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## $show=home 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 $$(2 \sin x-3)\left(4 \sin ^{2} x-6 \sin x+3\right)=1+3 \sqrt{6 \sin x-4}.$$ 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. ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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