2013 Issue 436

  1. Let \[n=1234567891011\ldots99100.\] Delete $100$ digits so that the remaining digits in the original order, is greatest possible.
  2. Find all integers $x,y,z$ such that \[(x-y)^{3}+3(y-z)^{2}+5|z-x|=35.\]
  3. For what values of $a$ is the numbers $a+\sqrt{15}$ and $\dfrac{1}{a}-\sqrt{15}$ are both intergers?.
  4. Solve the equation \[\sqrt{6-x}+\sqrt{2x+6}+\sqrt{6x-5}=x^{2}-2x-5.\]
  5. Let $(O)$ be a circle of diameter $AB$. Point $I$ is outside the circle, $IH$ is the perpendicular line to $AB$ through $I$ ($H$ lies between $O$ and $A$). $IA$, $IB$ meet $(O)$ at points $E$ and $F$ respectively; $EF$ meets $AB$ at $P$; $EH$ meets $(O)$ at the second point $M$; $PM$ intersects $(O)$ at the second point $N$. Let $K$ denote the midpoint of $EF$, $O'$ is the circumcenter of triangle $HMN$. Prove that $O'H||OK$.
  6. Prove that in any triangle $ABC$, the following inequality holds \[\left(\frac{h_{a}}{l_{a}}-\sin\frac{A}{2}\right)\left(\frac{h_{b}}{l_{b}}-\sin\frac{B}{2}\right)\left(\frac{h_{c}}{l_{c}}-\sin\frac{C}{2}\right)\leq\frac{r}{4R},\] where $h_{a},h_{b},h_{c}$ and $l_{a},l_{b},l_{c}$ are the length of the altitudes and the internal angle bisector from $A,B$ and $C$ respectively; any $R,r$ are its circumradius and inradius.
  7. Let $S.ABC$ be a triangular pyramid where $\widehat{BAC}=90^{0}$, $BC=2a$, $\widehat{ACB}=\alpha$. The plane $(SAB)$ is perpendicular to $(ABC)$. Find the volume of this pyramid, given that $SAB$ is an isosceles triangle at vertex $S$ and $SBC$ is a right triangle.
  8. Solve the system of equations \[\begin{cases}x^{8}y^{8}+y^{4} & =2x\\ 1+x & =x(1+y)\sqrt{xy} \end{cases}.\]
  9. Let $A=2013^{30n^{2}+4n+2013}$ where $n\in\mathbb{N}$. Let $X$ be the set of remainders when dividing $A$ by $21$ for some natural number $n$. Determine the set $X$.
  10. Let $(x_{n})$ be a sequence of natural numbers with \[x_{1}=2,\quad x_{n+1}=\left[\frac{3}{2}x_{n}\right],\forall n=1,2,3,\ldots,\] where $[x]$ denotes the greates integer not exceeding $x$. Prove that in the sequence $(x_{n})$, there are infinitely many odd numbers and infinitely many even numbers.
  11. Given a real number $x\geq1$. Find the limit \[\lim_{n\to\infty}(2\sqrt[n]{x}-1)^{n}.\]
  12. Let $S$ denote the area of a given triangle, and let $P$ be an arbitrary point. $A',B',C'$ are the midpoints of $BC$, $CA$ and $AB$ respectively; $h_{a},h_{b},h_{c}$ denote the corresponding altitudes. Prove that \[PA^{2}+PB^{2}+PC^{2}\geq\frac{4}{\sqrt{3}}S\max\left\{ \frac{PA+PA'}{h_{a}},\frac{PB+PB'}{h_{b}},\frac{PC+PC'}{h_{c}}\right\}.\]




Mathematics & Youth: 2013 Issue 436
2013 Issue 436
Mathematics & Youth
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