2017 Issue 486

  1. Find all natural number of the form $\overline{abba}$ such that \[\overline{abba}=\overline{ab}^{2}+\overline{ba}^{2}+a-b.\]
  2. Given a right isosceles triangle $ABC$ with the right angle $A$. Inside the triangle, choose a point $D$ such that $\angle ABD=15^{0}$, $\angle BAD=30^{0}$. Prove that
    a) $BC=2BD$.
    b) $\angle BCD>\angle ACD$. 
  3. Find all integer solutions of the equation \[\sqrt{3x+4}=\sqrt[3]{y^{3}+5y^{2}+7y+4}.\]
  4. On a semicircle $O$ with the diameter $AB$ choose two points $E$, $F$ ($E$ is on the arc $BF$). A point O varies on the opposite ray of the ray $EB$. The circumcircle of $ABP$ intersects the line through $BF$ at the second point $Q$. Let $R$ be the midpoint of $PQ$. Prove that the circle with the diameter $AR$ always goes through a fixed point.
  5. Solve the system of equations \[\begin{cases} x^{3}-7x+\sqrt{x-2} & =y+4\\ y^{3}-7y+\sqrt{y-2} & =z+4\\ z^{3}-7z+\sqrt{z-2} & =x+4\end{cases}\]
  6. Given three non-negative numbers $a,b,c$ such that $a+b+c=3$, $a^{2}+b^{2}+c^{2}=5$. Prove that \[a^{3}b+b^{3}c+c^{3}a\leq8.\]
  7. Solve the following equation with $m,n,k\in\mathbb{N}$, $n\geq m$, $k\geq2$. \[\frac{1}{4}(|\sin x|^{n}+|\cos x|^{n})=\frac{|\sin x|^{m}+|\cos x|^{m}}{|\sin2x|^{k}+|\cos2x|^{k}}\]
  8. Given a triangle $OBC$ with $\angle AOB=120^{0}$, $OA=a$, and $OB=b$. Let $H$ be the perpendicular projection of $O$ on $AB$. Prove that \[aHA+bHB\leq\sqrt{3}ab.\]
  9. Prove that there exists infinitely many positive integers $n$ such that $2018^{n-2017}-1$ is divisible by $n$.
  10. Find all natural numbers $n$ satisfying $4^{n}+15^{2n+1}+19^{2n}$ is divisible by $18^{17}-1$.
  11. Find all funtions $f:\mathbb{R}\to\mathbb{R}$ such that $f(0)$ is rational and \[f(x+f^{2}(y))=f^{2}(x+y),\,\forall x,y\in\mathbb{R}.\]
  12. Given a triangle $ABC$. Prove that \[\cos\frac{A}{2}+\cos\frac{B}{2}+\cos\frac{C}{2}\geq\sqrt{\frac{3}{2}}\left(\sqrt{\sin\frac{A}{2}}+\sqrt{\sin\frac{B}{2}}+\sqrt{\sin\frac{C}{2}}\right).\]




Mathematics & Youth: 2017 Issue 486
2017 Issue 486
Mathematics & Youth
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