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2019 Issue 509

  1. Compare the following numbers $$A=\frac{1}{5}+\frac{2}{5^{2}}+\frac{3}{5^{3}}+\cdots+\frac{2018}{5^{2018}} ; \quad B=\frac{2018}{2019}$$
  2. Suppose that $P$ is a point inside a triangle $A B C$ so that $\widehat{P B C}=30^{\circ}$, $\widehat{P B A}=8^{\circ}$ and $\widehat{P A B}=\widehat{P A C}=22^{\circ}$. Find the value of the angle $\widehat{A P C}$.
  3. Find positive solutions of the equation $$\frac{1}{5 x^{2}-x+3}+\frac{1}{5 x^{2}+x+7} +\frac{1}{5 x^{2}+3 x+13}+\frac{1}{5 x^{2}+5 x+21}=\frac{4}{x^{2}+6 x+5}.$$
  4. Given a square $A B C D$ with the length of a side $a$. On the sides $A D$ and $C D$ respectively choose two points $M$, $N$ so that $M D+D N=a$. Let $E$ be the intersection of two lines $B N$ and $A D$. Let $F$ be the intersection of two lines $B M$ and $C D$. Show that $$M E^{2}-N E^{2}+N F^{2}-M F^{2}=2 a^{2}.$$
  5. Given positive numbers $a, b, c$ Find the minimum value of the expression $$P=\frac{a}{\sqrt[3]{a}+\sqrt[3]{b c}}+\frac{b}{\sqrt[3]{b}+\sqrt[3]{c a}}+\frac{c}{\sqrt[3]{c}+\sqrt[3]{a b}} + \frac{9 \sqrt[3]{(a+1)(b+1)(c+1)}}{4(a+b+c)}.$$
  6. Let $a, b, c, d$ be positive numbers such that $$\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c d}+\frac{1}{a d}=1 .$$ Show that $$\frac{a b c d}{8}+2 \geq \sqrt{(a+c)\left(\frac{1}{a}+\frac{1}{c}\right)}+\sqrt{(b+d)\left(\frac{1}{b}+\frac{1}{d}\right)}.$$
  7. Find real solutions of the following system of equations $$\begin{cases} x^{3}+2 y^{3} &=2 x^{2}+z^{2} \\ 2 x^{3}+3 x^{2} &=3 y^{3}+2 z^{2}+7 \\ x^{3}+x^{2}+y^{2}+2 x y &=2 x z+2 y z+2\end{cases}$$
  8. Given a triangle $A B C$ inscribed in a circle $(O)$. Assume that $B O$ and $C O$ intersect the altitude $A D$ of the triangle respectively at $E$ and $F$. Let $I$ and $J$ respectively be the centers of the circles $(A C F)$ and $(A B E)$. Two points $K$, $H$ are on $A B$, $A C$ respectively so that $J K \parallel A O \parallel  I H$. Suppose that $I J$ intersects $A B$ and $A C$ at $M$ and $N$. Show that the intersection between $M H$ and $N K$ is on the midsegment, which is opposite to the vertex $A$, of the triangle $A B C$.
  9. Solve the equation $$8^{x}+27^{\frac{1}{x}}+2^{x+1} \cdot 3^{\frac{x+1}{x}}+2^{x} \cdot 3^{\frac{2 x+1}{x}}=125.$$
  10. Let $[x]$ be the maximal integer which does not exceed $x$ and let $\{x\}=x-[x]$. Consider the sequence $\left(u_{n}\right)$ with $$u_{n}=\left\{\frac{2^{2 n+1}+n^{2}+n+2}{2^{n+1}+2}\right\}.$$ Find the number of terms of the sequence $\left(u_{n}\right)$ satisfying $$\frac{2526.2^{n-99}}{2^{n}+1} \leq u_{n} \leq \frac{23}{65}.$$
  11. Find all functions $f: \mathbb{N}^{*} \rightarrow \mathbb{R} \backslash\{0\}$ so that $$f(1)+f(2)+\cdots+f(n)=\frac{f(n) f(n+1)}{2}, \forall n \in \mathbb{N}^{*}.$$
  12. Given a triangle $A B C$ with $A B+A C=2 B C$. Let $I_{a}$ be the center of the excircle corresponding to the angle $A$. The circle $\left(A, A I_{a}\right)$ intersects $B C$ at $E$ and $F$ with $E$ is on the ray $C B$ and $F$ is on the ray $B C$. The circle $\left(E B I_{\alpha}\right)$ meets $A B$ at $M$ and the circle $\left(F C I_{\alpha}\right)$ meets $A C$ at $N$. Show that $B C N M$ is both a cyclic and tangential quadrilateral.

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Mathematics & Youth: 2019 Issue 509
2019 Issue 509
Mathematics & Youth
https://www.molympiad.org/2020/09/2019-issue-509.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2019-issue-509.html
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