2016 Issue 464

  1. Find all pairs of natural number $\left(m.n\right)$ satisfying $3^{m}-2^{n}=5$.
  2. Given a right triangle $ABC$ with the right angle $A$ and $AB<AC$. Let $AH$ be the altitude from the vertex $A$. On $AC$ choose $D$ such that $AD=AB$. Let $I$ be the midpoint of $BD$. Prove that $\widehat{BIH}=\widehat{ACB}$.
  3. Can we cover a square-shape area of the size $3,5{\rm m}\times3,5{\rm m}$ by rectangle-shape tiles of the size $25{\rm cm}\times100{\rm cm}$ without cutting any tile?.
  4. Construct a triangle $ABC$ given three lines $d_{a},d_{b}$ and $d_{c}$ which contain perpendicular bisector of $ABC$ (assume that they are concurrent at $O$) and given the length of $AH$ where $H$ is the orthocenter of $ABC$.
  5. Solve the equation $$2010-\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{2016-x}}}=x.$$
  6. Solve the system of equations $$\begin{cases} \frac{12y}{x} & =3+x-2\sqrt{4y-x}\\ \sqrt{y+3}+y & =x^{2}-x-3 \end{cases}.$$
  7. Let $x,y,z$ be real numbers such that $x^{2}+y^{2}+z^{2}=8$. Find the maximum and minimum values of the expression $$H=\left|x^{3}-y^{3}\right|+\left|y^{3}-z^{3}\right|+\left|z^{3}-x^{3}\right|.$$
  8. Given an acute triangle $ABC$ with the circumcenter $O$. Let $P$ be an arbitrary point on the circumcircle of the triangle $ABC$ and $P$ os different from $B$ and $C$. The bisector of the angles $\widehat{CPA}$ and $\widehat{APB}$ respectively intersects $CA$ and $AB$ at $E$ and $F$. Let $I$, $L$ and $K$ respectively be the incenters of the incircles of the triangles $PEF$, $PCA$ and $PAB$. Prove that $I$, $K$ and $L$ are colinear.
  9. Find positive integers $x,y$ such that $x^{3}+y^{3}=x^{2}+12xy+y^{2}$.
  10. Given $n$ real numbers $a_{1},a_{2},\ldots,a_{n}$ $\left(n\geq3\right)$ satisfying $$a_{1}+a_{2}+\ldots+a_{n}\geq n,\quad a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}\geq n^{2}.$$ Prove that $\max\left\{ a_{1},a_{2},\ldots a_{n}\right\} \geq2$.
  11. Consider the following sequence of real numbers $\left(a_{n}\right)$: $$\begin{cases} a_{1} & \geq 0 \\ a_{n+1} & =10^{n}a_{n}^{2},\quad n\geq1 \end{cases}.$$ Find all possible values for $a_{i}$ so that ${\displaystyle \lim_{n\to\infty}a_{n}=0}$.
  12. Given an acute triangle $ABC$. Let $E$ and $F$ respectively be the perpendicular projecions of $B$ and $C$ on $AC$ and $AB$. Let $I$ and $J$ respectively be the excenters of the excircles relative to the vertices $F$ and $E$ of the triangles $AFC$ and $AEB$. Assume that $BJ$ intersects $CI$ at $K$. Choose $Q$ on the circumcircle of the triangle $BKC$ such that circumcircle of the $BKC$ such that $\widehat{AQK}=90^{0}$. Prove that $AQ,BI$ and $CJ$ are concurrent.




Mathematics & Youth: 2016 Issue 464
2016 Issue 464
Mathematics & Youth
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