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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 $A B C D$.

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2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
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Mathematics & Youth: 2008 Issue 370
2008 Issue 370
Mathematics & Youth
https://www.molympiad.org/2020/09/2008-issue-370.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2008-issue-370.html
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