2008 Issue 370

  1. Prove the inequality $$\frac{3}{2}+\frac{7}{4}+\frac{11}{8}+\frac{15}{16}+\ldots+\frac{4 n-1}{2^{n}}<7$$ where $n$ is an arbitrary positive integer.
  2. Let $A B C$ be a right triangle with right angle at $A$. Suppose $A B=5cm$ and $I C=6cm$ where $I$ is the incenter of $A B C$. Determine the length of $B C$.
  3. Find all positive integers $x, y, z$ such that the following equality holds $$5 x y z=x+5 y+7 z+10.$$
  4. Solve for $x$ $$x^{4}-2 x^{2}-16 x+1=0.$$
  5. Let $A B C$ be an acute triangle whose altitudes $A A^{\prime}$, $B B^{\prime}$, $C C^{\prime}$ meet at $H$ Denote by $A_{1}$, $B_{1},$ and $C_{1}$ the othocenters of the triangles $A B^{\prime} C^{\prime}$, $B C^{\prime} A^{\prime}$ and $C A^{\prime} B^{\prime}$ respectively. Suppose that $H$ is the incenter of the triangle $A_{1} B_{1} C_{1}$. Prove that $A B C$ is an equilateral triangle.
  6. Let $A B C$ be an isosceles triangle with $A B=B C=a$ and $\widehat{A B C}=140^{\circ} .$ Let $A N$ and $A H$ respectively be the anglebisector and the altitude from $A$. Prove that $2 B H \cdot C N=a^{2}$
  7. Find the minimum value of the function $$f(x)=\left(32 x^{5}-40 x^{3}+10 x-1\right)^{2006}+\left(16 x^{3}-12 x+\sqrt{5}-1\right)^{2008}.$$
  8. Prove that the following equation $$A \cdot a^{x}+B \cdot b^{x}=A+B$$ where $a>1$, $0<b<1$, $A, B \in \mathbb{R}$ has at most two solutions.
  9. Let $a_{1}=\dfrac{1}{2}$ and for each $n$ greater than $1,$ let $$a_{n}=\frac{1}{d_{1}+1}+\frac{1}{d_{2}+1}+\ldots+\frac{1}{d_{k}+1}$$ where $d_{1}, d_{2}, \ldots, d_{k}$ is the collection of all distinct positive divisors of $n .$ Prove the inequality $$n-\ln n<a_{1}+a_{2}+\ldots+a_{n}<n.$$
  10. Given $n$ numbers $a_{1}, a_{2}, \ldots, a_{n}$ in $[-1 ; 2]$ such that their total sum is 0. Let $U_{k}=\dfrac{a_{k} \sqrt{4 k-1}}{(4 k-3)(4 k+1)}$ for $k=1,2, \ldots, n .$ Prove that $$\left|U_{1}+U_{2}+\ldots+U_{n}\right|<\frac{\sqrt{n}}{2}$$
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x+y)=x^{2} f\left(\frac{1}{x}\right)+y^{2} f\left(\frac{1}{y}\right),\,\forall x, y \in \mathbb{R}^{*}.$$
  12. Let $P$ be a point on the insphere of a tetrahedron $A B C D$ and let $G_{a}$, $G_{b}$, $G_{c}$, $G_{d}$ be the centroids of the tetrahedra $P B C D$, $PCDA$, $PDAB$, $PABC$, respectively. Prove that $A G_{a}$, $B G_{b}$, $C G_{c}$ and $D G_{d}$ pass through a common point; and find the orbit of this common point when $P$ moves on the insphere of the given tetrahedron $A B C D$.




Mathematics & Youth: 2008 Issue 370
2008 Issue 370
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