2016 Issue 474

  1. Prove that there do not exist two coprime integers $a$ and $b$ which are greater than $1$ and satisfy the fact that $a^{2007}+b^{2007}$ is divisible by $a^{2006}+b^{2006}$.
  2. Find prime numbers $a,b,c,d,e$ such that $$a^{4}+b^{4}+c^{4}+d^{4}+e^{4}=abcde.$$
  3. Given $f\left(x\right)=a^{2016}x^{2}+bx+a^{2016}c-1$ where $a,b,c\in\mathbb{Z}$. Suppose that the equation $f\left(x\right)=-2$ has two positive integral solutions. Prove that $$A=\frac{f^{2}\left(1\right)+f^{2}\left(-1\right)}{2}$$ is a composite number.
  4. Given a triangle $ABC$ and $\left(I\right)$ is its incircle. Let $D$ be the point of tangency between $\left(I\right)$ and $BC$. The line which goes through $D$ and is perpendicular to $AI$ intersects $IB$ and $IC$ respectively at $P$ and $Q$. Prove that all $B,C,P$ and $Q$ lie on a circle whose center is on $AD$.
  5. Find all the real numbers $k$ such that for all non-negative numbers $a,b,c$ we always have the following inequality $$\left[a+k\left(b-c\right)\right]\left[b+k\left(c-a\right)\right]\left[c+k\left(a-b\right)\right]\leq abc.$$
  6. Find the minimum value of the expression $$\begin{align*}
    f= & \sqrt{x^{2}-2x+2}+\sqrt{x^{2}-8x+32}+\sqrt{x^{2}-6x+25}\\
    & +\sqrt{x^{2}-4x+20}+\sqrt{x^{2}-10x+26}.
  7. How many solutions does the following equations have $$\log_{\frac{5\pi}{2}}x=\cos x.$$
  8. Given a tetrahedron $SABC$ such that $SA,SB$ and $SC$ are mutually perpendicular. A moving plane $\left(P\right)$ which contains the centroid of the given tetrahedron intersects $SA,SB$ and $SC$ respectively at $A_{1},B_{1}$, and $C_{1}$. Prove that $$\frac{1}{SA_{1}^{2}}+\frac{1}{SB_{1}^{2}}+\frac{1}{SC_{1}^{2}}\geq\frac{4}{R^{2}},$$ where $R$ is the radius of the circumscribed sphere of the tetrahedron $SABC$.
  9. Find the integral part of the following number $$T=\frac{2}{1}\cdot\frac{4}{3}\cdot\frac{6}{5}\ldots\frac{2016}{2015}.$$
  10. A positive interger is called a "HV2015'' number if in its decima representation there are $2015$ consecutive digits $9$. Let a stricly increasing sequence of positive integers such that the sequence $\left(\dfrac{a_{n}}{n}\right)$, $n=1,2,3,\ldots$ is bounded. Prove that there are infinitely many "HV2015'' numbers in the given sequence $\left(a_{n}\right),n=1,2,3,\ldots$.
  11. Given a grid of the size $1\times n$ $\left(n>2\right)$. Now we fill in each square with $0$ or $1$. A way to fill the grid is called "good'' if any three consecutive squares do not contain the same number. Let $a_{n}$ be the number of "good'' ways to fill the grid. Compute $a_{n}$.
  12. Given a triangle $ABC$. Prove that $$\sin\frac{A}{2}+\sin\frac{B}{2}+\sin\frac{C}{2}\geq\frac{1}{\sqrt{2}}\left(\sqrt{\sin\frac{A}{2}}+\sqrt{\sin\frac{B}{2}}+\sqrt{\sin\frac{C}{2}}\right).$$




Mathematics & Youth: 2016 Issue 474
2016 Issue 474
Mathematics & Youth
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