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## $show=home 1. The number$(\overline{9 x})^{8}$, where$x \in\{0,1,2, \ldots, 9\}$contains how many digits when written in decimal form? 2. Let$H$be the orthocenter of an acute triangle$A B C$and denote by$M$the midpoint of$B C$. The perpendicular line to$M H$through$H$meets$A B$and$A C$at$P$and$Q$respectively. Prove that$H P=H Q$. 3. Prove that if$a, b, c$and$d$are distinct positive integers such that the sum $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}$$ is also an integer then the product$a b c d$is a perfect square. 4. Compare the values of the following two numbers $$A=\min _{|y| \leq 1} \max _{|x| \leq 1}\left(x^{2}+y x\right) ,\quad B=\max _{|x| \leq 1} \min _{|y| \leq 1}\left(x^{2}+y x\right).$$ 5. Solve the system of equations $$\begin{cases}\sqrt{x}+\sqrt{y}+\sqrt{z}-\dfrac{1}{\sqrt{x}}-\dfrac{1}{\sqrt{y}}-\dfrac{1}{\sqrt{z}} &=\dfrac{8}{3} \\ x+y+z+\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} &=\dfrac{118}{9} \\ x \sqrt{x}+y \sqrt{y}+z \sqrt{z}-\dfrac{1}{x \sqrt{x}}-\dfrac{1}{y \sqrt{y}}-\dfrac{1}{z \sqrt{z}} &=\dfrac{728}{27}\end{cases}.$$ 6. Let$A B C D$be a parallelogram. Let$M$be a point in the plane spanned by the parallelogram$A B C D$such that$\widehat{M D A}=\widehat{M B A}$. Prove that the two triangles$M A B$and$M C D$share a common orthocenter. 7. Let$AD$,$BE$,$CF$be the three bisectors of a triangle$A B C$($D \in B C$,$ E \in C A$,$F \in A B)$. Denote by$O$the circumcenter of this triangle and by$R$its. Let$O_{1}$,$O_{2}$and$O_{3}$be the circumcenters of the triangles$A B D$.$B C E$and$A C F$respectively. Prove that $$\frac{3}{2} R \leq O O_{1}+O O_{2}+O O_{3}<2 R.$$ 8. Let$a, b, c$be positive real numbers such that$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} \leq 1 .$Find the smallest possible value of $$P=[a+b]+[b+c]+[c+a]$$ where$[x]$is the largest integers which is smaller than$x$. 9. Find all funtions$f: \mathbb{R} \rightarrow \mathbb{R}$such that the following equality holds for all$x, y \in \mathbb{R}$$$f\left(x^{3}-y\right)+2 y\left(3 f^{2}(x)+y^{2}\right)=f(y+f(x)).$$ 10. Prove that in any triangle$A B C$, the following inequality holds $$-1<6 \cos A+3 \cos B+2 \cos C<7$$ 11. Let ABC be a triangle with circumcircle$(O ; R)$. Let$E$be the midpoint of$A B$and denote by$F$the point on$A C$such that$\dfrac{A F}{A C}=\dfrac{1}{3} .$Construct a parallelogram$A E M F$Prove that $$M A+M B+M C \leq \sqrt{11\left(R^{2}-O M^{2}\right)}.$$ Now suppose that$(O ; R)$is fixed. Construct a triangle$A B C,$inscribed in$(O, R)$such that $$M A+M B+M C=\sqrt{11\left(R^{2}-O M^{2}\right)}.$$ 12. Let$A B C D$be a tetrahedron whose sides$D A$,$D B$and$D C$are pairwise perpendicular. Denote by$x, y, z$the angles formed from sides$A B$,$B C$and$C A$respectively. Prove that $$\left(2+\tan ^{2} x\right)\left(2+\tan ^{2} y\right)\left(2+\tan ^{2} z\right) \geq 64.$$ ##$type=three$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: 2007 Issue 361
2007 Issue 361
Mathematics & Youth