2008 Issue 377

  1. Let $$A=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots+\frac{1}{2007}+\frac{1}{2008},$$ $$B=\frac{2007}{1}+\frac{2006}{2}+\frac{2005}{3}+\ldots+\frac{2}{2006}+\frac{1}{2007}.$$ Determine $\dfrac{B}{A}$.
  2. Let $A B C$ be a right isosceles triangle with right angle at $A$. $M$ is an arbitrary point on the side $B C$ (M differs from $B$, $C$ as well as the midpoint of $B C$). The altitudes from $B$ and $C$ onto $A M$ meet $A M$ at $H$ and $K$, respectively. The line through $C$ and parallel to $A M$ meets $B H$ at $N$, $A N$ meets $C K$ at $P$, $B P$ intersects with $A M$ at $I$ Prove that $I B=I P$.
  3. Let $a, b, c, d$ and $e$ be natural numbers such that $$a^{4}+b^{4}+c^{4}+d^{4}+e^{4}=2009^{2008}.$$ Prove that abcde is a multiple of $10^{4}$.
  4. Let $a, b, c$ be positive real numbers such that $a \geq b \geq c .$ Prove the inequality $$a^{2} b(a-b)+b^{2} c(b-c)+c^{2} a(c-a) \geq 0.$$
  5. Let $A M$ and $B N$ be two tangent lines from two points $A$, $B$ ($A$ differs from $B$) outside the circle $(O)$ ($M$, $N$ are on the circle, $M$ and $N$ are different). Prove that if $A M=B N$, then the line $M N$ is either parallel to $A B$ or passes through its midpoint.
  6. A number is said to be a beautiful number if it is a composite number and but it is not a multiple of either $2$, $3$ or $5$ (for example, the three smallest beautiful numbers are $49,77$ and $91$). How many beautiful numbers which are less than $1000 ?$
  7. Let $D$ and $E$ be two points on the side $B C$ of a triangle $A B C$ such that $\dfrac{B D}{C D}=2\dfrac{C E}{B E}$. The circumcircle of $A D E$ meets $A B$ and $A C$ at $M$ and $N,$ respectively. Prove that regardless of the positions of the points $D$ and $E$ on $B C,$ the centroid of the triangle $A M N$ lies on a fixed line.
  8. Find the smallest real number $k$ such that the following inequality holds for all nonnegative real numbers $a, b, c$ $$\frac{a+b+c}{3} \leq \sqrt[3]{a b c}+k \cdot \max \{|a-b|, b-c|,| c-a \mid\}.$$
  9. Determine all triple of real numbers $x, y, z$ such that $$x^{6}+y^{6}+z^{6}-6\left(x^{4}+y^{4}+z^{4}\right)+10\left(x^{2}+y^{2}+z^{2}\right) - 2\left(x^{3} y+y^{3} z+z^{3} x\right)+6(x y+y z+z x)=0.$$
  10. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f\left(x^{3}-y\right)+2 y\left(3 f^{2}(x)+y^{2}\right)=f(y+f(x)),\, \forall x, y \in \mathbb{R}.$$
  11. Consider the sequence $\left(u_{n}\right)$ $(n=1,2,\ldots)$ given by the following recursive formula $$u_{1}=u_{2}=1,\quad u_{n+1}=4 u_{n}-5 u_{n-1},\,\forall n \geq 2.$$ Prove that $\displaystyle \lim_{n \rightarrow+\infty}\left(\frac{u_{n}}{a^{n}}\right)=0$ for all real number $a>\sqrt{5}$.
  12. Choose three points $A_{1}$, $B_{1}$, $C_{1}$ on the sides of a triangle $A B C$, $A_{1} \in B C$, $B_{1} \in A C$, $C_{1} \in A B$ such that $A A_{1}$, $B B_{1}$, $C C_{1}$ meet at a common point. Again, choose three points $A_{2}$, $B_{2}$, $C_{2}$ on the sides of the triangle $A_{1} B_{1} C_{1}$, $A_{2} \in B_{1} C_{1}$, $B_{2} \in A_{1} C_{1}$, $C_{2} \in A_{1} B_{1}$. Prove that the three lines $A A_{2}$, $B B_{2}$, $C C_{2}$ meet at a common point if and only if so do $A_{1} A_{2}$, $B_{1} B_{2}$, $C_{1} C_{2}$.




Mathematics & Youth: 2008 Issue 377
2008 Issue 377
Mathematics & Youth
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