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.




Mathematics & Youth: 2008 Issue 376
2008 Issue 376
Mathematics & Youth
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS CONTENT IS PREMIUM Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy