2012 Issue 423

  1. Find all numbers abcde, where all five digits are distinct and $\overline{a b c d}=(5 e+1)^{2}$
  2. Find all positive integers $x, y, z$ such that $x+3=2^{y}$ và $3 x+1=4^{z}$
  3. Find the last digit of the sum $$S=1^{2}+2^{2}+3^{3}+\ldots+n^{n}+\ldots+2012^{2012}.$$
  4. Given a function $f$ such that $$f\left(1+\frac{\sqrt{2}}{x}\right)=\frac{(1+2011) x^{2}+2 \sqrt{2 x}+2}{x^{2}}$$ for all nonzero $x$. Determine $f(\sqrt{2012-\sqrt{2011}})$
  5. Let $A B C$ be a triangle inscribed in the circle $(O)$. The tangents of $(O)$ at $B$ and $C$ meet at $T$. The line passing through $T$ and parallel to $B C$ cuts $A B$ and $A C$ respectively at $B_{1}$ and $C_{1}$ Prove that $\widehat{B_{1} O C_{1}}$ is an acute angle.
  6. On the outside of triangle $A B C$, construct equilateral triangles $A B C_{1}$, $B C A_{1}$, $CAB_{1}$ and inside of $A B C$ construct equilateral triangles $A B C_{2}$, $B C A_{2}$, $C A B_{2}$. Let $G_{1}$, $G_{2}$, $G_{3}$ be respectively the centroids of $A B C_{1}$, $B C A_{1}$, $C A B_{1}$ and let $G_{4}$, $G_{5}$, $G_{6}$ be respectively the centroids of triangles $A B C_{2}$, $BCA_{2}$ and $CAB_{2}$. Prove that the centroids of triangle $G_{1} G_{2} G_{3}$ and of triangle $G_{4} G_{5} G_{6}$ coincide.
  7. Solve the equation $$3^{3 x}+3^{x}=\log _{3}\left(2^{x}+x\right)+2^{x}+3^{2^{x}+x}.$$
  8. Let $A$, $B$, $C$ be the three angles of an acute triangle. Prove the inequality $$\sqrt{\frac{\cos A \cos B}{\cos C}}+\sqrt{\frac{\cos B \cos C}{\cos A}}+\sqrt{\frac{\cos C \cos A}{\cos B}}>2.$$
  9. Find the largest positive integer $n$ $(n \geq 3)$ such that there exists a sequence of positive integers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying the condition $$a_{k+1}+1=\frac{a_{k}^{2}+1}{a_{k-1}+1},\, k \in\{2,3, \ldots, n-1\}.$$
  10. Let $p$ be an odd prime number, $n$ is a positive integer so that $p-1$, $p$, $n$ and $n+1$ are pairwise coprime. Find all positive integers $x$, $y$ satisfying $$x^{p-1}+x^{p-2}+\ldots+x+2=y^{n+1}.$$
  11. Solve the system of equations $$\begin{cases}\sqrt{5 x^{2}+2 x y+2 y^{2}}+\sqrt{2 x^{2}+2 x y+5 y^{2}} &=3(x+y) \\ \sqrt{2 x+y+1}+2 \sqrt[3]{7 x+12 y+8} &=2 x y+y+5\end{cases}.$$
  12. Let $A B C$ be a triangle inscribed in the circle $(O)$ and let $I$ be its incenter. $A I$, $B I$, $Cl$ cut the circle $(O)$ at $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$ respectively; $A^{\prime} C^{\prime}$, $A^{\prime} B^{\prime}$ cut $B C$ at $M$, $N$; $B^{\prime} A^{\prime}$; $B^{\prime} C^{\prime}$ cut $C A$ at $P$, $Q$; $C^{\prime} B^{\prime}$, $C^{\prime} A$ cut $A B$ at $R$, $S$. Prove that $$\frac{2}{3} S_{A B C} \leq S_{M N P Q R S} \leq \frac{2}{3} S_{A^{\prime} B^{\prime} C^{\prime}}.$$




Mathematics & Youth: 2012 Issue 423
2012 Issue 423
Mathematics & Youth
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