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2020 Issue 511

  1. Find all $6$-digit natural numbers which are both a perfect square and a cube.
  2. 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 $\widehat{B C D}$.
  3. 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) $7 b \leq 24 a$.
  4. 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 $E D$.
  5. Suppose that $x, y, z$ are positive numbers satisfying $x+y \leq 2 z$. Find the minimum value of the expression $$P=\frac{x}{y+z}+\frac{y}{x+z}-\frac{x+y}{2 z}.$$
  6. Show the inequality $$\left(\frac{x+y}{x-y}\right)^{2020}+\left(\frac{y+z}{y-z}\right)^{2020}+\left(\frac{z+x}{z-x}\right)^{2020}>\frac{2^{1010}}{3^{1009}}$$ where $x, y, z$ are different numbers.
  7. Solve the system of equations $$\begin{cases}x_{2} &=x_{1}^{3}-3 x_{1} \\ x_{3} &=x_{2}^{3}-3 x_{2} \\ \ldots  & \ldots \\ x_{2020} &=x_{2019}^{3}-3 x_{2019} \\ x_{1} &=x_{2020}^{3}-3 x_{2020}\end{cases}$$
  8. 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.
  9. Let $x$, $y$ be real numbers. Find the minimum value of the expression $$P=\sin ^{4} x\left(\sin ^{4} y+\cos ^{4} y+\frac{9}{8} \cos ^{2} x \cdot \sin ^{2} 2 y\right)+\cos ^{4} x.$$
  10. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(2 f(x)+2 y)=x+f(2 f(y)+x),\, \forall x, y \in \mathbb{R}.$$
  11. 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?
  12. 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 $A G D$

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Mathematics & Youth: 2020 Issue 511
2020 Issue 511
Mathematics & Youth
https://www.molympiad.org/2020/09/2020-issue-511.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2020-issue-511.html
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