2013 Issue 427

  1. Let $a=123456789$. Which number is greater $2012^{9^{9^{a}}}$ or $2013^{a^{a^{9}}}$?.
  2. Let $ABC$ ($AB<AC$) be a triangle, with two altitudes $BD,CE$ and $AB=c$, $AC=b$, $BD=h_{b}$, $CE=h_{c}$. Prove that \[c^{n}+h_{c^{n}}<b^{n}+h_{b}^{n},\quad\forall n\in\mathbb{N}^{*}.\]
  3. Find all positive integers $n$ such that \[A=\left[\frac{n^{2}+n-5}{2}\right]\] is a prime number, where $[a]$ is the largest integer not exceeding $a$.
  4. Find all postive integer soltuions of the equation \[\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.\]
  5. Let $ABC$ be a right triangle, right angle at $A$. The bisectors $BD$ and $CE$ intersect at $O$. The area of $BOC$ is $a$. Determine the product $BD\cdot CE$ in terms of $a$.
  6. Solve the system pf equations \[\begin{cases} 2\sqrt[4]{\frac{x^{4}}{3}+4} & =1+\sqrt{\frac{3}{2}}|y|\\ 2\sqrt[4]{\frac{y^{4}}{3}+4} & =1+\sqrt{\frac{3}{2}}|x| \end{cases}.\]
  7. The side lengths of a traingle $ABC$ are $AB=9$, $BC=\sqrt{39}$, $CA=\sqrt{201}$. Find a point $M$ on the circle $(C;\sqrt{3})$ such that the sum $MA+MB$ is the maximum. 
  8. Prove that in any traingle $ABC$, \begin{align*} & \sqrt{\left(\tan\frac{A}{2}+\tan\frac{B}{2}\right)\left(\tan\frac{B}{2}+\tan\frac{C}{2}\right)}\\  + &\sqrt{\left(\tan\frac{B}{2}+\tan\frac{C}{2}\right)\left(\tan\frac{C}{2}+\tan\frac{A}{2}\right)}\\ + &\sqrt{\left(\tan\frac{C}{2}+\tan\frac{A}{2}\right)\left(\tan\frac{A}{2}+\tan\frac{B}{2}\right)}\\ \leq & 2(\cot A+\cot B+\cot C).\end{align*}
  9. Let $N=1+10+10^{2}+\ldots+10^{4023}$. Find the 2013-th digit after the decimal comma of $\sqrt{N}$. 
  10. Find the maximum and minimum values of the expression \[P=\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b},\] where $a,b,c$ are positive real numbers satisfying the condition \[\min\{a,b,c\}\geq\frac{1}{4}\max\{a,b,c\}.\] Let $\{S_{n}(x)\}$ be a sequence of real-valued functions defined by \[S_{n}(x)=\cos^{3}x-\frac{1}{3}\cos^{3}3x+\frac{1}{3^{2}}\cos^{3}3^{2}x-\ldots+\left(\frac{-1}{3}\right)^{n}\cos^{3}3^{n}x.\]
  11. Find all real values of $x$ such that \[\lim S_{n}(x)=\frac{3-3x}{4}.\] 
  12. In a con-cyclic quadrilateral $ABCD,$ let $A',B',C',D'$ be the circumcenters of triangles $BCD$, $CDA$, $DAB$ and $ABC$ respectively. Let $A'',B'',C'',D''$ be the centers of the Euler circles of triangle $BCD$, $CDA$, $DAB$, $ABC$ respectively. Prove that the two quadrilateral $A'B'C'D'$, $A''B''C''D''$ are both convex and inversely similar.




Mathematics & Youth: 2013 Issue 427
2013 Issue 427
Mathematics & Youth
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