2011 Issue 403

  1. Given the sum $$S=\frac{5}{1.2 .3}+\frac{8}{2.3 .4}+\frac{11}{3.4 .5}+\ldots+\frac{6026}{2008.2009 .2010}.$$ Compare $S$ with $2$.
  2. Let $A B C$ be a triangle with $\widehat{B A C}=50^{\circ}$, $\widehat{A B C}=72^{\circ}$. Outside of the triangle $A B C,$ draw a triangle $B D C$ such that $\widehat{C B D}=28^{\circ}$; $\widehat{B C D}=22^{\circ}$. Find the measure of the angle $A D B$.
  3. Find all possible pair of integers $x$, $y$ satisfying the following condition $$x^{2}+y^{2}=(x-y)(x y+2)+9$$
  4. Given $a, b, c, d \in[0 ; 1]$ satisfies the following condition $$a+b+c+d=x+y+z+t=1.$$ Prove the inequality $$a x+b y+c z+d t \geq 54 a b c d.$$
  5. Let $A B C$ be a triangle with $\widehat{B A C}=45^{\circ}$. The attitudes $B D$ and $C E$ intersect at $H .$ Let $I$ be a midpoint of $D E .$ Prove that the line $H I$ goes through the centroid of the triangle $A B C$.
  6. Solve the equation $$\sqrt{x}+\sqrt[4]{x}+4 \sqrt{17-x}+8 \sqrt[4]{17-x}=34.$$
  7. A number is said to be an interesting number if it has $10$ digits, all are distinct, and is a multiple of $11111$. How many interesting numbers are there?
  8. Let $A_{1} A_{2} A_{3} \ldots A_{n}$ be a convex polygon $(n \geq 3)$ on the plane $(P)$ and let $S$ be a point outside $(P)$. Another plane $(\alpha)$ intersects the sides $S A_{1}, S A_{2}, \ldots, S A_{n}$ at $B_{1}, B_{2},\ldots, B_{n}$ respectively such that $$\frac{S A_{1}}{S B_{1}}+\frac{S A_{2}}{S B_{2}}+\ldots+\frac{S A_{n}}{S B_{n}}=a$$ where $a$ is a given positive number. Prove that such a plane $(\alpha)$ always contains a fixed point.
  9. Two circles $\omega_{1}$, $\omega_{2}$ intersect at points $A$, $B$. $C D$ is a common tangent line of $\omega_{1}$, $\omega_{2}$ $(C \in \omega_{1}, D \in \omega_{2}$) where point $B$ is closer to $C D$ than point $A$. $C B$ cuts $A D$ at $E$, $D B$ cuts $CA$ at $F$ and $E F$ cuts $A B$ at $N$. $K$ is the orthogonal projection of $N$ onto $C D$.
    a) Prove that $\widehat{C A B}=\widehat{D A K}$.
    b) Let $O$ be the circumcenter of the triangle $A C D$ and $H$ is the orthocenter of the triangle $K E F .$ Prove that $O$, $B$, $H$ are collinear
  10. Let $\left(x_{n}\right)$ be the sequence where $$x_{1}=5,\quad x_{n+1}=\frac{x_{n}^{2010}+3 x_{n}+16}{x_{n}^{2009}-x_{n}+11},\,n=1,2, \ldots$$ For each positive number $n,$ put $\displaystyle y_{n}=\sum_{i=1}^{n} \frac{1}{x_{i}^{2009}+7}$. Determine $\displaystyle \lim_{n\to\infty}y_{n}$.
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfy the following equation $$f(f(x-y))=f(x) \cdot f(y)+f(x)-f(y)-x y,\,\forall x, y \in \mathbb{R}.$$
  12. For each $n \in \mathbb{N}$, let $a_{n}$ be a number of bijections $f:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, n\}$ such that $f(f(k))=k$ for all $k \in\{1,2, \ldots, n\}$.
    a) Prove that $a_{n}$ is an even number for every $n \geq 2$.
    b) Prove that for every $n \geq 10$ and $n$ is divisible by $3$ then $a_{n}-a_{n-9}$ is divisible by $3$.




Mathematics & Youth: 2011 Issue 403
2011 Issue 403
Mathematics & Youth
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