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2020 Issue 520

  1. Find all prime numbers $p$ so that $2^{p}+p^{2}$ is a prime.
  2. Given an acute triangle $A B C$. Let $D$ be the point which is equidistant to $3$ vertices of the triangle. Suppose that $E$, $F$ respectively are the midpoints of $B C$, $A C$. Let $G$ be the perpendicular projection of $E$ on $A B,$ then call $J$ the midpoint of $D E .$ Show that the triangle $B G J$ is isosceles.
  3. Solve the following system of equations $$\begin{cases}x^{3}+y &=2 \\ y^{3}+z &=2 \\ z^{3}+t&=2 \\ t^{3}+x&=2\end{cases}.$$
  4. Given a quadrilateral $A B C D$ with $\widehat{A B D}=\widehat{A C D}=90^{\circ} .$ Draw $B H \perp A D$ at $H$. On the diangonal $A C$ we choose the point $I$ so that $A I=A B .$ Let $O$ be the midpoint of $A D .$ The line perpendicular to $O I$ at $I$ intersects $B H$ and $C D$ at $E$ and $F$ respectively. Show that $I F=2 I E$.
  5. Solve the equation $$x+\frac{2 x \sqrt{6}}{\sqrt{x^{2}+1}}=1$$
  6. Solve the following system of equations $$\begin{cases}\sqrt{x}+\sqrt{y}+\sqrt{z}+1&=4 \sqrt{x y z} \\ \dfrac{1}{2 \sqrt{x}+1}+\dfrac{1}{2 \sqrt{y}+1}+\dfrac{1}{2 \sqrt{z}+1}&=\dfrac{3 \sqrt{x y z}}{x+y+z}\end{cases}.$$
  7. Given non-zero numbers $a, b, c$ $d$ whose sum is 4 and each of them is greater or equal to $-2 .$ Show that $$\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{d^{2}} \geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$$
  8. Given a triangle $A B C$ with $B C=a$, $C A=b$, $A B=c .$ Prove that $$\frac{b}{a^{2} c^{2}}\left[a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b)\right] \geq \frac{6 \sin ^{3} B}{\cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}}.$$ When does the equality happen?
  9. Solve the equation $$\sum_{r=1}^{\infty}(-1)^{-1} \frac{x(x-1) \ldots(x-r+1)}{(x+1) \ldots(x+r)}=\frac{1}{2}$$ where $n$ is a given positive integer.
  10. The sequence $\left(u_{e}\right)$ is determined as follows $$ u_1=3,\quad u_n = 4u_{n-1}+3 n^{2}-12 n^{3}+12 n-4 ,\,\forall n=2,3, \ldots$$ Show that for any odd prime number $p$, $\displaystyle 2019 \sum_{i=1}^{n-1} u_{i}$ is always divisble by $p$.
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(f(x)-2 y)=6 x+f(f(y)+x),\, \forall x, y \in \mathbb{R}$$
  12. Given a triangle $A B C$ and let $(O)$ be it circumcircle. Let $A_{0}$, $B_{0}$, $C_{0}$ respectively be the midpoints of $B C$, $C A$, $A B$. Assume that $A_{1}$, $B_{1}$, $C_{1}$ respectively are the perpendicular projections of $A$, $B$, $C$. Let $(O_a)$ be the circle passing through $B_0$, $C_{0}$ and is internally tangent to $(O)$ at $A_{2}$ which is different from $A$; $\left(O_{b}\right)$ the circle passing through $C_{0}$, $A_{0}$ and is internally tangent to $(O)$ at $B,$ which is different from $B$; and $\left(O_{c}\right)$ the circle passing through $A_{0}$, $B_{0}$ and is internally tangent to $(O)$ at $C_{2}$ which is different from $C$. Let $A_{3}$ be the intersection between $B_{1} C_{1}$ and $B_{2} C_{2}$, and similarly we get the points $B_{3}$, $C_{3}$. Show that $A_{3}, B_{3}$, $C_{3}$ belong to a line which is perpendicular to the Euler line of the triangle $A B C$.

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Mathematics & Youth: 2020 Issue 520
2020 Issue 520
Mathematics & Youth
https://www.molympiad.org/2020/12/2020-issue-520.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/12/2020-issue-520.html
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