2012 Issue 415

  1. Let $$A=\dfrac{2011^{2011}}{2012^{2012}},\quad B=\dfrac{2011^{2011}+2011}{2012^{2012}+2012}.$$ Which number is greater, $A$ or $B$?.
  2. Given \[A=\sqrt{6+\sqrt{6+\ldots+\sqrt{6}}},\:B=\sqrt[3]{6+\sqrt[3]{6+\ldots+\sqrt[3]{6}}},\] where there are exactly $n$ square roots in $A$ and $n$ cube roots in $B$. Write $[x]$ for the greatest integer not exceeding $x$. Determine the value of $\left[\dfrac{A-B}{A+B}\right]$.
  3. Find all pairs of natural numbers $x,y$ such that \[x^{2}-5x+7=3^{y}.\]
  4. Prove the inequality \[\left(1+\frac{1}{2}\right)\left(1+\frac{1}{2^{2}}\right)\ldots\left(1+\frac{1}{2^{n}}\right)<3.\]
  5. Let $ABCD$ be aparallelogram. Points $H$ and $K$ are chosen on lines $AB$ and $BC$ such that triangles $KAB$ and $HCB$ are isosceles ($KA=AB$, $HC=CB$). Prove that
    a) Triangle $KDH$ is also isosceles.
    b) Triangle $KAB$, $BCH$ and $KDH$ are similar.
  6. In a triangle $ABC$ with $a=BC$, $b=CA$, $c=AB$, $A_{1}$ is the midpoint of $BC$; $O$ and $I$ are its circumcenter and incenter respectively. Prove that if $AA_{1}$ isperpendicular to $OI$ then \[\min\{b,c\}\leq a\leq\max\{b,c\}.\]
  7. The real numbers $x,y$ and $z$ are such that \[\begin{cases}\sqrt{x}\sin\alpha+\sqrt{y}\cos\alpha-\sqrt{z} & =-\sqrt{2(x+y+z)}\\ 2x+2y-13\sqrt{z} & =7 \end{cases},\quad\pi\leq\alpha\leq\frac{3\pi}{2}.\]Determine the value of $(x+y)z$.
  8. Solve the following system of equations in two variables \[\begin{cases}\log_{2}x & =2^{y+2}\\ 2\sqrt{1+x}+xy\sqrt{4+y^{2}} & =0 \end{cases}.\]
  9. A collection of prime numbers (each prime can be repeated) is said to be beautiful if their product is exactly ten times their sum. Find all beautiful collections. 
  10. Points $A,B,C,D,E$ in clockwise order, lie on the same circle. $M,N,P,Q$ are the feet of perpendicular lines from $E$ onto $AB$, $BC$, $CD$, $DA$. Prove that $MN$, $NP$, $PQ$, $QM$ are tangent lines to a certain parabole whose focus point if $E$. 
  11. The sequence $(a_{n})$ is defined recursively by the following rules \[a_{1}=1,\quad a_{n+1}=\frac{1}{a_{1}+\ldots+a_{n}}-\sqrt{2},\:n=1,2,\ldots.\] Find the limit of the sequence $(b_{n})$ where \[b_{n}=a_{1}+\ldots+a_{n}.\]
  12. Let $\alpha$ and $\beta$ be two real roots of the equation \[4x^{2}-4tx-1=0\] where $t$ is a parameter. Let $f(x)=\dfrac{2x-t}{x^{2}+1}$ be a funtion defined on the interval $[\alpha;\beta]$, and let \[g(t)=\max_{x\in[\alpha;\beta]}f(x)-\min_{x\in[\alpha;\beta]}f(x).\] Prove that if a triple $a,b,c\in\left(0;\frac{\pi}{2}\right)$ are such that $\sin a+\sin b+\sin c=1$, then \[\frac{1}{g(\tan a)}+\frac{1}{g(\tan b)}+\frac{1}{g(\tan c)}<\frac{3\sqrt{6}}{4}.\]




Mathematics & Youth: 2012 Issue 415
2012 Issue 415
Mathematics & Youth
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