2008 Issue 371

  1. Which number is greater? $A=\dfrac{1}{2006}$ or $$B =\frac{1}{2008}+\left(\frac{1}{2008}+\frac{1}{2008^{2}}\right)^{2}+\ldots +\left(\frac{1}{2008}+\frac{1}{2008^{2}}+\ldots+\frac{1}{2008^{2007}}\right)^{2007}.$$
  2. In a triangle $A B C$ with altitude $A D,$ one has $A D=D C=3 B D$. Let $O$ and $H$ be the circumcenter and the orthocenter, respectively. Prove that $\dfrac{O H}{B C}=\dfrac{1}{4}$.
  3. Find all pairs of natural numbers $x$ and $y$ such that $$x^{3}=y^{3}+2\left(x^{2}+y^{2}\right)+3 x y+17.$$
  4. Let $a, b, c$ be positive real numbers. Prove the inequality $$\frac{a^{2}-b^{2}}{\sqrt{b+c}}+\frac{b^{2}-c^{2}}{\sqrt{c+a}}+\frac{c^{2}-a^{2}}{\sqrt{a+b}} \geq 0.$$ When does equality occur?
  5. A circle $\left(S_{1}\right)$ passing through the vertices $A$ and $B$ of a triangle $A B C$ meets $B C$ at another point $D .$ Another circle, $\left(S_{2}\right),$ passing through $B$ and $C$ meets $A B$ at another point $E$ and meets $\left(S_{1}\right)$ at $F$. Prove that if all four points $A$, $C$, $D$ and $E$ lie on the same circle with center at $O$, then $\widehat{B F O}=90^{\circ}$.
  6. Solve the system of equations $$\begin{cases}\dfrac{x}{a-30}+\dfrac{y}{a-4}+\dfrac{z}{a-14}+\dfrac{t}{a-10} &=1 \\ \dfrac{x}{b-30}+\dfrac{y}{b-4}+\dfrac{z}{b-14}+\dfrac{t}{b-10} &=1 \\ \dfrac{x}{c-30}+\dfrac{y}{c-4}+\dfrac{z}{c-14}+\dfrac{t}{c-10} &=1 \\ \dfrac{x}{d-30}+\dfrac{y}{d-4}+\dfrac{z}{d-14}+\dfrac{1}{d-10} &=1\end{cases}$$ where $a, b, c, d$ are distinct numbers, none of which belong to the set $\{4 ; 10 ; 14 ; 30\} .$
  7. Let $a, b, c$ be positive numbers. Prove that $$a^{b+c}+b^{c+a}+c^{a+b} \geq 1$$
  8. Choose six points $D$, $E$, $F$, $G$, $H$, $K$ in that order, on a circle with radius $R$ and center at $O$ such that $D E=F G=H K=R$. $K D$ and $E F$ meet at $A$, $E F$ and $G H$ meet at $B$ and $G H$ meets $K D$ at $C$. Prove that $$OA \cdot B C=O B \cdot C A=O C \cdot A B.$$
  9. Let $A B C D$ be a cyclic quadrilateral, inscribed in a circle centered at $I .$ Prove the following inequality $$(A I+D I)^{2}+(B I+C I)^{2} \leq(A B+C D)^{2}$$ and determine when equality occurs.
  10. Consider the quadratic equation $x^{2}-a x-1=0$ where $a$ is a positive integer. Let $\alpha$ be a positive root of this equation and construct a sequence $\left(x_{n}\right)$ by the following recursive rule $$x_{0}=a, x_{n+1}=\left[\alpha x_{n}\right], n=0,1,2, \ldots$$ Prove that there exists infinitely many integer $n$ such that $x_{n}$ is a multiple of $a$. 
  11. Given a function $f: \mathrm{N} \rightarrow \mathrm{N}$ such that for all $n \in \mathbb{N} $ $$(f(2 n)+f(2 n+1)+1)(f(2 n+1)-f(2 n)-1)=3(1+2 f(n)),\quad f(2 n) \geq f(n).$$ Denote $M=\{m \in f(\mathrm{N}): m \leq 2008\} .$ Find the number of elements of $M$.
  12. For what kind of triangle $A B C$ that the following relation among its sides and its angles holds $$\frac{b c}{b+c}(1+\cos A)+\frac{c a}{c+a}(1+\cos B)+\frac{a b}{a+b}(1+\cos C) \\ = \frac{3}{16}(a+b+c)^{2}+\cos ^{2} A+\cos ^{2} B+\cos ^{2} C.$$




Mathematics & Youth: 2008 Issue 371
2008 Issue 371
Mathematics & Youth
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