# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 1. Which of the following two numbers is greater? $$A=\frac{326}{1955}+\frac{988}{1975}+\frac{662}{1985},\quad B=\dfrac{3951}{3950}+\dfrac{1}{5955}+\dfrac{1}{11730}.$$ 2. Let$A B C$be an isosceles triangle with$\widehat{B A C}=96^{\circ} .$A point$M$is inside the triangle such that$\widehat{M B C}=12^{\circ}$,$\widehat{M C B}=24^{\circ} .$Prove that$M A=M C$. 3. Find the maximum value of the expression$P=\max \{a, b, c\}-\min \{a, b, c\}$where$a, b, c$are real numbers satisfying the condition $$a+b+c=a^{3}+b^{3}+c^{3}-3 a b c=2.$$ 4. Solve the equation $$21 x-25+2 \sqrt{x-2}=19 \sqrt{x^{2}-x+2}+\sqrt{x+1}$$ 5.$A B C$is a triangle inscribed the circle$(O)$with$\widehat{B A C}=60^{\circ}$,$A K$is the angle-bisector of$\widehat{B A C}$($K$is on the circle$(O)$). Let$F$be the midpoint of$A K,$the ray$O F$meets the altitude$C E$of triangle$A B C$at$H .$Prove that$B H$is perpendicular to$A C$. 6. Find the minimum value of the expression $$P=\left(5 a+\frac{2}{b+c}\right)^{3}+\left(5 b+\frac{2}{c+a}\right)^{3}+\left(5 c+\frac{2}{a+b}\right)^{3}$$ where$a, b, c$are positive real numbers satisfying$a^{2}+b^{2}+c^{2}=3$7.$OABC$is a trirectangular tetrahedron at vertex$O$.$O H$is the altitude from$O$of tetrahedron. Let$R$be the circumradius of triangle$A B C .$Prove that$O H \leq \dfrac{R \sqrt{2}}{2} .$When does equality occur? 8. a) Find all distinct permutations of the word$TOANHOCTUOITRE$. b) How many permutations are there that has three consecutive$T - TTT$? c) How many permutations are there without adjacent$T$s? 9. Let$k$be a positive integer,$\alpha$is an arbitrary real number. Find the limit of sequence$\left(a_{n}\right)$where $$a_{n}=\frac{\left[1^{k} \cdot \alpha\right]+\left[2^{k} \cdot \alpha\right]+\ldots+\left[n^{k} \cdot \alpha\right]}{n^{k+1}},\, n=1,2, \ldots$$ here the notation$[x]$is the largest integer that does not exceed$x .$10. Find all functions$f: \mathbb{N} \rightarrow \mathbb{N}$which satisfy the following conditions •$f$is strictly increasing; •$f(f(n))=4 n+9$for all$n \in \mathbb{N}^{*}$•$f(f(n)-n)=2 n+9$for all$n \in \mathbb{N}^{*}$11. Does there exist a positive integer$n \geq 2$so that $$f(x)=1+4 x+4 x^{2}+\ldots+4 x^{2 n}$$ is a perfect square polynomial? 12. Let$A B C$be a triangle inscribed the circle$(O)$and$A^{\prime}$is a fixed point on$(O)$.$P$moves on$B C$,$K$belongs to$A C$so that$P K$is always parallel to a fixed line$d$. The circumcircle of triangle$A P K$cuts the circle$(O)$at a second point$E$.$A E$cuts$B C$at$M$.$A^{\prime} P$cuts the circle$(O)$at a second point$N .$Prove that the line$M N$passes through a fixed point. ##$type=three$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 405
2011 Issue 405
Mathematics & Youth