2018 Issue 487

  1. Find all pairs of prime numbers $(p,q)$ satisfying \[p^{q}-q^{p}=79.\]
  2. Find integers $a,b,c,d$ satisfying \[\sqrt[3]{a^{2}+b^{2}+c^{2}}=\sqrt{a+b+c}=d.\]
  3. Let $a,b,c$ be positive numbers such that $a+b+c=1$. Prove that \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq\frac{21}{1+36abc}.\]
  4. Given a triangle $ABC$ circumscribing a circle $(O)$. The sides $AB$, $BC$ and $CA$ are tangent to $(O)$ at $D$, $E$ and $F$ respectively and assume furthermore that $EC=2EB$. Suppose that $EI$ is a diameter of $(O)$. Through $D$ draw a line which is parallel to $BC$. This line intersects the line segment $EF$ at $K$. Prove that $A$, $I$,$K$ are collinear.
  5. Find the maximal number $M$ such that the inequality $x^{2}\geq M[x]\{x\}$ holds for every $x$ (where $[x]$, $\{x\}$ respectively are the integral part and the fractional part of $x$).
  6. Solve the equation \[x^{3}+x+6=2(x+1)\sqrt{3+2x-x^{2}}.\]
  7. Solve the system of equations \[\begin{cases}\dfrac{1}{\sqrt{x}}+\dfrac{1}{\sqrt{y}}+\dfrac{1}{\sqrt{z}} & =3\\ \dfrac{x^{2}}{y}+\dfrac{y^{2}}{z}+\dfrac{z^{2}}{x} & =\sqrt[4]{\dfrac{x^{4}+y^{4}}{2}}+\sqrt[4]{\dfrac{y^{4}+z^{4}}{2}}+\sqrt[4]{\dfrac{z^{4}+x^{4}}{2}}\end{cases}.\]
  8. Given a triangle $ABC$ with the exradii $r_{a}$, $r_{b}$, $r_{c}$, the medians $m_{a}$, $m_{b}$, $m_{c}$ and the area $S$. Prove that \[r_{a}^{2}+r_{b}^{2}+r_{c}^{2}\geq3\sqrt{3}S+(m_{a}-m_{b})^{2}+(m_{b}-m_{c})^{2}+(m_{c}-m_{a})^{2}.\]
  9. Given a positive integer $n$ and positive numbers $a_{1},a_{2},\ldots a_{n}$. Find a real number $\lambda$ such that \[a_{1}^{x}+a_{2}^{x}+\ldots+a_{2}^{x}\geq n+\lambda x,\quad\forall x\in\mathbb{R}.\]
  10. a) Prove that for every positive integer $n$ the equation $2012^{x}(x^{2}-n^{2})=1$ has a unique solution (denoted by $x_{n}$).
    b) Find ${\displaystyle \lim_{n\to\infty}(x_{n+1}-x_{n})}$.
  11. Let $T$ be the set of all positive factors of $n=2004^{2010}$. Suppose that $S$ be an arbitrary nonempty subset of $T$ satisfying the fact that for all $a$, $b$ belong to $S$ and $a>b$ then $a$ is not divisible by $b$. Find the maximal number of elements of such subset $S$.
  12. Given a triangle $ABC$ whose the incircle $(I)$ is tangent to $BC$ at $D$. Let $H$ be the perpendicular projection of $A$ on $BC$. Let $N$ be the midpoint pf $AH$. The line through $D$ and $N$ intersects $CA$, $AB$ respectively at $J$ and $S$. Assume that $BJ$ intersects $CS$ at $P$. Suppose that $DA$, $DP$ intersect $(I)$ respectively at $G$, $L$. Prove that $B$, $C$, $G$, $L$ lie on some circle.




Mathematics & Youth: 2018 Issue 487
2018 Issue 487
Mathematics & Youth
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