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2008 Issue 376

  1. Write the numbers $8^{2008}$ and $125^{2008}$ consecutively. What is the number of decimal digits of the resulting number?
  2. Find a rational number $\dfrac{a}{b}$ such that the following three conditions are satisfied
    • $-\dfrac{1}{2}<\dfrac{a}{b}<-\dfrac{2}{5}$
    • $11 a+5 b=26$
    • $200<|a|+|b|<230$
  3. Find all non-zero natural numbers $n$ such that the number $A=\dfrac{1.3 .5 .7 \ldots(2 n-1)}{n^{n}}$ is an integer, here the numerator of $A$ is the product of the first $n$ odd numbers.
  4. Prove that $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{3}{2}(a+b+c-1)$$ where $a, b, c$ are positive real numbers such that $a b c=1 .$ When does equality hold? $?$
  5. In a right triangle $A B C$ with right angle at $A,$ the altitude $A H,$ the median $B M,$ and the angle-bisector $C D$ meet at a common point. Determine the ratio $\dfrac{A B}{A C}$
  6. Solve for $x$ $$\sqrt[3]{x+6}+\sqrt{x-1}=x^{2}-1$$
  7. Let $S$ denote the area of a given triangle $A B C,$ and denote $B C=a$, $C A=b$, $A B=c$. Prove the inequality $$a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2} \geq 16 S^{2}+\frac{1}{2} a^{2}(b-c)^{2}+\frac{1}{2} b^{2}(c-a)^{2}+\frac{1}{2} c^{2}(a-b)^{2}.$$ When does equality hold?
  8. Given a triangle $A B C$ with three sides $B C=a$, $A C=b$, $A B=c$ such that $a+c=2 b$ let $h_{a}$, $h_{c}$ be the altitudes from $A$ and $C$ respectively; and let $r_{a}$, $r_{c}$ denote the $A$-exradius and $C$-exradius respectively. Prove that $$\frac{1}{r_{a}}+\frac{1}{r_{c}}=\frac{1}{h_{a}}+\frac{1}{h_{c}}.$$
  9. The positive real numbers $a$, $b$, $c$, $x$, $y$ and $z$ are such that $$\begin{cases} c y+b z &=a \\ a z+c x &=b \\ b x+a y &=c\end{cases}.$$ Find the smallest possible value of the expression $$P=\frac{x^{2}}{1+x}+\frac{y^{2}}{1+y}+\frac{z^{2}}{1+z}.$$
  10. Let $f$ be a continuous function on $\mathbb{R}$ such that $f(2010)=2009$ and $f(x) \cdot f_{4}(x)=1$ for all $x \in \mathbb{R}$ (where $\left.f_{4}(x)=f(f f(f(x)))\right)$. Determine the value of $f(2008)$.
  11. Let $u_{1}, u_{2}, \ldots, u_{n}$ $(n>2)$ be a sequence of positive real numbers such that
    • $\dfrac{1004}{k}=u_{1} \geq u_{2} \geq \ldots \geq u_{n} \quad$ for some positive integer $k$;
    • $u_{1}+u_{2}+\ldots+u_{n}=2008$. Show that it is possible to select $k$ elements from the set $\left(u_{n}\right)$ such that in this collection of $k$ numbers, the smallest one is at least half of the largest.
  12. Consider the circle $(O)$ and three colinear points $X$, $Y$, $H$ that are not on this circle such that $\overline{H X} \cdot \overline{H Y} \neq \mathscr{P}_{H/(O)} .$ A straight line $d$ through $H$ meets $(O)$ at two points $M$ and $N$. $M X$ and $N Y$ intersect with $(O)$ again at $P$ and $Q$ respectively. Show that as the line $d$ through $H$ varies, the line connecting $P$ and $Q$ always passes through a fixed point.

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Mathematics & Youth: 2008 Issue 376
2008 Issue 376
Mathematics & Youth
https://www.molympiad.org/2020/09/2008-issue-376.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2008-issue-376.html
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