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2019 Issue 499

  1. Find positive integers $x, y$ such that $x^{y}+y^{x}=100$
  2. Given an acute triangle $A B C$. Outside the triangle, we construct two isosceles right triangles with the right angles $A A B D$ and $A C E .$ Let $J$ and $K$ respectively be the perpendicular projections of $D$ and $E$ on $B C$. Prove that $A J K$ is an isosceles right triangle. 
  3. Solve the equation $$\frac{1}{\sqrt{3 x^{2}+x^{3}}}+2 \sqrt{\frac{x}{3 x+1}}=\frac{3}{2}.$$
  4. Given a right triangle $A B C$ with the right angle $A$ and $A C=2 A B$. Let $A H$ be an altitude of $A B C$. On the opposite ray of $A H$ choose the point $K$ such that $A H=2 A K$. Find the sum $\widehat{A K B}+\widehat{H A B}$.
  5. Given real numbers $x, y$ satisfying $y-2 x+4<0$. Find the minimum value of the expression $$P=x^{2}-4 y+\frac{4\left(x^{2}-4 y\right)}{(y-2 x+4)^{2}}.$$
  6. Find all positive numbers $x, y, z$ satisfying $$\begin{cases}x^{2}+y^{2}+z^{2}+x y z &=4 \\ \left(\dfrac{1}{x^{9}}+\dfrac{1}{y^{9}}+\dfrac{1}{z^{9}}\right)(1+2 x y z) &=9\end{cases}.$$
  7. Given positive numbers $a, b, c$ such that $a^{3}+b^{3}+c^{3}=3$. Find the minimum value of the expression $$M=\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{a}.$$
  8. Let $A B C$ be inscribed in a circle $(O)$ and assume that $I$ is the incenter of $A B C$. Assume that $M$ is the midpoint of $A I$. Choose $K$ on $B C$ such that $I K \perp I O$. Choose $Q$ on $(O)$ such that $A Q \parallel I K$. Suppose that $Q M$ intersects $B C$ at $H$. Choose $G$ on $A K$ such that $H G \parallel Q I$. $Q K$ intersects $(O)$ at $L$ other than $Q$. Show that $\widehat{G I K}=\widehat{I A L}$.
  9. For each natural number $n$, let $$A_{n}=\left[\frac{n+3}{4}\right]+\left[\frac{n+5}{4}\right]+\left[\frac{n}{2}\right]$$ (the notation $[x]$ denotes the maximal integer which does not exceed $x$). Find all natural numbers $n$ such that $\dfrac{n^{6}-1}{A_{n}}$ is a perfect square.
  10. We call the trace of a triangle the ratio between the length of its shortest side and the length of its longest side. Given a square with the side equals 2019 length units. Does there exist a positive integer $n$ and a decomposition of this square into $n$ triangles satisfying both following conditions
    • Any side of any of these triangles is less than and equal the side of the square,
    • The sum of all the traces of these $n$ triangles does not exceed $\dfrac{n^{2}+6 n-9}{n^{2}}$.
  11. The sequence $\left\{u_{n}\right\}_{n \in \mathbb{N}^{*}}$ is determined as follows $$u_{1}=1,\quad u_{n+1}=\sqrt{n(n+1)}\frac{u_{n}}{u_{n}^{2}+n},\, \forall n=1,2 \ldots.$$ a) Find $\displaystyle \lim _{n \rightarrow+\infty} u_{n}$.
    b) Find $\left[\dfrac{\sqrt{2018}}{u_{2018}}\right]$ where $[x]$ denotes the maximal integer which does not exceed $x$. 
  12. Given an acute triangle $A B C$. The points $M$, $N$ are on the line segment $B C$ and the points $P$, $Q$ respectively are on the line segments $CA$, $A B$ such that $MNPQ$ is a square. The incircles of $A P Q$, $B Q M$, $C P N$ are respectively tangent to $P Q$, $Q M$, $P N$ at $X$, $Y$, $Z$. Prove that $A X \perp Y Z$.

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Mathematics & Youth: 2019 Issue 499
2019 Issue 499
Mathematics & Youth
https://www.molympiad.org/2020/09/2019-issue-499.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2019-issue-499.html
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