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2019 Issue 506

  1. Find all integers $x$, $y$, $z$ which satisfy $$3 x^{2}+6 y^{2}+2 z^{2}+3 y^{2} z^{2}-18 x=6.$$
  2. Given an isosceles triangle $A B C$ with the vertex angle $A$. Let $H$ be the point in the interior domain determined by the angle $A$ such that $H B \perp B A$, $H C \perp C A$. On the line segment $B C$ we choose $M$ such that $B M=\dfrac{1}{4} B C$. Let $N$ be the midpoint of $A C$ Calculate the angle $\widehat{H M N}$.
  3. Find all pairs of integers $(x ; y)$ satisfying $$y^{3}-2(x-4) y^{2}+\left(x^{2}-9 x-1\right) y+3 x^{2}+x=0.$$
  4. Given an acute triangle $A B C$ and suppose that $B E$ and $C F$ are the two altitudes. Draw $F H$ and $E K$ perpendicular to $B C$ $(H, K \in B C)$. Draw $H M$ parallel to $A C$ and $K N$ parallel to $A B$ $(M \in A B, N \in A C)$. Show that $E F \parallel M N$.
  5. Solve the equation $$\sqrt{\frac{x-1}{x+1}}+\frac{2 x+6}{(\sqrt{x-1}+\sqrt{x+3})^{2}}=2.$$
  6. Given two positive numbers $a$ and $b$ such that $a<b$ and $a^{b}=b^{a}$. Show that there exists a positve number $c$ such that $$a=\left(1+\frac{1}{c}\right)^{c},\quad b=\left(1+\frac{1}{c}\right)^{c+1}.$$
  7. Solve the system of equations $$\begin{cases}\tan x-\tan y &=(1+\sqrt{x+y})^{y}-(1+\sqrt{x+y})^{x} \\ 3^{\sqrt{1-x}}+5^{\sqrt{1-y}} &=2(1+\sqrt{9-10 x+y})\end{cases}.$$
  8. Show that for any triangle $A B C$ we always have $$\frac{(b+c) a}{m_{a}^{2}}+\frac{(c+a) b}{m_{b}^{2}}+\frac{(a+b) c}{m_{c}^{2}} \geq 8$$ where $a$, $b$, $c$, $m_{a}$, $m_{b}$, $m_{c}$ respectively are the lengths of the sides $B C$, $C A$, $A B$ and the corresponding medians.
  9. Let $a$, $b$, $c$ be positive numbers such that $a+b+c=3$. Find the minimum value of the expression $$M=\sqrt[3]{\frac{a^{5}}{b^{4}}}+\sqrt[3]{\frac{b^{5}}{c^{4}}}+\sqrt[3]{\frac{c^{5}}{a^{4}}}.$$
  10. Find all natural numbers $n$ so that $2^{n}+n^{2}+1$ is a perfect square.
  11. Given a strictly increasing sequence of positive integers $\left(a_{n}\right)$. Let $$S_{n}=\frac{\sqrt{a_{1}}}{\left[a_{1}, a_{2}\right]}+\frac{\sqrt{a_{2}}}{\left[a_{2}, a_{3}\right]}+\ldots+\frac{\sqrt{a_{n}}}{\left[a_{n}, a_{n+1}\right]},\, \forall n=1,2, \ldots$$ (for positive integers $x, y$ we denote $[x, y]$ the least common multiple (l.c.m.) of $x$ and $y$.) Show that the sequence $\left(S_{n}\right)$ has the finite limit when $n \rightarrow+\infty$.
  12. Given an acute triangle $A B C$ $(A B < A C)$. Two altitudes $B E$ and $C F$ intersect at $H$. Let $I$ be the center of the circle which passes through $A$, $B$ and is tangent to $B C$ and $J$ the center of the circle which passes through $B$, $H$ and is tangent to $B C$. Let $M$ be the midpoint of $A H$, and $S=E F \cap B C$ Show that $S M$ bisects $I J$.

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Mathematics & Youth: 2019 Issue 506
2019 Issue 506
Mathematics & Youth
https://www.molympiad.org/2020/09/2019-issue-506.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2019-issue-506.html
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