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2013 Issue 432

  1. The first $2013$ natural numbers from 1 to 2013 are writeen in a line in some order. Substract one from the first number, two from the second ... and 2013 from the $2013^{\text{th}}$ number. Is the product of the resulting $2013$ numbers odd or even?.
  2. Let $ABC$ be an acute triangle with orthocenter $O$. $AO$ meets $BC$ at $D$. Points $E$ and $F$ are on sides $AB$ and $AC$ respectively such that $DE=DB$, $DF=DC$. Prove that $DA$ is the angle bisector of angle $EDF$.
  3. Find all positive integers $a,b$ ($a\geq2$, $b\geq2$) so that $a+b$ is amultiple of $4$ and \[\frac{a(a-1)+b(b-1)}{(a+b)(a+b-1)}=\frac{1}{2}.\]
  4. Find $x,y$ such that \[\begin{cases} x\sqrt{x}+y\sqrt{y} & =2\\ x^{3}+2y^{2} & \leq y^{3}+2x\end{cases}.\]
  5. Given a circle centered at $O$, and diameter $AB$. Point $C$, different from $A$ and $B$, is chosen on circle $(O)$. Point $P$ on $AB$ such that $BP=AC$. The perpendicular from $P$ to $AC$ meet $AC$ at $H$. The intenal angle bisector of angle $CAB$ intersects circle $(O)$ at $E$ and intersects $PH$ at $F$. $CF$ meets circle $(O)$ at $N$. Prove that $CN$ passes through the midpoint of $AP$.
  6. Let $a,b,c$ the positive real number. Prove the following inequality \begin{align*} & \left(\frac{1}{a}+\frac{2}{b+c}+\frac{3}{a+b+c}\right)^{2}+\left(\frac{1}{b}+\frac{2}{c+a}+\frac{3}{a+b+c}\right)^{2}\\ & +\left(\frac{1}{c}+\frac{2}{a+b}+\frac{3}{a+b+c}\right)^{2}\geq\frac{81}{a^{2}+b^{2}+c^{2}}.\end{align*}
  7. Triangle $ABC$ is inscribed in circle $(O)$, another circle $(O')$ touches $AB$, $AC$ at $P,Q$ respectively and touches circle $(O)$ at other points $M$, $N$. Points $E$, $D$, $F$ are the perpendicular feet of point $S$ on $AM$, $MN$, $NA$ respectively. Prove that $DE=DF$.
  8. The real numbers $a,b,c$ satisfying the condition that the polynomial \[P(x)=x^{4}+ax^{3}+bx^{2}+cx+1\] has at least one real root. Determine all triple $(a,b,c)$ such that $s^{2}+b^{2}+c^{2}$ is smallest possible.
  9. Let $a$ and $B$ be two real numbers such that $a^{p}-b^{p}$ is a positive integers for all prime number $p$. Prove that $a$ and $b$ are integers. 
  10. The sequence $\{u_{n}\}$ is given recursively as follows \[u_{1}=\frac{1}{1+a},\quad\frac{1}{u_{n+1}}=\frac{1}{u_{n}^{2}}-\frac{1}{u_{n}}+1,\,\forall n\geq1\] where $a\in\mathbb{R},$ $a\ne-1$. Let $$S_{n}=u_{1}+u_{2}+\ldots+u_{n},\quad P_{n}=u_{1}u_{2}\ldots u_{n}.$$ Determine the value of the following expression $aS_{n}+P_{n}$.
  11. Determine all funtions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that
    a) $f$ is a decreasing function on $\mathbb{R}^{+}$.
    b) $f(2x)=2012^{-x}f(x)$, $\forall x\in\mathbb{R}^{+}$
    where $\mathbb{R}^{+}=(0,+\infty)$.
  12. Let $ABC$ be a triangle inscribed in circle $(O)$. $AD$ is a diameter of $(O)$. Point $E$ belongs to the opposite ray of ray $DA$. The perpendicular through $E$ o $AD$ meets $BC$ at $T$. $TP$ is a tangent line to $(O)$ such that $P$ and $A$ are on opposite sides of $BC$; $AP$ meets $TE$ at $Q$. $M$ is the midpoint of $AQ$; $TM$ meets $AB$, $AC$ at $X$, $Y$ respectively. Prove that $M$ is the midpoint of $XY$.

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2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,4,Anniversary,4,
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Mathematics & Youth: 2013 Issue 432
2013 Issue 432
Mathematics & Youth
https://www.molympiad.org/2017/09/mathematics-and-youth-magazine-problems_87.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/09/mathematics-and-youth-magazine-problems_87.html
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