2009 Issue 390

  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 $f(0)=0$ $f(1)=1 .$ 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 $B C$.




Mathematics & Youth: 2009 Issue 390
2009 Issue 390
Mathematics & Youth
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