2014 Issue 449

  1. Find the minimum value of the products of $5$ different integers among which the sum of any $3$ arbitrary numbers is always greater than the sum of the remains.
  2. Let $ABC$ be a triangle with $AB>AC$ and $AB>BC$. On the side $AB$ choose $D$ and $E$ such that $BC=BD$ and $AC=AE$. Choose $K$ on $CA$ and $I$ on $CB$ such that $DK$ is parallel to $BC$ and $EI$ is parallel to $CA$. Prove that $CK=CI$.
  3. Solve the follwowing equation \[\frac{1}{\sqrt{x+3}}+\frac{1}{\sqrt{3x+1}}=\frac{2}{1+\sqrt{x}}.\]
  4. Given an acute triangle $ABC$ with the orthocenter $H$. Let $M$ be a point inside the triangle such that $\widehat{MAB}=\widehat{MCA}$. Let $E$ and $F$ respectively be the orthogonal projections of $M$ on $AB$ and $AC$. Let $I$ and $J$ respectively be the midpoints of $BC$ and $MA$. Prove that 3 lines $MH$, $EF$ and $IJ$ are concurrent.
  5. Find all pairs of integers $(x,y)$ satisfying \[x^{4}+y^{3}=xy^{3}+1.\]
  6. Solve the following equation \[ 8^{x}-9|x|=2-3^{x}.\]
  7. Given a triangle $ABC$ with the sides $AB=c$, $CA=b$, $BC=a$. Assume that the radius of the circumscribed circle is $R$ and the radius of the inscribed circle is $r$. Show that \[ \frac{r}{R}\leq\frac{3(ab+bc+ca)}{2(a+b+c)^{2}}.\]
  8. Let $x,y,z$ be 3 positive real numbers with $x\geq z$. Find the minimum value of the expression \[P=\frac{xz}{y^{2}+yz}+\frac{y^{2}}{xz+yz}+\frac{x+2z}{x+z}.\]
  9. Find the integer part of the expression \[B=\frac{1}{3}+\frac{5}{7}+\frac{9}{13}+\ldots+\frac{2013}{2015}.\]
  10. Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ is a divisor of $3^{n}-1$ for every positive integer $n$. 
  11. Let $\{x_{n}\}$ be a sequence satisfying \[x_{0}=4,\,x_{1}=34,\,x_{n+2}\cdot x_{n}=x_{n+1}^{2}+18\cdot10^{n+1},\,\forall n\in\mathbb{N}.\] Let ${\displaystyle S_{n}=\sum_{k=0}^{26}x_{n+k}}$, $n\in\mathbb{N}^{*}$. Prove that, for every odd natural number $n$, $66|S_{n}$.
  12. Given a triangle $ABC$. The point $E$ and $F$ respectively vary on the sides $CA$ and $AB$ such that $BF=CE$. Let $D$ be the intersection of $BE$ and $CF$. Let $H$ and $K$ respectively be the orthocenters of $DEF$ and $DBC$. Prove that, when $E$ and $F$ change, the line $HK$ always passes through a fixed point.




Mathematics & Youth: 2014 Issue 449
2014 Issue 449
Mathematics & Youth
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