2010 Issue 397

  1. Compare the following two fractions $$A=\frac{1010^{1010}}{2010^{2010}} \text { and } B=\frac{2010^{2010}}{3010^{3010}}$$
  2. Let $A B C$ be a triangle with $\widehat{B A C} \geq 60^{\circ}$. Prove that $A B+A C \leq 2 B C$.
  3. Find the remainder when dividing $3^{2^{n}}$ by $2^{n+3}$ where $n$ is a positive integer.
  4. Let $A B C$ be a triangle. Construct a parallelogram $AMPN$ so that the points $M$, $N$ are in $A B$, $A C$ respectively; $P$ lies inside the triangle $A B C$. Let $Q$ be the intersection of the line $A P$ and $B C$. Prove that $$\frac{A M \cdot A N \cdot P Q}{A B \cdot A C \cdot A Q} \leq \frac{1}{27}.$$ Find the position of $P$ when the equality occurs.
  5. Solve the equation $$\left(x+\frac{5-x}{\sqrt{x}+1}\right)^{2}=\frac{-192(\sqrt{x}+1)}{5 \sqrt{x}-x \sqrt{x}}$$
  6. Two circles $\left(O_{1}\right)$ and $\left(O_{2}\right)$ meet at points $K$ and $L$ such that their centers $O_{1}$ and $O_{2}$ lie on the same side of the line $K L .$ The tangent line to $\left(O_{1}\right)$ at $K$ meets $\left(O_{2}\right)$ at $A .$ The tangent line to $\left(O_{2}\right)$ at $K$ meets $\left(O_{1}\right)$ at $B$. Find the area of the triangle $A K B,$ given that $A L=3$, $B L=6$ and $\tan \widehat{A K B}=-\dfrac{1}{2}$.
  7. Let $a, b, c$ be positive real numbers satisfying $a+b+c=3$. Prove the following inequality $$\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}+5 \geq(a+b)(b+c)(c+a).$$ When does equality occur?
  8. Solve the system of equations $$\begin{cases}x&=3 z^{3}+2 z^{2} \\ y&=3 x^{3}+2 x^{2} \\ z&=3 y^{3}+2 y^{2}\end{cases}$$
  9. Let $A B C$ be an acute triangle. Let $a$, $b$, $c$ be the side lengths of the triangle and $h_{a}$, $h_{b}$, $h_{c}$ be the length of the corresponding altitudes. Let $r$, $R$ be respectively the inradius and circumradius of this triangle. Prove the inequality $$\frac{9 R}{a^{2}+b^{2}+c^{2}} \leq \frac{1}{h_{a}+\sqrt{h_{b} h_{c}}}+\frac{1}{h_{b}+\sqrt{h_{c} h_{a}}}+\frac{1}{h_{c}+\sqrt{h_{a} h_{b}}} \leq \frac{1}{2 r}$$
  10. Let $A$ be the set of $n$ distinct points on the plane $(n \geq 2)$ and $T(A)$ be the set of vectors whose endpoints are in $A$. Find the maximum and minimum value of $|T(A)|$. (where $|T(A)|$ denotes the cardinality of $T(A)$.)
  11. Let $f$ be a continuous function on $\mathbb{R}$ satisfying the following two conditions $$f(2012)=2011,\quad f(x)f_{4}(x)=1,\, \forall x \in \mathbb{R}.$$ Denote $f_{n}(x)=\underbrace{f(f \ldots f(x)))}_{n \text { times } f}$. Find $f(2010)$
  12. Let $\left(x_{n}\right)$ $(n=1,2, \ldots)$ be a sequence given by $$x_{1}=2,1;\quad x_{n+1}=\frac{x_{n}-2+\sqrt{x_{n}^{2}+8 x_{n}-4}}{2}.$$ For each positive integer $n$, let $\displaystyle y_{n}=\sum_{i=1}^{n} \frac{1}{x_{i+1}^{2}-4}$. Find $\displaystyle\lim_{n \rightarrow+\infty} y_{n}$.




Mathematics & Youth: 2010 Issue 397
2010 Issue 397
Mathematics & Youth
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