2014 Issue 442

  1. Find two whole numbers of the form $\overline{ab}$ and $\overline{ba}$ ($a\ne b$) such that \[\frac{\overline{ab}}{\overline{ba}}=\frac{\underset{2014\text{ digits}}{\overline{a\underbrace{3\ldots3}b}}}{\underset{2014\text{ digits}}{\overline{b\underbrace{3\ldots3}a}}}.\]
  2. The sum $A$ below consists of 2014 summands \[A=\frac{1}{19^{1}}+\frac{2}{19^{2}}+\frac{3}{19^{3}}+\ldots+\frac{2014}{19^{2014}}.\] Compare the number $A^{2013}$ with $A^{2014}$.
  3. Let $ABCD$ be a quadriteral whose diagonals $AC$ and $BD$ are perpendicular. $M$ and $N$ are the midpoints of line segments $AB,AD$ respectively. Points $E,F$ are the feet of perpendicular lines from $M$ and $N$ onto $CD,BC$ respectively. Prove that $MNEF$ is a cyclic quadrilateral.
  4. Solve for $x$ \[4x^{3}+4x^{2}-5x+9=4\sqrt[4]{16x+8}.\]
  5. The real numbers $x,y,z$ satisfy $x+y+z=1$. Prove the inequality \[44(xy+yz+zx)\leq(3x+4y+5z)^{2}.\] 
  6. Prove that the following equation has no real solutions \[9x^{4}+x(12x^{2}+6x-1)+(x+1)(9x^{2}+12x+5)+1=0.\]
  7. Triangle $ABC$ inscribed in a circle centerd at $O$ and radius $R$, where $CA\ne CB$, $\widehat{ACB}=90^{0}$. The circumcircle centered at $S$ of triangle $AOB$ meets $CA,CB$ at points $M,N$ respectively. Let $K$ be the reflection of $S$ in the line $MN$. Prove that $SK=R$. 
  8. The real numbers $x,y,z$ satisfy $x^{2}+y^{2}+z^{2}=8$. Determine the largest and smallest values of the following expression \[P=(x-y)^{5}+(y-z)^{5}+(z-x)^{5}.\]




Mathematics & Youth: 2014 Issue 442
2014 Issue 442
Mathematics & Youth
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