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

## $show=home 1. Find all triples$(a, b, c)$of positive integers such that$(a+b+c)^{2}-2 a+2 b$is a perfect square. 2. Given a triangle$A B C$with$A B=A C$and$\widehat{B A C}=80^{\circ} .$Choose a point$I$inside the triangle so that$\widehat{I A C}=10^{\circ}$,$\widehat{I C A}=20^{\circ} .$Find the measure of the angle$\widehat{C B I}$. 3. Let$G$and$I$be respectively the centroid and the incenter of a given triangle$A B C .$Prove that if$A B^{2}-A C^{2}=2\left(I B^{2}-I C^{2}\right)$then$G I$is parallel to$B C$. 4. Solve the equation $$\left(x^{2}+1\right)\left|x^{2}+2 x-1\right|+6 x\left(1-x^{2}\right)=\left(x^{2}+1\right)^{2}$$ 5. Let$\left(a_{n}\right)$be a sequence given by $$a_{1}=1,\quad a_{n+1}=\sqrt{a_{n}\left(a_{n}+1\right)\left(a_{n}+2\right)\left(a_{n}+3\right)+2},\,\forall n \in \mathbb{N}^{*}.$$ Compare$\dfrac{1}{2}$with the sum $$S=\frac{1}{a_{1}+2}+\frac{1}{a_{2}+2}+\frac{1}{a_{3}+2}+\ldots+\frac{1}{a_{2009}+2}.$$ 6. Let$a, b, c$be positive real numbers such that$a+b+c=6 .$Prove that $$\frac{a}{\sqrt{b^{3}+1}}+\frac{b}{\sqrt{c^{3}+1}}+\frac{c}{\sqrt{a^{3}+1}} \geq 2$$ 7. Find all triangles whose inradius equal 3 and the side lengths form the first three terms of an arithmetic progression with common difference$d$distinct from$0 .$8. Let$A B C$be a triangle with$A B=3 R$,$B C=R \sqrt{7}, C A=2 R .$Let$M$be an arbitrary point on the spherical surface$(C ; R) .$Find the least value of$M A+2 M B$. 9. There are$294$people in a meeting. Those who are acquainted shake hands with each other. Knowing that if$A$shakes hands with$B$then one of them shakes hands at most 6 times. What is the greatest number of possible handshakes? 10. Given a positive integer$m$, find all functions$f: \mathbb{N} \rightarrow \mathbb{N}$such that for every$x, y \in \mathbb{N}$we have • If$f(x)=f(y)$then$x=y$•$f(f(f(\ldots)))=x+y$. Here,$f$appears$m$times on the left hand side. 11. Let$f(x)$be a continuous function on the closed interval$[0 ; 1],$and differentiable on the open interval$(0 ; 1)$such that$f(0)=0f(1)=1 .$Prove that for two arbitrary real numbers$k_{1}, k_{2},$there exist two distinct numbers$a, b$in the open interval$(0 ; 1)$such that $$\frac{k_{1}}{f(a)}+\frac{k_{2}}{f(b)}=k_{1}+k_{2}.$$ 12. Let$A B C$be a triangle with orthocenter$H$. Prove that the common tangent, distinct from$A H,$of the incircles of the triangles$A B H$and$A C H$passes through the midpoint of$B C$. ##$type=three$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: 2009 Issue 390
2009 Issue 390
Mathematics & Youth