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2010 Isue 401

  1. Let $n$ be a natural number greater than $11$. Does there exist a natural number $x$ so that $n^{2010}<x<n^{2011}$ and the last 2011 digits of $x$ are $0 ?$
  2. Let $p$ be a prime number, $a$ and $b$ are natural numbers $(a<b)$ such that the sum of all irreducible fractions with denominator $p$ which lies between $a$ and $b$ is equal to $2011 .$ Find the values of $p$, $a$, $b$.
  3. Let $a$ be an $n$ -digits natural number (in decimal system) and $a^{3}$ has $m$ digits. Can $n+m$ be equal to $2011 ?$ Why?
  4. Solve the equation $$x+y+z+\sqrt{x y z}=2(\sqrt{x y}+\sqrt{y z}+\sqrt{z x}-2).$$
  5. Let $A B C D$ be a parallelogram. The angle-bisector of $B A D$ meets $B C$, $D C$ at $M$, $N$ respectively. Let $E$ be the other intersection point of the circumcircles of the triangles $B C D$ and $C M N .$ Find the measure of angle $A E C$.
  6. Find the least value of $$A=\frac{2}{|a-b|}+\frac{2}{|b-c|}+\frac{2}{|c-a|}+\frac{5}{\sqrt{a b+b c+c a}}$$ where $a$, $b$, $c$ are real numbers satisfying $a+b+c=1$ and $a b+b c+c a>0$
  7. Solve the system of equations $$\begin{cases}4+9.3^{x^{2}-2 y} &= \left(4+9^{x^{2}-2 y}\right) \cdot 7^{2 y-x^{2}+2} \\ 4^{x}+4 &= 4 x+4 \sqrt{2 y-2 x+4}\end{cases}$$
  8. Let $A B C D$ be a square of side $4 a$. $M$, $N$ are points on the spheres $S(D ; a)$, $S(C ; 2 a)$ Determine the minimum value of the sum $M A+2 N B+4 M N$.
  9. Let $A B C$ be an acute triangle with $A B \neq A C$. Let $P$ be a point inside the triangle so that $\widehat{P B A}=\widehat{P C A},$ draw lines $P M$ and $P N$ perpendicular to $A B$ and $A C$ respectively. $O$ is the midpoint of $B C$. The angle-bisectors of $B A C$ and $M O N$ intersects at $R .$ Prove that the circumcircles of the triangles $B M R$ and $C N R$ meet at another point on the line segment $B C$.
  10. Let $a$ be a positive real number. Let $\left(x_{n}\right)(n=1,2, \ldots)$ be a sequence defined by $$x_{1}=a,\quad x_{n+1}=\frac{x_{n} \sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}}{x_{n}+1},\,\forall n=1,2, \ldots$$ (there are exactly $n$ numbers $2$ in the numerator). Prove that the sequence $\left(x_{n}\right)(n=1,2, \ldots)$ has a finite limit and find this limit.
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the condition $$f(x+f(y))=f^{4}(y)+4 x^{3} f(y)+6 x^{2} f^{2}(y)+4 x f^{3}(y)+f(-x)$$ for all $x$, $y$ in $\mathbb{R}$.
  12. Let $A B C$ be an equilateral triangle with circumradius $R$ and let $P$ be a point inside the triangle. Prove that $$P A \cdot P B \cdot P C \leq \frac{9}{8} R^{3}.$$

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Mathematics & Youth: 2010 Isue 401
2010 Isue 401
Mathematics & Youth
https://www.molympiad.org/2020/09/2010-isue-401.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2010-isue-401.html
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