2017 Issue 481

  1. Find natural numbers $a,b,c$ satisfying $9^{a}+952=(b+41)^{2}$ and $a=2^{b}\cdot c$.
  2. Prove that \[A=\frac{1}{1000}+\frac{1}{1002}+\ldots+\frac{1}{2000}<\frac{1}{2}.\]
  3. Find positive integral solutions of the equation \[x^{5}+y^{5}+2016=(x+2017)^{5}+(y-2018)^{5}.\]
  4. Given a quadrilateral $ABCD$ inscribed in a circle $(O)$ with $AB=BD$. The lines $AB$ and $DC$ intersect at $N$. The line $CB$ and the tangent line of the circle $(O)$ at the point $A$ intersect at $M$. Prove that $\widehat{AMN}=\widehat{ABD}$.
  5. Solve the equation \[(1-2x)\sqrt{2-x^{2}}=x-1.\]
  6. Solve the system of equations \[ \begin{cases} \dfrac{x+\sqrt{1+x^{2}}}{1+\sqrt{1+y^{2}}} & =x^{2}y\\ 1+y^{4}-\dfrac{4y}{x}+\dfrac{3}{x^{4}} & =0 \end{cases}.\]
  7. Given an integer $n>1$. Prove that \[\frac{2^{2n}}{2\sqrt{n}}<C_{2n}^{n}<\frac{2^{2n}}{\sqrt{2n+1}}.\]
  8. Given a quadrilateral $ABCD$ circumsribing a circle $(O)$. Let $E,F$ and $G$ respectively be the intersection of three pairs of line $(AB,CD)$, $(AD,CB)$ and $(AC,BD)$. Let $K$ be the intersection between $EF$ and $OG$. Prove that $\widehat{AKG}=\widehat{CKG}$ and $\widehat{BKG}=\widehat{DKG}$.
  9. Given three nonnegative numbers $x,y,z$ such that $2^{x}+4^{y}+8^{z}=4$. Find the maximum and minimum values of the expression \[S=\frac{x}{6}+\frac{y}{3}+\frac{z}{2}.\]
  10. Find all pairs of natural number $(m,n)$ so that $2^{m}\cdot3^{n}-1$ is a perfect square.
  11. Let $(u_{n})$ be a sequence defined as follows \[u_{1}=0,\quad\log_{\frac{1}{4}}u_{n+1}=\left(\frac{1}{4}\right)^{u_{n}}\quad\forall n\in\mathbb{N}^{*}.\] Prove that the sequence has a finite limit and find that limit.
  12. Given a triangle $ABC$ and its circumcircle $(O)$. Let $(O_{B})$ and $(O_{C})$ respectively be the mixtilinear incircles corresponding to $B$ and $C$ (recall that a mixtilinear incircle corresponding to $B$ is the circle which is internally tangent to $BA$, $BC$ and the circumcircle $(O)$). Let $M$ (resp. $N$) be the tangent point between $(O)$ and $(O_{B})$ (resp. $(O_{C})$). Let $(M)$ and $(N)$ be the circles with centers at $M$ and $N$ and are tangent to $AC$ and $AB$ respectively. Prove that the center of dilation of $(M)$ and $(N)$ belongs to $BC$.




Mathematics & Youth: 2017 Issue 481
2017 Issue 481
Mathematics & Youth
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