2016 Issue 471

  1. Let $a,b$ and $c$ be three integers such that $abc=2015^{2016}$. Find the remainder when $19a^{2}+5b^{2}+1890c^{2}$ is divided by $24$.
  2. Let $a,b\in\mathbb{Z}$. Prove that if $10\mid\left(a^{2}+ab+b^{2}\right)$ then $1000\mid\left(a^{3}-b^{3}\right)$.
  3. Solve the equation $$\sqrt{2x-2}+\sqrt[3]{x-2}=\frac{9-x}{\sqrt[3]{8x-16}}.$$
  4. Given a triangle $ABC$. Outside the triangle, we draw two equilateral triangles $ABD$ and $ACE$. Let $O$ be the centroid of $ACE$. On the opposite ray of the ray $OE$ choose $F$ such that $OF=OE$. Prove that $DF=BO$.
  5. Consider the quadratic function $f\left(x\right)=20x^{2}-11x+2016$. Show that there exists an integer $\alpha$ such that $2^{20^{11^{1960}}}\mid f\left(\alpha\right)$
  6. Solve the equation $$\frac{\log_{2}x}{x^{4}-10x^{2}+26}=\frac{5-x^{2}}{\log_{2}^{2}x+1}.$$
  7. Let $x$ and $y$ be positive numbers such that $\left(\sqrt{x}+1\right)\left(2\sqrt{y}+4\right)+y\geq13$. Find the minimum value of the expression $$P=\frac{x^{4}}{y}+\frac{y^{3}}{x}+y.$$
  8. Given an acute triangle $ABC$. Prove that $$\sqrt[3]{\cot A+\cot B}\geq\frac{16}{3}\cot A\cot B\cot C.$$
  9. Let $a,b$ and $c$ be three positive numbers such that $a+b+c=abc$. Prove that the following inequality $$\begin{align*} & \sqrt{\left(1+a^{2}\right)\left(1+b^{2}\right)}+\sqrt{\left(1+b^{2}\right)\left(1+c^{2}\right)}+\sqrt{\left(1+c^{2}\right)\left(1+c^{2}\right)}\\ \geq & \sqrt{\left(1+a^{2}\right)\left(1+b^{2}\right)\left(1+c^{2}\right)}+4.
  10. A school organizes 7 summer classes. Each student in the school attend at lease one class and each class has exactly 40 students. Besides, for any two classes, there are no more 9 students attending both of them. Prove that the number of students in that school is at least 120.
  11. Find all continuous functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(xy\right)=f\left(\frac{x^{2}+y^{2}}{2}\right)+\left(x-y\right)^{2}$$ for all $x,y\in\mathbb{R}$.
  12. Given a triangle $ABC$ and let $I$ be its incenter. Assume that $\Delta$ is the line which goes through $I$ and is perpendicular to $AI$. The points $E$ and $F$ belong to $\Delta$ such that $\widehat{EBA}=\widehat{FCA}=90^{0}$. The points $M$ and $N$ belong to $BC$ such that $ME\parallel NF\parallel AI$. Prove that the circumcircles of the triangles $ABC$ and $AMN$ are tangent to each other.




Mathematics & Youth: 2016 Issue 471
2016 Issue 471
Mathematics & Youth
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