}=\frac{1+x_{n+1}^{4}}{x_{n}},\,\forall n=1,2,3, \ldots$
Solve the system of equations $\begin{cases}\log _{2}(1+3 \cos x) &=\log _{3}(\sin y)+2 \\ \log _{2}(1+3 \sin y) &=\log _{3}(\cos x)+2\end{cases}$ $\begin{cases}\log _{2}(1+3 \cos x) &=\log _{3}(\sin y)+2 \\ \log _{2}(1+3 \sin y) &=\log _{3}(\cos x)+2\end{cases}$
Let $n$ be a given positive integer. Find the least number $t=t(n)$ such that for all real numbers $x_{1}, x_{2}, \ldots, x_{n}$ , holds the following inequality $\sum_{k=1}^{n}\left(x_{1}+x_{2}+\ldots+x_{k}\right)^{2} \leq t\left(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}\right)$ $\sum_{k=1}^{n}\left(x_{1}+x_{2}+\ldots+x_{k}\right)^{2} \leq t\left(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}\right)$
Prove that for arbitrary triangle $A B C$ , $\sin A \cos B+\sin B \cos C+\sin C \cos A \leq \frac{3 \sqrt{3}}{4}.$ $\sin A \cos B+\sin B \cos C+\sin C \cos A \leq \frac{3 \sqrt{3}}{4}.$ When does equality occur?
In space, let be given $n$ distinct points $A_{1}, A_{2}, \ldots, A_{n}$ $(n \geq 2)$ and $n$ points $K_{1}, K_{2}, \ldots, K_{n}$ ( $K_{i}$ does not coincide with $A_{i}$ for every $i=1,2, \ldots, n)$ . Prove that there exist $n$ spheres (with positive radii) $S_{i}$ $(i=1,2, \ldots, n)$ satisfying simultaneously the following two conditions
a) the spheres do not intersect each others,
b) the products $P_{A_{i} / S_{i}} \cdot P_{K_{i} / S_{i}}$ are negative numbers for all $i=1,2, \ldots, n$ , where $P$ denotes the power of a point with respect to a sphere.

PROBLEM OF THE WEEK | ЗАДАЧА НА НЕДЕЛАТА

PW 120. Consider the equation

x^{2} - 2 k x y^{2} + k (y^{3} - 1) = 0

where

k

is a positive integral parameter. Prove that this equation has integral solution

(x, y)

with

x > 0

y > 1

when and only when

k

is a perfect square.

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June 23 to 28
Tirana, Albania

July 2 to 13
Chiba, Japan