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2014 Issue 440

  1. Which number is greater? $P$ or $Q$, given that $$\begin{align*} P & =\frac{20}{30}+\frac{20}{70}+\frac{20}{126}+\ldots+\frac{20}{798};\\ Q & =\left(\frac{31}{2}\cdot\frac{32}{2}\cdot\frac{33}{2}\ldots\frac{60}{2}\right):(1.3.5\ldots59). \end{align*}$$
  2. Given $2014$ points $A_{1},A_{2},\ldots,A_{2014}$ and a circle with radius $1$ on the plane, prove that there always exists a point $M$ on the circle such that \[MA_{1}+MA_{2}+\ldots+MA_{2014}\geq2014.\]
  3. Prove that for any natural number $n$, \[n^{4}-5n^{3}-2n^{2}-10n+4\] is not divisible by $49$.
  4. Let $R$ denote the radius of the circumcircle of a given triangle $ABC$. The internal and external angle bisector of angle $\widehat{ACB}$ meet $AB$ at $E$ and $F$ respectively. Prove that if $CE=CF$, then $AC^{2}+BC^{2}=4R^{2}$.
  5. Solve the following system of equations \[\begin{cases} 2x\left(1+\frac{1}{x^{2}-y^{2}}\right) & =5\\ 2(x^{2}+y^{2})\left(1+\frac{1}{(x^{2}-y^{2})^{2}}\right) & =\frac{17}{2}\end{cases}.\]
  6. Determine the funtion \[f(x)=ax^{2}+bx+c \] where $a,b,c$ are integers such that $f(0)=2014$, $f(2014)=0$ and $f(2^{n})$ is a multiple of $3$ for any natural number $n$.
  7. The positive real numbers $x,y,z$ satisfy the equation $xy=1+z(x+y)$. Find the greatest value of \[P=\frac{2xy(xy+1)}{(1+x^{2})(1+y^{2})}+\frac{z}{1+z^{2}}.\]
  8. In an acute triangle $ABC$, the three altitudes $AA_{1},BB_{1},CC_{1}$ meet at $H$. Prove that $ABC$ is an equilateral triangle if and only if \[ HA^{2}+HB^{2}+HC^{2}=4(HA_{1}^{2}+HB_{1}^{2}+HC_{1}^{2}).\]

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Mathematics & Youth: 2014 Issue 440
2014 Issue 440
Mathematics & Youth
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_95.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/08/mathematics-and-youth-magazine-problems_95.html
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