2011 Issue 414

  1. Do there exist two natural numbers $a, b$ such that $$(3 a+2 b)(7 a+3 b)-4=\overline{22} * 12 * 2011 ?$$
  2. Equilateral triangles $A B E$ and $B C F$ are constructed outside triangle $A B C .$ Let $G$ be the centroid of triangle $A B E$ and $I$ be the midpoint of $A C .$ Find the measure of angle $G I F$.
  3. Find the smallest positive integer $n$ such that $2^{n}-1$ is divisible by $2011$.
  4. Prove that for all integers $k$, the equation $$x^{4}-2010 x^{3}+(2009+k) x^{2}-2007 x+k=0$$ does not have two distinct integer roots.
  5. From a point $M$ outside the cycle $(O),$ draw the tangents $M A, M B$ and the secant $M C D$ to $(O), C$ lies between $M$ and $D .$ $A B$ cuts $C D$ at $N .$ Prove that $$\frac{1}{M D}+\frac{1}{N D}=\frac{2}{C D}$$
  6. Let $A B C$ be a right triangle, right angle at $A,$ satisfying $A B+\sqrt{3} A C=2 B C$. Find the position of point $M$ such that $$4 \sqrt{3} \cdot M A+3 \sqrt{7} \cdot M B+\sqrt{39} \cdot M C$$ is smallest possible.
  7. Solve the equation $$\log _{3}\left(7^{x}+2\right)=\log _{5}\left(6^{x}+19\right)$$
  8. Let $A B C$ be a triangle satisfying $$\tan \frac{A}{2} \tan \frac{B}{2}=\frac{1}{2}.$$ Prove that $A B C$ is a right triangle iff $$\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}=\frac{1}{10}.$$
  9. Let $A B C$ be a triangle inscribed the circle $(O ; R), M$ is a point not on the circle respectively. Let $r$, $r_{1}$ be respectively the radii of the incircles of triangles $A B C$ and $A_{1} B_{1} C_{1}$. Prove that $$\left|R^{2}-O M^{2}\right| \geq 4 r \cdot r_{1}$$
  10. Find the greatest positive constant $k$ satisfying the inequality $$ \frac{k}{a^{3}+b^{3}}+\frac{1}{a^{3}}+\frac{1}{b^{3}} \geq \frac{16+4 k}{(a+b)^{3}}.$$
  11. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x f(y)+y)+f(x y+x)=f(x+y)+2 x y.$$
  12. For each positive integer $n$ consider a function $f_{n}$ in $\mathbb{R}$ defined by $$f_{n}(x)=\sum_{i=1}^{2 n} x^{i}+1.$$ Prove the following statements
    a) $f_{n}$ obtains its minimum value at a unique point $x_{n},$ for each positive integer $n .$ Put $S_{n}=f_{n}\left(x_{n}\right)$.
    b) $S_{n}>\dfrac{1}{2}$ for all $n \in \mathbb{N}^{*} .$ Moreover, $\dfrac{1}{2}$ is the best constant possible in the sense that there does not exist any real number $a>\dfrac{1}{2}$ such that $S_{n}>a$ for all $n \in \mathbb{N}^{*}$.
    c) The sequence $\left(S_{n}\right)$ $(n=1,2, \ldots)$ is decreasing and $\displaystyle\lim_{n\to\infty} S_{n}=\dfrac{1}{2}$.
    d) $\displaystyle\lim_{n\to\infty} x_{n}=-1$.




Mathematics & Youth: 2011 Issue 414
2011 Issue 414
Mathematics & Youth
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