2014 Issue 439

  1. Find all possible ways of inserting three distinct digits into the positions represented by a star in $\overline{155*710*4*16}$ so that the resulting number is divisible by 396.
  2. The triangles $XBC$, $YCA$ and $ZAB$ are constructed externally on the sides of a triangle $ABC$ such that triangle $XBC$ is isosceles with angle $BXC$ equals $120^{0}$ and $YCA$, $ZAB$ are both equilateral. Prove that $XA$ is perpendicular to $YZ$.
  3. Find all positive integer solutions $x,y$ of the equation \[(x^{2}-9y^{2})^{2}=33y+16.\]
  4. Solve the following system of equations \[\begin{cases}6(1-x)^{2} & =\frac{1}{y}\\ 6(1-y)^{2} & =\frac{1}{x} \end{cases}.\]
  5. Point $C$ lies on a half-circle $(O)$ with diameter $AB=2R$, $CH$ is the altitude from $C$ to $AB$ ($H$ differs from $O$). The points $E,F$ move on the half-circle such that $\widehat{CHE}=\widehat{CHF}$. Prove that the line $EF$ always passes through a fixed point.
  6. Solve for $x$ \[\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+x}}}}=\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+}x}}}.\]
  7. Solve the following system of equations \[\begin{cases} 4x^{2} & =(\sqrt{x^{2}+1}+1)(x^{2}-y^{3}+3y-2)\\ (x^{2}+y^{2})^{2}+1 & =x^{2}+2y \end{cases}.\]
  8. Let $BC=a$, $CA=b$, $AB=c$ be the side lengths of a triangle $ABC$; $R$ and $r$ denote its circumradius and inradius respectively. If $S$ is the area of triangle $ABC$, prove that \[\frac{R}{r}\geq\max\left\{ \frac{1}{2};\sqrt{\frac{ab^{3}+bc^{3}+ca^{3}}{3S^{2}}};\sqrt{\frac{ab^{3}+bc^{3}+ca^{3}}{3S^{2}}}\right\} .\]
  9. Find all odd positive integers $n$ such that $15^{n}+1$ is divisible by $n$.
  10. Determine all possible pairs of functions $f:\mathbb{R}\to\mathbb{R}$; $g:\mathbb{R}\to\mathbb{R}$ such that for any $x,y\in\mathbb{R}$, the following identity holds \[f(x+g(y))=xf(y)-yg(y)+g(x).\]
  11. $n$ students ($n\geq2$) are standing in a straigh line. Each time the teacher blow a whistle, exactly two students exchange their positions.  Can it be possible that after an odd number of such whistles, all students returned to their original positions?.
  12. Let $AH$ ($H\in BC$) be the altitude of an acute triangle $ABC$. Point $P$ moves on the segment $AH$. Let $E,F$ denote the feet of the perpendicular from $P$ to $AB,AC$ respectively.
    a) Prove that the points $B,R,F,C$ are concyclic.
    b) Let $O'$ denote the center of the circle containing $B,E,F,C$. Prove that $PO'$ always passes through a fix point, independent of the position of point $P$ chosen on $AH$.




Mathematics & Youth: 2014 Issue 439
2014 Issue 439
Mathematics & Youth
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