2007 Issue 362

  1. Let $$S=\frac{1}{2^{2}}+\frac{2}{2^{3}}+\ldots+\frac{n}{2^{n+1}}+\ldots+\frac{2006}{2^{2007}}.$$ Compare $S$ with $1 .$
  2. Let $A H$ be the altitude of a right triangle $A B C,$ right angle at $A .$ Let $P$ and $Q$ be the incenters of $A B H$ and $A C H$ respectively. $P Q$ meets with $A B$ at $E$ and with $A C$ at $F .$ Prove that $A E=A F$.
  3. Find all possible pairs of positive integers $(x ; y)$ such that $4 x^{2}+6 x+3$ is a multiple of $2 x y-1$.
  4. Solve the following equation $$\sqrt{2 x^{2}+4 x+7}=x^{4}+4 x^{3}+3 x^{2}-2 x-7.$$
  5. Find the maximum value of the following expression $$A=\frac{x}{x^{2}+y z}+\frac{y}{y^{2}+2 x}+\frac{z}{z^{2}+x y},$$ where $x, y, z$ are positive real numbers such that $x^{2}+y^{2}+z^{2}=x y z$.
  6. $(O)$ is a point chosen arbitrarily inside a triangle $A B C$. $A O$, $B O$ and $C O$ meet $B C$, $C A$ and $A B$ at $M$, $N$ and $P$ respectively. Prove that the value of ratio $$\left(\frac{O A \cdot A P}{O P}\right)\left(\frac{O B \cdot B M}{O M}\right)\left(\frac{O C \cdot C N}{O N}\right)$$ does not depend on the position of the point $O$.
  7. Let $M$ be a point inside a circle $(O)$ with center at $O$ and radius $R .$ Draw two chords $C D$ and $E F$ through $M$ but not passing through $O$. The tangent lines with $(O)$ at $C$ and $D$ intersects at $A$, and the tangent lines at $E$ and $F$ meets at $B$. Prove that $O M$ and $A B$ are orthogonal.
  8. Let $p>2007$ be a prime number and $n$ is an integer exceeding $2006 p .$ Prove that $\mathrm{C}_{n}^{2006 \mathrm{p}}-\mathrm{C}_{k}^{2006}$ is a multiple of $p,$ where $k=\left[\dfrac{n}{p}\right]$ is the largest integer which is not larger than $\dfrac{n}{p}$
  9. Let $\left(u_{n}\right)$ be a sequence given by $u_{1}$ and the formula $u_{n+1}=\dfrac{k+u_{n}}{1-u_{n}},$ where $k>0$ where $n=1,2, \ldots .$ Given that $u_{13}=u_{1},$ find the value of $k$.
  10. Suppose $a, b,$ and $c$ are the three lengths of sides of a triangle. Prove that $$\frac{a}{\sqrt{a^{2}+3 b c}}+\frac{b}{\sqrt{b^{2}+3 c a}}+\frac{c}{\sqrt{c^{2}+3 a b}} \geq \frac{3}{2}$$
  11. Let $P$ be a arbitrary point in a quadrilateral $A B C D$ such that $\widehat{P A B}$, $\widehat{P B A}$, $\widehat{P B C}$, $\widehat{P C B}$, $\widehat{P C D}$, $\widehat{P D C}$, $\widehat{P A D}$ and $\widehat{P D A}$ are all acute angles. Let $M$, $N$, $K$, $L$ denote the feet of the altitude from $P$ on $A B$, $B C$, $C D$ and $D A$ respectively. Find the smallest value of the sum $$\frac{A B}{P M}+\frac{B C}{P N}+\frac{C D}{P K}+\frac{D A}{P L}$$
  12. Let $A B C . A^{\prime} B^{\prime} C^{\prime}$ be a triangular prism. $M$ and $N$ are two points, chosen on $A A^{\prime}$ and $C B$ respectively such that $$\frac{A M}{A A^{\prime}}=\frac{C N}{C B} .$$ Prove that the length of $M N$ is no less than the distance from $A^{\prime}$ to the straight line $C^{\prime} B$.




Mathematics & Youth: 2007 Issue 362
2007 Issue 362
Mathematics & Youth
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