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2020 Issue 518

  1. Find prime numbers $x$, $y$, $z$ which satisfy the equality $$x^{5}+y^{3}-(x+y)^{2}=3 z^{3}$$
  2. Given a triangle $A B C$. Let $M$ be the point on the side $A B$ so that $M B=\dfrac{1}{4} A B,$ and $I$ the point on the side $B C$ so that $I C=\dfrac{3}{8} B C$. The point $N$ is the reflection of $A$ in the point $I$. Show that $M N \parallel A C$.
  3. Solve the equation $$\sqrt[3]{x+2020}+\sqrt[3]{x+2021}+\sqrt[3]{x+2022}=0$$
  4. Given an acute triangle $A B C$ $(A B>B C)$ inscribed in a circle $(O)$. Let $A D$ and $C E$ be two altitudes of the triangle $A B C$. Let $I$ be the midpoint of $D E$. The ray $A I$ intersects $(O)$ at $K$ $(K \neq A)$. Show that the circumcenter of the triangle $I D K$ lies on $B D$.
  5. Find the minimum value of the expression $$\frac{a^{3}}{1-a^{2}}+\frac{b^{3}}{1-b^{2}}+\frac{c^{3}}{1-c^{2}}$$ where $a, b, c$ are positive numbers satisfying $a+b+c=1$.
  6. Solve the system of equations $$\begin{cases}\dfrac{1}{\sqrt{4 x^{2}+8 x+5}}+\dfrac{1}{\sqrt{4 y^{2}-8 y+5}} &= \dfrac{2}{\sqrt{(x+y)^{2}+1}} \\ \dfrac{1}{\sqrt{x-1}} +\dfrac{1}{\sqrt{y-3}} &= \dfrac{2 \sqrt{5}}{5}\end{cases}.$$
  7. Given real numbers $a$, $b$, $c$ satisfying $a+b+c=1$. Show that $$8 a b c-8 \leq(a b+b c+c a+1)^{2}$$
  8. Given a triangle $ABC$ inscribed in a circle $(O)$. Let $H$ be the orthocenter of the triangle. Let $X$, $Y$, $Z$ respectively be the second intersection between the circles $(A O H)$, $(B O H)$, $(C O H)$ with $(O)$. Show that $H$ is the incenter of the triangle $X Y Z$. (The notation $(U V W)$ denotes the circumcircle of the triangle $U V W$).
  9. Find all real numbers $x$, $y$, $z$ such that $$2^{x^{2}-3 y+z}+2^{y^{2}-3 z+x}+2^{z^{2}-3 x+y}=\frac{3}{2}$$
  10. The integers from $51$ to $150$ are aranged in a chess board of the size $10 \times 10$. Does there exist an arrangment so that for any pair of numbers $(a ; b)$ which are adjacent in a row or in a column at least one of the following equations $$x^{2}-a x+b=0 \quad \text{and} \quad x^{2}-b x+a=0$$ has an integral solution?
  11. A sequence $\left(a_{n}\right)$ $\left(n \in \mathbb{N}^{*}\right)$ is determined as follows $$a_{1}=1,\, a_{2}=2,\quad a_{n+2}=2 a_{n+1}-p a_{n},\, n \in \mathbb{N}^{*}$$ where $p$ is some prime number. Find all possible values for $p$ so that there exists a positive integer $m$ satisfying $a_{m}=-3$.
  12. Given a non-isosceles triangle $A B C$ and let $I$, $J$ respectively be the center of the incircle of $A B C$ and the excircle corresponding to the angle $A$. Let $D$ be the second intersection between the circle with the diameter $A I$ and the circumcircle of $A B C$. Assume that $E$ is the intersection between $A I$ and $B C$ and $P$ is the midpoint of the arc $B A C$. Show that the intersect between $P J$ and $D E$ lies on the circumcircle of $A B C$.

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Mathematics & Youth: 2020 Issue 518
2020 Issue 518
Mathematics & Youth
https://www.molympiad.org/2020/09/2020-issue-518.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2020-issue-518.html
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