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2009 Issue 389

  1. Prove that the number of decimal digits in $2008^{2009}+2^{2009}$ and $2008^{2009}$ are equal.
  2. Find the least value of the expression $A=1-x y,$ where $x$ and $y$ are real numbers satisfying the following condition $$x^{2009}+y^{2009}=2 x^{1004} y^{1004}$$
  3. Does there exist a positive integer number $n$ such that $n^{6}+26^{n}=21^{2009} ?$
  4. Let $A B C$ be a right triangle with right angle at $A .$ On the sides $A B, B C$ and $C^{\prime} A$, choose $D, E$ and $F$ respectively such that $D E \perp B C$ and $D E=D F$. $M$ is the midpoint of EF. Prove that $\widehat{B C M}=\widehat{B F E}$.
  5. Let $x, y, z$ be real numbers in the interval $(0 ; 1)$. Prove that $$\frac{1}{x(1-y)}+\frac{1}{y(1-z)}+\frac{1}{z(1-x)} \geq \frac{3}{x y z+(1-x)(1-y)(1-z)}.$$ When does the equality occur?
  6. Solve the equation $$\sqrt[3]{x^{2}+4 x+3}+\sqrt[3]{4 x^{2}-9 x-3} -\sqrt[3]{3 x^{2}-2 x+2}+\sqrt[3]{2 x^{2}-3 x-2}$$
  7. Let $A_{1} A_{2} \ldots A_{n}$ be a convex polygon $(n \geq 3)$ circumscribed around the circle centered at $J$. Prove that for any point $M$, $$\sum_{i=1}^{n} \cos \frac{A_{i}}{2}\left(M A_{i}-J A_{i}\right) \geq 0$$
  8. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satifying the condition $$f(x y+f(z))=\frac{x f(y)+y f(x)}{2}+z$$ for all $x, y, z$ in $\mathbb{R}$.
  9. Let $a, b, c$ be positive numbers. Prove that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{a} \geq \sqrt{a^{2}-a b+b^{2}}+\sqrt{b^{2}-b c+c^{2}}+\sqrt{c^{2}-c a+a^{2}}.$$
  10. Let $A B C$ be an acute triangle and the altitudes $A D, B E, C F$ meet at $H .$ On $D E$ choose a point $K$ such that $D K=D H$. On $D H,$ choose a point $I$ such that $\widehat{I K D}=90^{\circ} .$ Prove that the circle centered at $I$ with radius $I K$ touches the circle whose diameter is $B C$.
  11. Let $\left(u_{n}\right)$ be a sequence of positive numbers. Put $S_{n}=u_{1}^{3}+u_{2}^{3}+u_{3}^{3}+\ldots+u_{n}^{3}$ for $n=1,2, \ldots$. Assume that $$u_{n+1} \leq\left(\left(S_{n}-1\right) u_{n}+u_{n-1}\right) \frac{1}{S_{n+1}},$$ for any $n=2,3, \ldots$ Find $\displaystyle\lim_{n\to\infty} u_{n}$
  12. Let $p$ be a prime number and $m, n, q$ be natural numbers satisfying $2 \leq n \leq m$ and $(p, q)=1 .$ Prove that $C_{q p^{n}}^{n}$ is divisible by $p^{n+n+1}$ (The binomial coefficient $C_{n}^{k}$ is the number of ways of picking $k$ unordered elements from a set of $n$ elements).

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Mathematics & Youth: 2009 Issue 389
2009 Issue 389
Mathematics & Youth
https://www.molympiad.org/2020/09/2009-issue-389.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2009-issue-389.html
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