2016 Issue 465

  1. Let $$S=1\cdot2^{0}+2\cdot2^{1}+3\cdot2^{2}+\ldots+2016\cdot2^{2015}$$ where $2^{0}=1$. Compare $S$ and $2015\cdot2^{2016}$.
  2. Given a right triangle $ABC$ with the right angle $A$ and $AB<AC$. On the opposite ray of the ray $AB$ choose $D$ so that $BD=AC$ and so the opposite ray of the ray $CA$ choose $E$ so that $CE=AD$. The ray $DC$ intersects $BE$ at $F$. Find the angle $\widehat{CFB}$.
  3. Let $a,b,c$ be positive integers such that $$\frac{a^{2}+1}{bc}+\frac{b^{2}+1}{ca}+\frac{c^{2}+1}{ab}$$ is also a posotive integer. Prove that $$\left(a,b,c\right)\leq\left[\sqrt[3]{a+b+c}\right],$$ where $\left(a,b,c\right)$ is the greatest common divisor of $a,b,c$ and $\left[x\right]$ is the integer part of $x$.
  4. Given a right triangle $ABC$ with the right angle $A$ with $AB<AC$, $BC=2+2\sqrt{3}$, and the inradius is equal to $1$. Find the angles $B$ and $C$.
  5. Solve the equation $$\frac{1}{\sqrt{x+1}}+\frac{1}{\sqrt{2x+1}}+\frac{1}{\sqrt{1-2x}}=\frac{4\sqrt{10}}{5}.$$
  6. Solve the system of equations $$e^{x}=y+\sqrt{z^{2}+1}$$ $$e^{y}=z+\sqrt{x^{2}+1}$$ $$e^{z}=x+\sqrt{y^{2}+1}$$
  7. Which number is bigger $$\sin\left(\cos x\right)\text{ or }\cos\left(\sin x\right)?$$
  8. Let $A,B$ and $C$ be the angles of a triangle. Prove that $$\frac{1}{\sin A}+\frac{1}{\sin B}+\frac{1}{\sin C}\leq\frac{2}{3}\left(\cot\frac{A}{2}+\cot\frac{B}{2}+\cot\frac{C}{2}\right).$$
  9. Find the maximal positive value of $T$ such that the following inequality $$\frac{a+b}{b\left(a+1\right)}+\frac{b+c}{c\left(b+1\right)}+\frac{c+a}{a\left(c+1\right)}\geq T$$ always holds for all positive numbers $a,b,c$ satisfying $abc=1$.
  10. Find all pairs of positive integers $x,y$ so that $x^{2}$ is divisible by $2xy^{2}-y^{2}+1$.
  11. Find all monotonic funtions $f:\left(0,+\infty\right)\to\mathbb{R}$ such that $$f\left(x+y\right)=x^{2016}f\left(\frac{1}{x^{2015}}\right)+y^{2016}f\left(\frac{1}{y^{2015}}\right)$$ for all positive numbers $x$ and $y$.
  12. Given an isosceles triangle $ABC$ with the vertex angle $\hat{A}<90^{0}$. Let $CD$ be the altitude from $C$. Let $E$ be the midpoint of $BD$ and let $M$ be the midpoint of $CE$. The angle bisector of $\widehat{BDC}$ intersects $CE$ at $P$. The circle with center at $C$ and with radius $\left|CD\right|$ intersects $AC$ at $Q$. Let $K=PQ\cap AM$. Prove that $CKD$ is a right triangle.




Mathematics & Youth: 2016 Issue 465
2016 Issue 465
Mathematics & Youth
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