2008 Issue 367

  1. Let $S$ be the sum of $100$ terms $$S=\frac{1}{1.1 .3}+\frac{1}{2.3 .5}+\frac{1}{3.5 .7}+\frac{1}{4.7 .9}+\ldots+\frac{1}{100.199 .201}.$$ Compare $S$ with $\dfrac{2}{3}$.
  2. Let $A B C$ be an isosceles triangle $(A B=A C)$ such that $\widehat{B A C}<90^{\circ} .$ Let $B D$ and $A H$ be the altitudes. Choose a point $K$ on $B D$ such that $B K=B A$ Find the measure of angle $HAK$.
  3. Given $0<b<a \leq 4$ and $2 a b \leq 3 a+4 b$. Find the maximum value of the expression $a^{2}+b^{2}$.
  4. The equation $$5 x^{6}-16 x^{4}-33 x^{3}-40 x^{2}+8=0$$ has two roots which are reciprocal. Find these roots.
  5. Let $A B C$ be a right triangle with right angle at $A$ and $A C>A B$. Let $O$ be the midpoint of $B C,$ and $I$ be the incenter of the triangle $A B C .$ Suppose that $\widehat{O I B}=90^{\circ},$ find the ratio between three edges of the triangle $A B C$.
  6. Find all pairs of positive integers $(x ; y)$ such that $$y^{x}-1=(y-1) !$$ where $y$ is a prime number.
  7. Let $a, b, c$ be non-negative real numbers. Prove the inequality $$\left(a^{2}+b^{2}+c^{2}\right)^{2} \geq 4(a-b)(b-c)(c-a)(a+b+c).$$ When does equality occur?
  8. Let $A B C$ be an isosceles triangle at vertex $A$ and $B C \leq A C$. Choose a point $M$ on $A B$ (but not the vertices $A$ or $B$) and a point $N$ on $A C$ such that $M N$ touch the incircle of the triangle $A B C$. Find the maximum value of the ratio $\dfrac{A M}{B M \cdot C N}$ when $M$ moves along the edge $A B$.
  9. Let $O$ be the circumcenter of a triangle $A B C .$ Choose a point $P,$ different from $B$ and $C$ on the line connecting $B C$. The circumcircle of $A B C$ meets $A P$ at a point $N$ and the circle whose diameter is $A P$ at a point $E$ ($N$, $E$ are both different from $A$). $B C$ and $A E$ intersect at $M .$ Prove that $M N$ always passes through a fixed point.
  10. A sequence $\left(u_{n}\right)(n=1,2, \ldots)$ is determined by the following recursive formula $$u_{1}=1,\quad u_{n+1}=\frac{16 u_{n}^{3}+27 u_{n}}{48 u_{n}^{2}+9}.$$ Find the largest integer which is smaller than the sum $S$ of $2008$ summands $$S=\frac{1}{4 u_{1}+3}+\frac{1}{4 u_{2}+3}+\ldots+\frac{1}{4 u_{2008}+3}$$
  11. Find the smallest value of the following expression $$P=\frac{1}{\cos ^{6} a}+\frac{1}{\cos ^{6} b}+\frac{1}{\cos ^{6} c}$$ where $a$, $b$ and $c$ form an arithmetic sequence whose common difference is $\dfrac{\pi}{3}$.
  12. Let $f(x)$ be a function defined on $[0 ; 1]$ such that the following properties hold
    • $f(1)=1$
    • $f(x)=\dfrac{1}{3}\left(f\left(\dfrac{x}{3}\right)+f\left(\dfrac{x+1}{3}\right)+f\left(\dfrac{x+2}{3}\right)\right)$ for all $x \in[0 ; 1]$
    • for every $\varepsilon$ positive but can be arbitrarily small, there exists a positive number $\delta_{\varepsilon}\left(\delta_{\varepsilon}\right.$ depends on $\varepsilon$) such that: For all $x, y \in[0 ; 1]$ such that $|x-y|<\delta_{\varepsilon},$ we have $|f(x)-f(y)|<\varepsilon$.
      Prove that $f(x)=1$ for all $x \in[0 ; 1]$.




    Mathematics & Youth: 2008 Issue 367
    2008 Issue 367
    Mathematics & Youth
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