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

## $show=home 1. Find a natural number$x$and two decimal digits$y$,$z$such that $$\left(5.10^{n}-2\right) x=3 . \overline{y \ldots y z}$$ for any natural number$n>1,$where$\overline{y \ldots y z}$(in the decimal system) contains$n-1$digits$y$. 2. Prove that for any$x$and$y$$$\frac{|x|}{2008+|x|}+\frac{|y|}{2008+|y|} \geq \frac{|x-y|}{2008+|x-y|}$$ 3. Consider an integer$n>2008$such that both$2 n-4015$and$3 n-6023$are perfect squares. Find the remainder of the division of$n$by 40 4. Solve the quation $$\sqrt{x-\frac{1}{x}}-\sqrt{1-\frac{1}{x}}=\frac{x-1}{x}.$$ 5. Let$A B C$be an isosceles triangle with the apex angle at$A$and$\widehat{B A C}=150^{\circ}$. Construct the triangles$A M B$and$A N C$such that the rays$A M$and$A N$lie in the angle$B A C$and$\widehat{A B M}=\widehat{A C N}=90^{\circ}$,$\widehat{M A B}=30^{\circ}$,$\widehat{N A C}=60^{\circ} .$Let$D$be a point on$M N$such that$N D=3 M D$.$B D$intersects with$A M$and$A N$at$K$and$E,$respectively.$B C$and$A N$meets at$F$. Prove that a)$NCE$is an isosceles triangle; b)$K F$and$C D$are parallel. 6. Find all pairs of integers$m$and$n$, both greater than$1$such that the following equality $$a^{m+n}+b^{m+n}+c^{m+n}=\frac{a^{m}+b^{m}+c^{m}}{m} \cdot \frac{a^{n}+b^{n}+c^{n}}{n}$$ is true for all real numbers$a$,$b$,$c$satisfying$a+b+c=0$. 7. Let$k$be a positive integer and let$a$,$b$,$c$be positive real numbers such that$a b c \leq 1$. Prove the equality $$\frac{a}{b^{k}}+\frac{b}{c^{k}}+\frac{c}{a^{k}} \geq a+b+c.$$ 8. Let$C$be a point on a fixed circle whose diameter is$A B=2 R$($C$is different from$A$and$B$). The incircle of$A B C$touches$A B$and$A C$at$M$and$N$, respectively. Find the maximum value of the length of$M N$when$C$moves on the given fixed circle. 9. Let$\left(x_{n}\right)(n=0.1,2, \ldots)$be a sequence such that $$x_{0}=2,\quad x_{n+1}=\frac{2 x_{n}+1}{x_{n}+2},\,\forall n=0,1,2, \ldots.$$ Determine$\displaystyle \left[\sum_{k=1}^{n} x_{k}\right]$where$[x]$denote the largest integer not exceeding$x$. 10. Prove that if$a$,$b$,$c$are positive numbers whose product$a b c=1,$then $$\frac{a}{\sqrt{8 c^{3}+1}}+\frac{b}{\sqrt{8 a^{3}+1}}+\frac{c}{\sqrt{8 b^{3}+1}} \geq 1.$$ 11. Let$f: \mathbb{R} \rightarrow \mathbb{R}$be a function such that$f(0)=0$and$\dfrac{f(t)}{t}$is a monotonic function on$\mathbb{R} \backslash\{0\}$. Prove that $$x \cdot f\left(y^{2}-z x\right)+y \cdot f\left(z^{2}-x y\right)+z \cdot f\left(x^{2}-y z\right) \geq 0$$ for all positive numbers$x$,$y$and$z$. 12. Let$A_{1} A_{2} A_{3} A_{4}$be a tetrahedron. Denote by$B_{i}(i=1,2,3,4)$the feet of the altitude from a given point$M$onto$A_{i} A_{i+1}$(where we consider$A_{5}$as identical to$A_{1}$). Find the smallest value of$\displaystyle\sum_{1 \leq i \leq 4} A_{i} A_{i+1} . A_{i} B_{i}$##$type=three$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: 2008 Issue 375
2008 Issue 375
Mathematics & Youth