2016 Issue 470

  1. Find all integers $x,y,z$ and $t$ such that $$38\left(xyzt+xy+xt+zt+1\right)=49\left(yzt+y+t\right).$$
  2. Given an isosceles triangle $ABC$ with the vertex angle $A$. Let $M$ be a point inside the triangle such that $\widehat{AMB}>\widehat{AMC}$, and $N$ is a point between $A$ and $M$. Prove that $\widehat{ANB}>\widehat{ANC}$.
  3. Find the maximal $k$ such that there exist $k$ positive integers which do not exceed 100 and have the property that any number among them cannot be a power of any remain one.
  4. Assume that $ABC$ be a triangle inscribed in the semicircle with the center $O$ and the diameter $BC$. Two tangent lines to the semicircle $\left(O\right)$ at $A$ and $B$ intersect at $D$. Prove that $DC$ goes through the midpoint $I$ of the altitude $AH$ of $ABC$.
  5. Find positive integer solutions of the equation $$xyz=x+2y+3z-5.$$
  6. Let $a,b$ and $c$ are real numbers such that $\left|a+b\right|+\left|b+c\right|+\left|c+a\right|=8$. Find the maximum and minimum values of the expression $P=a^{2}+b^{2}+c^{2}$.
  7. Solve the system of equations $$\begin{cases} x & =2^{1-y},\\ y & =2^{1-x}. \end{cases}$$
  8. Let $ABC$ be a triangle without any obtuse angle. Prove that $$\cos A\cos B+\cos B\cos C+\cos C\cos A\geq2\sqrt{\cos A\cos B\cos C}.$$
  9. Given $n$ non-negative real numbers $x_{i}$, $i=1,2,\ldots,n$ $\left(n\geq2\right)$ satisfying $x_{1}+x_{2}+\ldots+x_{n}=1$. Show that $$\frac{1}{n}\left(\frac{x_{1}}{1+x_{1}}+\frac{x_{2}}{1+x_{2}}+\ldots\frac{x_{n}}{1+x_{n}}\right)<\frac{x_{1}^{2}}{1+x_{1}^{2}}+\frac{x_{2}^{2}}{1+x_{2}^{2}}+\ldots\frac{x_{n}^{2}}{1+x_{n}^{2}}.$$
  10. A natural number is called $k$-\emph{success} if the sum of its digits equals to $k$. Let $a_{ik}$ be the number of all $k$-success numbers which have $i$ digits. Find the sum $\sum_{k=1}^{m}\sum_{i=1}^{n}a_{ik}$ where $k,i,m,n$ are positive integers.
  11. Find all triples $\left(x,y,p\right)$ of two non-negative integers $x,y$ and a prime number $p$ such that $p^{x}-y^{p}=1$.
  12. Given a triangle ABC and $H$ is a point inside the triangle such that $\widehat{ABH}=\widehat{ACH}$. A circle which goes through $B$ and $C$ intersects the circle with the diameter $AH$ at two different points $X$ and $Y$, $AH$ intersects $BC$ at $D$. Let $K$ be the perpendicular projection of $D$ on $XY$. Prove that $\widehat{BKD}=\widehat{CKD}$.




Mathematics & Youth: 2016 Issue 470
2016 Issue 470
Mathematics & Youth
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