2008 Issue 369

  1. For each integer $n$ greater than $6,$ denote by $A_{n}$ the collection of integers which are less than $n$ and not less than $\dfrac{n}{2} .$ Find $n$ such that there are no perfect square in $A_{n}$.
  2. Let $A B C$ be an acute triangle with $\widehat{B A C}=60^{\circ} . E$ and $F$ are two points on $A C$ and $A B$ respectively such that $\widehat{E B C}=\widehat{F C B}=30^{\circ}$. Prove that $$B F=F E=E C \geq \dfrac{B C}{2}.$$
  3. Find four distinct integers $a, b, c, d$ in the set $\{10 ; 21 ; 37 ; 51\}$ such that $$a b+b c-a d=637.$$
  4. Solve for $x$ $$(x+3) \sqrt{(4-x)(12+x)}=28-x.$$
  5. Let $A B C$ be a triangle with $\widehat{B A C} \neq 45^{\circ}$ and $\widehat{A I O}=90^{\circ}$ where $(O)$ and $(I)$ are its circumcircle and incircle, respectively. Choose a point $D$ on the ray $B C$ such that $B D=A B+A C$. The tangent line through $D$ touches $(O)$ at $E$. The tangent line through $B$ of $(O)$ meets $D E$ at $F$; $C F$ meets $(O)$ at another point denoted by $K$. Let $G$ be the centroid of the triangle $A B C$. Prove that $I G$ is parallel to $E K$.
  6. Let $a_{1}, a_{2}, \ldots, a_{n}$ be positive integers such that the following two properties hold
    • $a_{i}<2008$ for all $i=1,2, \ldots, n$
    • The greatest common divisor of any pair of numbers is greater than $2008$. 
    Prove that $\displaystyle\sum_{i=1}^{n} \frac{1}{a_{i}}<2$.
  7. Prove the following inequality $$(3 a+2 b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \leq \frac{45}{2}$$ where $a, b,$ and $c$ are in the interval $[1 ; 2]$ When does equality occur?
  8. Given a triangle $A B C$ with circumcircle $(O)$ and three sides $B C=a$, $C A=b$, $A B^{\prime}=c .$ Denote by $A_{1}$, $B_{1},$ and $C_{1}$ the midpoints of $B C$, $C A,$ and $A B$ respectively; and $A_{2}$, $B_{2}$, $C_{2}$ are the midpoint of the arcs $\widehat{B C}$ (which does not contain $A$), $\widehat{C A}$ (which does not contains $B$), and $\widehat{A B}$ (which does not contain $C$). Draw the circles $\left(Q_{1}\right),\left(O_{2}\right),$ and $\left(O_{3}\right)$ whose diameters are $A_{1} A_{2}$, $B_{1} B_{2},$ and $C_{1} C_{2}$ respectively. Prove the inequality $$\mathcal{P}_{A/\left(O_{1}\right)}+\mathcal{P}_{B /\left(O_{2}\right)}+\mathcal{P}_{C /\left(O_{3}\right)} \geq \frac{(a+b+c)^{2}}{3}.$$ When does equality occur? 
  9. Find all positive integers $x, y, z, n$ such that $$x !+y !+z !=5 . n !$$ where $k !=1 \times 2 \times \ldots \times k$.
  10. Let $a$ and $b$ be two real numbers in the open interval $(0 ; 4) .$ A sequence $\left(a_{n}\right),$ $(n=0,1, \ldots)$ is constructed by the following recursive formula $$a_{0}=a,\, a_{1}=b,\quad a_{n+2}=\frac{2\left(a_{n+1}+a_{n}\right)}{\sqrt{a_{n+1}}+\sqrt{a_{n}}}.$$ Prove that the sequence $\left(a_{n}\right)$ (n=0,1, \ldots)$ converges and find its limit.
  11. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$f(f(x))+f(x)=\left(26^{3^{2008}}+\left(26^{32008}\right)^{2}\right) x.$$
  12. Let $S$ denote the surface area of a given tetrahedron. Prove that the sum of areas of the six angle-bisectors of this tetrahedron does not exceed $\dfrac{\sqrt{6}}{2} S$.




Mathematics & Youth: 2008 Issue 369
2008 Issue 369
Mathematics & Youth
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