2018 Issue 489

  1. Find all pairs of integers $(x,y)$ satisfying \[x^{2}+x=3^{2018y}+1.\]
  2. Given a triangle $ABC$ with $\angle B=45^{0}$, $\angle C=30^{0}$. Let $BM$ be one of the medians of $ABC$. Find the angle $\widehat{AMB}$.
  3. Given real numbers $x,y$ satisfying $0<x,y<1$. Find the minimum value of the expression \[F=x^{2}+y^{2}+\frac{2xy-x-y+1}{4xy}.\]
  4. Given a circle $(O)$ with a diameter $AB$. On $(O)$ pick a point $C$ ($C$ is different from $A$ and $B$). Draw $CH$ perpendicular to $AB$ at $H$. Choose $M$ and $N$ on the line segments $CH$ and $BC$ respectively such that $MN$ is parallel to $AB$. Through $N$ draw a line perpendicular to $BC$. This line intersects the ray $AM$ at $D$. On the line $DO$ choose two points $F$ and $K$ such that $O$ is the midpoint of $FK$. The lines $AF$ and $AK$ respectively intersect $(O)$ at $P$ and $Q$. Prove that $D$, $P$, $Q$ are colinear.
  5. Suppose that the polynomial \[f(x)=x^{3}+ax^{2}+bx+c\] has $3$ non-negative real solutions. Find the maximal real number $\alpha$ so that \[f(x)\geq\alpha(x-a)^{3},\,\forall x\geq0.\]
  6. Solve the equation \[(1-\sqrt{2}\sin x)(\cos2x+\sin2x)=\frac{1}{2}.\]
  7. Given the following system of equations \[\begin{cases}\dfrac{yz(y+z-x)}{x+y+z} & =a\\ \dfrac{zx(z+x-y)}{x+y+z} & =b\\ \dfrac{xy(x+y-z)}{x+y+z} & =c\end{cases}\] where $a,b,c$ are positive parameters.
    a) Show that the system always has a positive solution.
    b) Solve the system when $a=2$, $b=5$, $c=10$.
  8. Suppose that $a,b,c$ are the lengths of three sides of a triangle. Prove that \[ \frac{ab}{a^{2}+b^{2}}+\frac{bc}{b^{2}+c^{2}}+\frac{ca}{c^{2}+a^{2}}\geq\frac{1}{2}+\frac{2r}{R}\] with $R$, $r$ respectively are the inradius and the circumradius of the triangle.
  9. For any integer $n$ which is greater than $3$, let \[P=\sqrt[60]{3}\cdot\sqrt[120]{4}\ldots\sqrt[n^{3}-n]{n-1}.\] Show that \[\sqrt[24n^{2}+24n]{3^{n^{2}+n-12}}\leq P<\sqrt[8]{3}.\]
  10. Find natural numbers $n$ so that $4^{m}+2^{n}+29$ cannot be a perfect square for any natural number $m$.
  11. The sequence $(a_{n})$ is given as follows \[a_{1}=\frac{1}{2},\quad a_{n+1}=\frac{(a_{n}-1)^{2}}{2-a_{n}},\, n\in\mathbb{N^{*}.}\] a) Find ${\displaystyle \lim_{n\to\infty}a_{n}}$.
    b) Show that ${\displaystyle \frac{a_{1}+a_{2}+\ldots+a_{n}}{n}\geq1-\frac{\sqrt{2}}{2}}$ for all $n\in\mathbb{N^{*}}$.
  12. Given a triangle $ABC$ inscribed in a circle $(O)$. A point $P$ varies on $(O)$ but is different from $A$, $B$ and $C$. Choose $M$, $N$ respectively on $PB$, $PC$ so that $AMPN$ is a parallelogram.
    a) Prove that there exists a fixed point which is equidistant from $M$ and $N$.
    b) Prove that the Euler line of $AMN$ always goes through a fixed point.




Mathematics & Youth: 2018 Issue 489
2018 Issue 489
Mathematics & Youth
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