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2020 Issue 512

  1. Find all natural numbers $N$ so that the sum of its factors is equal to $2 N$ and the product of its factors is equal to $N^{2}$.
  2. Given natural numbers $a, b, c$ such that $\dfrac{a}{3}=\dfrac{b}{5}=\dfrac{c}{7}$. Prove that $$\frac{2019 b-2020 a}{2019 c-2020 b}>1.$$
  3. Solve the system of equations $$\begin{cases} x^{2} &=2 z-1 \\ y^{2} &=x z \\ z^{2} &=2 y-1\end{cases}.$$
  4. Given an acute triangle $A B C$. Draw the altitudes $C H$, $B K$ ($H$, $K$ is respectively on $A B$ and $A C$). Choose two points $P$ and $Q$ on the ray $C H$ and the ray $B K$ respectively such that $\widehat{P A Q}=90^{\circ}$. Draw $A M$ perpendicular to $P Q$ ($M$ is on $P Q$). Show that $M B$ is perpendicular to $M C$.
  5. Let $a, b, c$ be positive numbers satisfying $a b+b c+c a=8$. Find the minimum value of the expression $$P=3\left(a^{2}+b^{2}+c^{2}\right)+\frac{27(a+b)(b+c)(c+a)}{(a+b+c)^{3}}.$$
  6. Let $x, y, z$ be positive numbers so that $x \geq z$. Find the minimum value of the expression $$P=\frac{x z}{y^{2}+y z}+\frac{y^{2}}{x z+y z}+\frac{x+2 z}{x+z}.$$
  7. Find the integral solutions of the equation $$\tan \frac{3 \pi}{x}+4 \sin \frac{2 \pi}{x}=\sqrt{x}.$$
  8. Given a triangle $A B C$ inscribed in a circle $(O)$ and $(I)$ is the incircle of the triangle. Let $M$ be the midpoint of $B C$ and $X$ the midpoint of the arc $\widehat{B A C}$ of $(O)$. Let $P$, $Q$ respectively be the perpendicular projections of $M$ on $C I$, $B I$. Show that $X I \perp P Q$.
  9. Given a triangle $A B C$ with area $S$ and $B C=a$, $C A=b$, $A B=c$. Solve the system of equations (variables $x, y, z$) $$\begin{cases}a^{2} x+b^{2} y+c^{2} z &=4 S \\ x y+y z+z x &=1\end{cases}.$$
  10. For any positive integer $n$ show that $n$ and $2^{2^{n}}+1$ are coprime.
  11. The sequences $\left(x_{n}\right)$, $\left(y_{n}\right)$ are determined as follows $$x_{1}=3,\, x_{2}=17,\quad x_{n+2}=6 x_{n+1}-x_{n},\,\forall n \in \mathbb{N}^*,$$ $$y_{1}=4,\, y_{2}=24,\quad y_{n+2}=6 y_{n+1}-y_{n},\,\forall n \in \mathbb{N}^*.$$ Show that no term in these sequences $\left(x_{n}\right)$, $\left(y_{n}\right)$ is a cube number.
  12. Given a right triangle $A B C$, with the right angle $A$, inscribed in a circle $(O)$. The point $A^{\prime}$ is the reflection point of $A$ over $0$. The point $P$ is the perpendicular projection of $A^{\prime}$ on the perpendicular bisector of $B C$. Let $H_{a}$, $H_{b}$, $H_{c}$ respectively be the orthocenter of $A P A^{\prime}$, $B P A^{\prime}$, $C P A^{\prime}$. Show that the circle $(H_aH_bH_c)$ is tangent to the circle $(O)$.

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Mathematics & Youth: 2020 Issue 512
2020 Issue 512
Mathematics & Youth
https://www.molympiad.org/2020/09/2020-issue-512.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2020-issue-512.html
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