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2017 Issue 477

  1. Let $$S=\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+\ldots+\frac{1}{20.21.22.23}.$$ Compare $S$ and $\frac{1}{18}$.
  2. Given an isosceles triangle $ABC$ with the base $BC$. Let $BD$ be the angle bisector. On the ray $BA$, choose $E$ such that $BE=2CD$. Show that $\widehat{EDB}=90^{0}$.
  3. Let $x,y,z$ be positive numbers such that $x+y+z=xyz$. Find the minimum value of the expression $$ S=\frac{x}{y^{2}}+\frac{y}{z^{2}}+\frac{z}{x^{2}}.$$
  4. Given an isosceles triangle $ABC$ with base $BC$ and let $M$ be a point inside the triangle such that $\widehat{ABM}=\widehat{BCM}$. Let $H$, $I$ and $K$ respectively be the perpendicular projections of $M$ on $AB$, $BC$ and $CA$. Suppose that $E$ is the intersection between $MB$ and $IH$, and $F$ is the intersection between $MC$ and $IK$. Assume that the circumcircles of the triangles $MEH$ and $MFK$ intersect at $N\ne M$. Prove that the line $NM$ contains the midpoint of $BC$.
  5. Solve the equation $$ (x^{2}+x+1)(\sqrt[3]{(3x-2)^{2}}+\sqrt[3]{3x-2}+1)=9.$$
  6. Suppose that the equation $$ x^{3}-ax^{2}+bx-c=0$$ has $3$ positive solutions. Prove that if $$2a^{3}+3a^{2}-7ab+9c-6b-3a+2=0$$ then $1\leq a\leq2$.
  7. Let $a,b,c,d$ and $e$ be the real numbers such that $$\sin a+\sin b+\sin c+\sin d+\sin e = 0$$ $$\cos2a+\cos2b+\cos2c+\cos2d+\cos2e =-3.$$ Find the maximum and minimum values of the expression $$\cos a+2\cos2b+3\cos3c.$$
  8. Assume that $H$ is a point inside the triangle $ABC$. Let $K$ be the orthocenter of $ABH$. The line which goes through $H$ and is perpendicular to $BC$ intersects $AK$ at $E$. The line which goes through $H$ and is perpendicular to $AC$ intersects $HK$ at $F$. Prove that $CH\bot EF$.
  9. For $n\in\mathbb{N}$, let $S_{n}=1+\frac{1}{2}+\ldots+\frac{1}{n}$. Let $\{S_{n}\}$ denote the fractional part of $S_{n}$. Prove that for every $\epsilon\in(0,1)$, there always exists $n\in\mathbb{N}^{+}$ such that $\{S_{n}\}\leq\epsilon$.
  10. Find the smallest $k$ such that we can use $k$ colors to color the numbers $1,2,\ldots,20$ in the following way. For each number, we use exactly one color, we can use one color for more than one number, and no $3$ numbers with the same color forms an arithmetic sequence.
  11. Consider the sequence $\{x_{n}\}$ determined as follows $$x_{1}=-\pi,\,x_{2}=-1,$$ $$x_{n+2}=x_{n+1}+\log_{2}\frac{9+3(\cos x_{n+1}-\cos x_{n})-\cos x_{n+1}\cos x_{n}}{8+\sin^{2}x_{n}},\quad n=1,2,3,\ldots.$$ Prove that the sequence $(x_{n})$ has a finite limit and find that limit.
  12. Given a quadrilateral $ABCD$ inscribed a circle $(O)$. The external angle bisectors of the angles $\widehat{DAB}$, $\widehat{ABC}$, $\widehat{BCD}$, $\widehat{CDA}$ respectively intersects the external angle bisector of the angle $\widehat{ABC}$, $\widehat{BCD}$, $\widehat{CDA}$, $\widehat{DAB}$ at $X,Y,Z,T$. Let $E$ and $F$ respectively be the midpoints of $XZ$ and $YT$. Prove that
    a) $XYZT$ is a cyclic quadrilateral and $XZ\bot YT$.
    b) $O$, $E$ and $F$ are collinear.

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Mathematics & Youth: 2017 Issue 477
2017 Issue 477
Mathematics & Youth
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_44.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_44.html
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