2014 Issue 444

  1. Find the maximum calue of positive integer $n$ such that $2013$ can be written as the sum of $n$ compound numbers. How does the answer change if $2013$ is replaced by $2014$.
  2. Let $ABC$ be a right triangle, right angle at $A$, $\widehat{B}=60^{0}$. Point $E$ on side $AC$ such that $\widehat{ABE}=20^{0}$. Point $K$ on the half line $BE$ such that $EK=BC$. Find the measure of the angle $\widehat{BCK}$.
  3. Solve the inequality \[\frac{x^{2}+8}{x+1}+\frac{x^{3}+8}{x^{2}+1}+\frac{x^{4}+8}{x^{3}+8}+\ldots+\frac{x^{101}+8}{x^{100}+1}\geq800.\]
  4. The quadrilateral $ABCD$ is inscribed in circle $(O)$ where angle $\widehat{BAD}$ is obtuse. The rays through $A$ and perpendicular to $AD,AB$ meet $CB,CD$ at $P$ and $Q$ respectively. $PQ$ intersects $BD$ at $M$. Prove that $\widehat{MAC}=90^{0}$.
  5. Solve the system of equations \[\begin{cases} \sqrt{2x-3}-\sqrt{y} & =2x-6\\ x^{3}+y^{3}+7(x+y)xy & =8xy\sqrt{2(x^{2}+y^{2})} \end{cases}.\]
  6. The positive real numbers $a,b,c$ satisfy the equation $abc=1$. Prove the inequality \[a^{3}+b^{3}+c^{3}+\frac{ab}{a^{2}+b^{2}}+\frac{bc}{b^{2}+c^{2}}+\frac{ca}{c^{2}+a^{2}}\geq\frac{9}{2}.\]
  7. Let $ABC$ be a triangle. $D$ is the midpoint of side $BC$ and $M$ is an arbitrary point on segment $BD$. $MEAF$ is a parallellogram where vertex $E$ lies on $AB$, $F$ lies on $AC$, $MF$ and $AD$ intersect at $H$. The line through $B$ and parallel to $EH$ intersects $MF$ at $K$; $AK$ meets $BC$ at $I$. Find the ratio $\dfrac{IB}{ID}$.
  8. The sequence $\{v_{n}\}_{n}$ satisfies \[v_{1}=5,\quad v_{n+1}=v_{n}^{4}-4v_{n}^{2}+2.\] Find a closed formular for $v_{n}$.




Mathematics & Youth: 2014 Issue 444
2014 Issue 444
Mathematics & Youth
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