2014 Issue 448

  1. Let $m,n$ be two positive integers such that $3^{m}+5^{n}$ is divisible by $8$. Prove that $3^{n}+5^{m}$ is also divisible by $8$.
  2. Given a triangle $ABC$ with $A$ is an obtuse angle. Let $M$ be the midpoint of $BC$. Inside $\widehat{BAC}$, draw two rays $Ax$ and $Ay$ such that $\widehat{BAx}=\widehat{CAy}=22^{0}$. Let $H$ be the projection of $B$ on $Ax$, and $I$ the projection of $C$ on $Ay$. Find the angle $HMI$.
  3. Solve the following equation \[\sqrt[3]{x^{2}+3x+3}+\sqrt[3]{2x^{2}+3x+2}=6x^{2}+12x+8.\]
  4. Let $ABC$ be a right triangle with the right angle $A$ and let $AB=a$, $AC=b$. Two internal angle bisectors $BB_{1}$ and $CC_{1}$ intersect at $R$, $AR$ intersects $B_{1}C_{1}$ at $M$. Compute the distance from $M$ to $BC$ in terms of $a$ and $b$.
  5. Let $a,b,c$ be positive real numbers satisfying $a^{3}+b^{3}+c^{3}=1$. Prove that \[\frac{a^{2}+b^{2}}{ab(a+b)^{3}}+\frac{b^{2}+c^{2}}{bc(b+c)^{3}}+\frac{c^{2}+a^{2}}{ca(c+a)^{3}}\geq\frac{9}{4}.\]
  6. Express 2015 as a sum of integers $a_{1},a_{2},\ldots,a_{n}$ which are greater than $1$ such that ${\displaystyle \sum_{i=1}^{n}\sqrt[a_{i}]{a_{i}}}$ is maximal.
  7. Given a quadrilateral $ABCD$ and $a,b,c,d$ respectively are external angle bisectors of $\widehat{DAB}$, $\widehat{ABC}$, $\widehat{BCD}$, $\widehat{CDA}$. Denote $K=a\cap b$, $L=b\cap c$, $M=c\cap d$, $N=d\cap a$. Prove that the quadrilateral $KLMN$ inscribes a circle whose radius is \[\frac{KM\cdot LN}{AB+BC+CD+DA}.\]
  8. Suppose that the polynomial \[f(x)=x^{3}+ax^{2}+bx+c\] has three non-negative solutions. Find the maximal real number $\alpha$ such that \[f(x)\geq\alpha(x-a)^{2},\quad\forall x\geq0.\]
  9. Let $[x]$ be the greatest integer not exceeding $x$ and let $\{x\}=x-[x]$. Find $$\left\{ \frac{p^{2012}+q^{2016}}{120}\right\}$$ where $p,q$ are primes numbers which are greater than 5.
  10. Let $x,y,z$ be positive real numbers satisfying $x^{3}+y^{2}+z=2\sqrt{3}+1$. Find the minimum value of the expression \[P=\frac{1}{x}+\frac{1}{y^{2}}+\frac{1}{z^{3}}.\]
  11. Given a sequence $\{a_{n}\}$ whose terms are greater than 1 and satisfy \[\lim_{n\to\infty}\frac{\ln(\ln a_{n})}{n}=\frac{1}{2014}. \] Let $b_{n}=\sqrt{a_{1}+\sqrt{a_{2}+\ldots+\sqrt{a_{n}}}}$ ($n\in\mathbb{N}^{*}$). Prove that $\lim_{n\to\infty}b_{n}$ is a finite number.
  12. Given a triangle $ABC$ and $O$ is any point inside the triangle. Let $P,Q$ and $R$ respectively be the projections of $O$ on $BC$, $CA$ and $AB$ respectively. Let $A_{1},B_{1}$ and $C_{1}$ be arbitrary points other than $A,B,C$ on the lines $BC,CA$ and $AB$ respectively. Let $A_{2},B_{2}$and $C_{2}$ are the reflections of $A_{1},B_{1}$ and $C_{1}$ through the points $P,Q$ and $R$. Let \begin{align*} Z_{1} & \equiv(AB_{1}C_{1})\cap(BC_{1}A_{1})\cap(CA_{1}B_{1}),\\ Z_{2} & \equiv(AB_{2}C_{2})\cap(BC_{2}A_{2})\cap(CA_{2}B_{2}). \end{align*} Prove that $O$ is equidistant from $Z_{1}$ and $Z_{2}$.




Mathematics & Youth: 2014 Issue 448
2014 Issue 448
Mathematics & Youth
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