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2016 Issue 463

  1. Let $a_{1},a_{2},\ldots,a_{1964}$ be integers such that $$a_{1964}^{2}+a_{1963}^{2}=a_{1962}^{2}-a_{1961}^{2}+a_{1960}^{2}-a_{1959}^{2}+\ldots+a_{2}^{2}-a_{1}^{2}.$$ Prove that $a_{1}\cdot a_{2}\cdots a_{1964}+2015$ can be written as a difference of two perfect squares.
  2. Let $ABC$ be a triangle with its side $BC$ is fixed and its vertex $A$ varies such that the triangle is not isosceles at $A$. Draw the internal angle bisector $AD$ of the triangle. On the ray $CA$ choose $E$ such that $CE=AB$. Let $I$ be the midpoint of $AE$. Prove that the line which passes through a fixed point.
  3. Find all natural numbers $x,y$ such that $$x^{5}=y^{5}+10y^{3}+20y+1.$$
  4. Given a triangle $PQR$ inscribed in a circle $\left(O\right)$. Let $S$ be the midpoint of the arc $QR$ which does not contain $P$. Draw a circle with center $O'$ passing through $P$ and $S$. This circle intersects $PQ$ and $PR$ respectively at $M$ and $N$. Let $H$ and $K$ respectively be the midpoint of $MN$ and $QR$. Prove that $HK$ is perpendicular to $PS$.
  5. Solve the system of equations $$\begin{cases}x^{3}-x^{2}+x\left(y^{2}+1\right) &=y^{2}-y+1 \\ 2y^{3}+12y^{2}+18y-2+z&=0 \\ 3z^{3}-9z+x-7  &=0\end{cases}$$ where $x,y,z\in\mathbb{R}$.
  6. Given positive real numbers $x,y$ and $z$ such that $$\left(x+y\right)\left(y+z\right)\left(z+x\right)=1.$$ Show that $$\frac{\sqrt{x^{2}+xy+y^{2}}}{\sqrt{xy}+1}+\frac{\sqrt{y^{2}+yz+z^{2}}}{\sqrt{yz}+1}+\frac{\sqrt{z^{2}+zx+x^{2}}}{\sqrt{zx}+1}\geq\sqrt{3}.$$
  7. Solve the equation $$\left(\log\left(x^{2}\left(2-x\right)\right)\right)^{3}+\left(\log x\right)^{2}\cdot\log\left(x\left(2-x\right)^{2}\right)=0.$$
  8. Let $d_{a},d_{b},f_{c}$ respectively be the distances from the orthocenter $H$ of an acute triangle $ABC$ to the sides $BC,CA,AB$. Let $r$ and $R$ respectively be the inradius and circumradius of $ABC$. Prove that $$
    d_{a}+d_{b}+d_{c}\leq\frac{3}{4}\cdot\frac{R^{2}}{r}.$$
  9. Solve the system of equations $$\begin{cases}x\left(x-2\right)+\left(y-2\right)\left(2z+1\right)&=0 \\ x\left(y+1\right)+\left(y-2\right)\left(5z+1\right)&=0 \\ \sqrt{\left(y+1\right)^{2}+\left(5z+1\right)^{2}}&=2\sqrt{\left(x-2\right)^{2}+\left(2z+1\right)^{2}}\end{cases}$$
  10. Find the maximal number $m$ such that the following inequality holds for all non-negative real numbers $a,b,c,d$ $$\left(ab+cd\right)^{2}+\left(ac+bd\right)^{2}+\left(ad+bc\right)^{2}\geq ma\left(b+c\right)\left(b+d\right)\left(c+d\right).$$
  11. Given natural numbers $m,n>2$. Prove that the number $\dfrac{m^{2^{n}-1}-1}{m-1}$ always has a divisor of the form $m^{a}+1$ where $a$ is a natural number.
  12. Given an acute triangle $ABC$ with the circumcenter $O$. Let $D,E$ and $F$ respectively be the midpoints of $BC,CA$ and $AB$. Choose a point $M$ on the line through $BC$. Let $N$ be the intersection between $AM$ and $EF$ and $P$ the second intersection between $ON$ and the circumcircle of the triangle $ODM$. Let $Q$ be the reflection point of $M$ through the midpoint of $DP$. Prove that $Q$ belongs to the circumcircle of the triangle $DEF$.

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Mathematics & Youth: 2016 Issue 463
2016 Issue 463
Mathematics & Youth
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_91.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_91.html
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