2016 Issue 467

  1. Given the following increasing sequence $1,3,5,7,9,\ldots$ where all the digits of all the terms are odd. Find the $2016^{{\rm th}}$ term of the above sequence.
  2. Given an acute triangle $ABC$ and the median $AM$. Draw $BH\perp AC$. The line which goes through $A$ and is perpendicular to $AM$ intersects $BH$ at $E$. On the opposite ray of the ray $AE$ choose $F$ so that $AE=AF$. Prove that $CF\perp AB$.
  3. Let $a,b$ and $c$ be three different nonzero rational numbers such that $$\left(\frac{a}{b-c}\right)^{2}+\left(\frac{b}{c-a}\right)^{2}+\left(\frac{c}{a-b}\right)^{2}\leq2.$$ Prove that $$\sqrt{\left(\frac{b-c}{a}\right)^{2}+\left(\frac{c-a}{b}\right)^{2}+\left(\frac{c-a}{b}\right)^{2}}$$ is a rational number.
  4. Given a pentagon $ABCDE$ with $\widehat{ACB}=\widehat{ADE}=90^{0}$ and $\widehat{CAB}=\widehat{DAE}$. Let $O$ be the intersection of $BD$ and $CE$. Prove that $AO\perp DC$.
  5. Solve the equation $$x^{4}-2x^{3}+\sqrt{2x^{3}+x^{2}+2}-2=0.$$
  6. Given three real numbers $x,y$ and $z$ such that $x+y+z=3$ and $xy+yz+zx=-9$. Find the maximum and minimum values of $xyz$.
  7. Let $x,y$ and $x$ be real numbers satisfying $x^{2}+y^{2}+z^{2}=1$. Find the maximum value of the expression $$P=\left(x^{2}-yz\right)\left(y^{2}-zx\right)\left(z^{2}-xy\right).$$
  8. Let $p$ and $R$ respectively be the semiperimeter and the circuradius of the circumcircle of a triangle $ABC$. Prove that $$\left(\frac{p}{3R}\right)^{2}+\left(\frac{3R}{p}\right)^{2}\geq\frac{25}{12}.$$ When does the equality happen?.
  9. Find all real numbers $k>-2$ such that the inequality $$\frac{x}{x+y+kz}+\frac{y}{y+z+kx}+\frac{z}{z+x+ky}\geq\frac{3}{k+2}$$ holds for all nonnegative $x,y$ and $z$.
  10. Find all natural numbers $x$ and $y$ such that $$\left(\sqrt{x}-1\right)\left(\sqrt{y}-1\right)=2017.$$
  11. Determine the sequence $\left(x_{n}\right)$ as follows $$ x_{1}=1;\quad x_{n+1}=\frac{n+1}{n+2}x_{n}+n^{2},\quad n=1,2,\ldots$$ Find ${\displaystyle \lim_{n\to\infty}\left(\frac{\sqrt[3]{x_{n}}}{n+1}\right)}$.
  12. Assume that $ABC$ is a triangle inscribed in a circle $\left(O\right)$ and $AD,BE,CF$ are its altitudes. The line through $AO$ intersects $BC$ at $A'$. Let $M$ be the midpoint of $BC$ and let $S$ be the intersection between two tangent lines of $\left(O\right)$ at $B$ and $C$, The line through $EF$ intersects $SD$ and $SA'$ respectively at $I$ and $J$. Show that $BC$ is tangent to the circumcircle of the triangle $IJM$ at $M$.




Mathematics & Youth: 2016 Issue 467
2016 Issue 467
Mathematics & Youth
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