2018 Issue 491

  1. Consider all pairs of integers $x$, $y$ which are greater than $1$ and satisfy $x^{2017}=y^{2018}$. Find the pair with the smallest possible value for $y$.
  2. Given an isosceles triangle $ABC$ with the apex $A$. Suppose that $\hat{A}=180^{0}$, $BC=a$, $AC=b$. Outside $ABC$, construct the isosceles triangle $ABD$ with the apex $A$ and $\widehat{BAD}=36^{0}$. Find the perimeter of the triangle $ABD$ in terms of $a$ and $b$.
  3. Find positive integers $n$ such that if the positive integer $a$ is a divisor of $n$ then $a+2$ is also a divisor of $n+2$.
  4. Given a triangle $ABC$ inscribed the circle $(O)$ with diameter $AC$. Draw a line which is perpendicular to $AC$ at $A$ and intersects $BC$ at $K$. Choose a point $T$ on the minor $AB$ ($T$ is different from $A$ and $B$). The line $KT$ intersects $(O)$ at the second point $P$. On the tangent line to the circle $(O)$ at the point $T$ choose two points $I$ and $J$ such that $KIA$ and $KAJ$ are isosceles triangles with the apex $K$. Show that
    a) $\widehat{TIP}=\widehat{TKJ}$.
    b) The circle $(O)$ and the circumcircle of the triangle $KPJ$ are tangent to each other.
  5. Find integral solutions of the equation \[y^{2}+2y=4x^{2}y+8x+7.\]
  6. Solve the system of equations \[\begin{cases}\log_{2}x+\log_{2}y+\log_{2}z & =3\\ \sqrt{x^{2}+4}+\sqrt{y^{2}+4}+\sqrt{z^{2}+4} & =\sqrt{2}(x+y+z)\end{cases}\]
  7. Show that \[\lim_{n\to\infty}\sqrt{1+2\sqrt{1+3\sqrt{1+\ldots\sqrt{1+(n-1)\sqrt{1+n}}}}}=3.\]
  8. Let $m_{a}$, $m_{b}$, $m_{c}$ be the lengths of the medians of the triangle with the perimeter $2$. Show that \[\max\{1;3\sqrt[3]{r}\}\leq m_{a}+m_{b}+m_{c}<\frac{3}{\sqrt{2}}\] where $r$ is the inradius of the triangle.
  9. Assume that $a,b,c$ are three non-negative numbers such that $a+b+c=3$. Find the maximum value of the expression \[P=a\sqrt{b^{3}+1}+b\sqrt{c^{3}+1}+c\sqrt{a^{3}+1}.\]
  10. For every $n\in\mathbb N$, let $F_{n}=2^{2^{n}}+1$. For each $n\in\mathbb N$, let $q$ be a prime divisor of $F_{n}$. Show that $2^{n+1}\mid q-1$. Futhermore, if $n\geq 2$, show even more that $2^{n+2}\mid q-1$.
  11. Find all funtions $f:\mathbb{R\to\mathbb{R}}$satisfying \[f(x)f(y)+f(xy)+f(x)+f(y)=f(x+y)+2xy\] for all $x,y\in\mathbb{R}.$
  12. Given a convex hexagon $ABCDEF$ circumscribing a circle $(O)$. Assume that $O$ is the circumcenter of the triangle $ACE$. Prove that the circumcenter of the triangles $OAD$, $OBE$ and $OCF$ has another common point besides $O$.




Mathematics & Youth: 2018 Issue 491
2018 Issue 491
Mathematics & Youth
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