2011 Issue 413

  1. On the cardboards, Write each five-digit numbers, from $11111$ to $99999$, on a cardboard. After mixing the cardboards, place them in a sequence in certain order. Prove that the resulting number is not a power of $2 .$
  2. Find all triple of pairwise distinct prime numbers $a, b, c$ such that $$20 a b c<30(a b+b c+c a)<21 a b c.$$
  3. Point $O$ on the median $A D$ of triangle $A B C$ is chosen such that $\dfrac{A O}{A D}=k$ $(0<k<1)$. The rays $B O$, $C O$ cut $A C$, $A B$ at $E$, $F$ respectively. Determine the value of $k$ so that $$S_{A E O F}=\frac{1}{15} S_{A B C}.$$
  4. Solve the equation $$\sqrt{x^{2}+x+19}+\sqrt{7 x^{2}+22 x+28}+\sqrt{13 x^{2}+43 x+37}=3 \sqrt{3}(x+3).$$
  5. Let $A B C$ be a right triangle with right angle at $A$. $D$ is a point within the triangle so that $C D=C A$. Choose point $M$ on the edge $A B$ so that $\widehat{B D M}=\dfrac{1}{2} \widehat{A C D}$; $N$ is the intersection of $M D$ and the altitude $A H$ of triangle $A B C$. Prove that $D M=D N$.
  6. Let $A B C$ be a triangle inscribed the circle $(O)$. $F$ is an arbitrary point on the arc $\widehat{A B}$ (not containing $C$) $(F$ differs from $A$ and $B$). $M$ is the midpoint of the arc $\widehat{B C}$ (not containing $A$); $N$ is the midpoint of the arc $\widehat{A C}$ (not containing $B$). The line passing through $C$ and parallel to $M N$ cuts the circle $(O)$ at another point $P .$ Let $I$, $I_{1}$, $I_{2}$ be the incenters of triangles $A B C$, $F A C$, $F B C$. $P I$ cuts the circle $(O)$ at $G$. Prove that the four points $I_{1}$, $F$, $G$, $I_{2}$ are concyclic.
  7. Solve the equation $$(2 \sin x-3)\left(4 \sin ^{2} x-6 \sin x+3\right)=1+3 \sqrt[3]{6 \sin x-4}.$$
  8. $x, y, z,$ and $t$ are four real numbers longing to the interval $\left[\frac{1}{2} ; \frac{2}{3}\right]$. Find the least and greatest values of the expression $$P=9\left(\frac{x+z}{x+t}\right)^{2}+16\left(\frac{x+t}{x+y}\right)^{2}$$
  9. Prove that given any prime number $p,$ there exist natural numbers $x$, $y$, $z$, $t$ so that $x^{2}+y^{2}+z^{2}-t p=0$ and $0<t<p$.
  10. Let $\left(u_{n}\right)$ be a sequence given by $$u_{1}=a ,\quad u_{n+1}=\frac{(\sqrt{2}+1) u_{n}-1}{\sqrt{2}+1+u_{n}},\,\forall n \geq 1.$$ a) Find the condition of $a$ so that all terms in the sequence are well-defined.
    b) Find the value of $a$ such that $u_{2011}=2011$.
  11. Find all functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ satisfying the following condition $$\frac{f(x+y)+f(x)}{2 x+f(y)}=\frac{2 y+f(x)}{f(x+y)+f(y)}, \forall x, y \in \mathbb{N}^{*}$$
  12. Let $A B C$ be a triangle, $\widehat{B A C} \neq 90^{\circ}$. $D$ is a fixed point on the edge $B C$. $P$ is a point inside the triangle $A B C$. Let $B_{1}$, $C_{1}$ be respectively the projections of $P$ onto $A C$, $A B$. $D B_{1}$ cuts $A B$ at $C_{2}$, $D C_{1}$ cut $A C$ at $B_{2}$. $Q$ is the intersection differs from $A$ of the circumcircles of triangles $A B_{1} C_{1}$ and $A B_{2} C_{2}$. Prove that the line $P Q$ always go through a fixed point when $P$ is moving.




Mathematics & Youth: 2011 Issue 413
2011 Issue 413
Mathematics & Youth
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