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2018 Issue 498

  1. Let $$A=1^{2016}+2^{2016}+3^{2016}+\ldots+2015^{2016}+2016^{2016}.$$ Show that $A$ is not a perfect square.
  2. Find all natural numbers $k$, $m$, $n$ so that $2 . k !=m !-2 . n !$ where $n !=1.2 .3 \ldots n$, $0 !=1$.
  3. Find integral solutions $(x ; y)$ of the equation $$\left(y-\sqrt{y^{2}+2}\right)\sqrt{x}+\sqrt{4+2 x}=2$$
  4. Given a triangle $A B C$ with the angles satisfying $\hat{A}: \widehat{B}: \widehat{C}=8: 3: 1 .$ Let $A D$ be the angle bisector of the angle $A(D$ is on $B C)$. Assume furthermore that $$\frac{1}{B D^{2}}+\frac{1}{B C^{2}}=\frac{4}{3}.$$ Compute the length of $A D$. 
  5. Solve the equation $$2 x^{3}+20=9 x \sqrt[3]{x^{3}-7}$$
  6. Solve the system of equations $$\begin{cases}\sqrt{2\left(x^{2}+4 y^{2}-8\right)} &=\left(\sqrt{y^{2}+x-3}+1-y\right) \left(\sqrt{y^{2}+x-3}+1+y\right) \\ 2 \sqrt[4]{y^{4}+5} &=x  \end{cases}$$
  7. Solve the equation $$\sqrt{1+2 \log _{16} x^{2}}+\sqrt{4-\frac{3}{4} \log _{8} x^{4}} +\frac{\log _{2} x^{2}-3}{\log _{2}^{2} x-\frac{2}{3} \log _{2} x^{3}+2}=0$$
  8. Given a triangle $A B C$ $(A B<A C)$, inscribed in a given circle $(O)$ with the point $A$ can be varied and the points $B$, $C$ are fixed, and two points $A$ and $O$ are always on the same side determined by $B C$. A circle $\left(O^{\prime}\right)$ is internally tangent to $(O)$ at $T$ ($T$ is outside the triangle $A B C$) and is tangent to the sides $A B$, $A C$ respectively at $P$, $Q$. The line $P Q$ intersects $B C$ at $R$. The lines $T B$, $T C$ meet again $\left(O^{\prime}\right)$ respectively at $E$, $F$ $(E \neq T, F \neq T)$. Prove that
    a) The line $E F$ is parallel to the line $B C$.
    b) The line $R T$ always passes through a fixed point when $A$ varies. 
  9. Let $a, b, c$ be positive numbers such that $a+b+c+2=2 a b c$. Prove that $$\frac{a+2}{\sqrt{6\left(a^{2}+2\right)}}+\frac{b+2}{\sqrt{6\left(b^{2}+2\right)}}+\frac{c+2}{\sqrt{6\left(c^{2}+2\right)}} \leq 2$$
  10. a) Given rational numbers $a_{1}, a_{2}, \ldots, a_{n}$ $\left(n \in \mathbb{Z}^{+}\right) $. Show that if, for any positive integer $m$, $a_{1}^{m}+a_{2}^{m}+\ldots+a_{n}^{m}$ is interger then $a_{1}, a_{2}, \ldots, a_{n}$ are integers.
    b) Is the above conclusion still correct if we only assume that $a_{1}, a_{2}, \ldots, a_{n}$ are real?
  11. People come and sit on the bench, one by one. The first person can choose any seat. The next people firstly try to avoid to sit next to the previous ones. If they cannot avoid, then they can choose any empty seat. How many such arrangements are there?
  12. Given a triangle $A B C$. Let $m_{a}$, $m_{b}$, $m_{c}$ respectively be the lengths of the medians corresponding to the sides $B C=a$, $A C=b$, $A B=c$. Show that $$(a b+b c+c a)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geq 2 \sqrt{3}\left(m_{a}+m_{b}+m_{c}\right).$$ When does the equality hold?

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Mathematics & Youth: 2018 Issue 498
2018 Issue 498
Mathematics & Youth
https://www.molympiad.org/2020/09/2018-issue-498.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2018-issue-498.html
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