# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 1. Find all numbers abcde, where all five digits are distinct and$\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 $$\begin{cases}\sqrt{5 x^{2}+2 x y+2 y^{2}}+\sqrt{2 x^{2}+2 x y+5 y^{2}} &=3(x+y) \\ \sqrt{2 x+y+1}+2 \sqrt{7 x+12 y+8} &=2 x y+y+5\end{cases}.$$ 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 $$\frac{2}{3} S_{A B C} \leq S_{M N P Q R S} \leq \frac{2}{3} S_{A^{\prime} B^{\prime} C^{\prime}}.$$ ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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