2012 Issue 419

  1. Let $$A=\frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\ldots+\frac{1}{50^{2}}$$ and $B=\dfrac{165}{101}$. Compare $A$ and $B$.
  2. Let $A B C$ be a right isosceles triangle with right angle at $A .$ If there exists a point $M$ inside the triangle with $\widehat{M B A}=$ $\widehat{M A C}=\widehat{M C B}$. Find the ratio $M A: M B: M C$.
  3. Find the minimum values of the natural numbers $a, b, c$ satisfying $$\begin{align} & a+(a+1)+(a+2)+\ldots+(a+6) \\ =& b+(b+1)+(b+2)+\ldots+(b+8) \\ = &c+(c+1)+(c+2)+\ldots+(c+10).\end{align}$$
  4. Solve the following equation $$6(x-1) \sqrt{x+1}+\left(x^{2}+2\right)(\sqrt{x-1}-3)=x\left(x^{2}+2\right).$$
  5. Let $M$ be the midpoint of the arc $A B$ of a semicircle with center $O$ and diameter $A B$. $A C$ meets $M O$ at $D$. Prove that the circumcenter of triangle $M D C$ always lies on a fixed line when $C$ moves on the semicircle.
  6. Let $a, b, c$ be positive real numbers. Prove that $$6\left(a^{3}+b^{3}+c^{3}\right) \geq 18 a b c+\left(\sqrt[3]{a(b-c)^{2}}+\sqrt[3]{b(c-a)^{2}}+\sqrt[3]{c(a-b)^{2}}\right)^{3}.$$
  7. Let $A B C$ be an acute triangle which is not isosceles; and $H$, $O$ be its orthocenter and circumcenter respectively; let $D$, $E$ be respectively the foot of the altitude from $A$, $B$. The lines $O D$ and $B E$ intersect at $K$, $O E$ and $A D$ intersect at $L$. Let $M$ be the midpoint of edge $A B$. Prove that $K$, $L$, $M$ are collinear if and only if $C$, $D$, $O$, $H$ lies on the same circle.
  8. Find all pairs of positive integers $(n, k)$ satisfying $C_{3 n}^{n}=3^{n} n^{k},$ where $$C_{p}^{m}=\frac{p !}{m !(p-m) !} ; 0 \leq m \leq p, p \neq 0, m, p \in \mathbb{N}.$$
  9. Let $a, b, c$ be three positive real numbers satisfying $$15\left(\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}\right)=10\left(\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a}\right)+2012.$$ Find the largest possible value of the expression $$P=\frac{1}{\sqrt{5 a^{2}+2 a b+2 b^{2}}}+\frac{1}{\sqrt{5 b^{2}+2 b c+2 c^{2}}}+\frac{1}{\sqrt{5 c^{2}+2 c a+2 a^{2}}}.$$
  10. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$ we have $$f(x f(y))+f(f(x)+f(y))=y f(x)+f(x+f(y)).$$
  11. On the interval $[a ; b],$ pick $k$ distinct points $x_{1}, x_{2}, \ldots, x_{k}$. Let $d_{n}$ be the product of the distances from $x_{n}$ to the $k-1$ remaining points; $n=1,2,3 \ldots, k .$ Find the smallest value of $\displaystyle \sum_{n=1}^{k} \frac{1}{d_{n}}$.
  12. Given a triangle $A B C$ and an arbitrary point $M$. Prove that $$\frac{M A}{B C}+\frac{M B}{C A}+\frac{M C}{A B} \geq \frac{B C+C A+A B}{M A+M B+M C}$$




Mathematics & Youth: 2012 Issue 419
2012 Issue 419
Mathematics & Youth
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