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

## $show=home 1. Prove that the number of decimal digits in$2008^{2009}+2^{2009}$and$2008^{2009}$are equal. 2. Find the least value of the expression$A=1-x y,$where$x$and$y$are real numbers satisfying the following condition $$x^{2009}+y^{2009}=2 x^{1004} y^{1004}$$ 3. Does there exist a positive integer number$n$such that$n^{6}+26^{n}=21^{2009} ?$4. Let$A B C$be a right triangle with right angle at$A .$On the sides$A B, B C$and$C^{\prime} A$, choose$D, E$and$F$respectively such that$D E \perp B C$and$D E=D F$.$M$is the midpoint of EF. Prove that$\widehat{B C M}=\widehat{B F E}$. 5. Let$x, y, z$be real numbers in the interval$(0 ; 1)$. Prove that $$\frac{1}{x(1-y)}+\frac{1}{y(1-z)}+\frac{1}{z(1-x)} \geq \frac{3}{x y z+(1-x)(1-y)(1-z)}.$$ When does the equality occur? 6. Solve the equation $$\sqrt{x^{2}+4 x+3}+\sqrt{4 x^{2}-9 x-3} -\sqrt{3 x^{2}-2 x+2}+\sqrt{2 x^{2}-3 x-2}$$ 7. Let$A_{1} A_{2} \ldots A_{n}$be a convex polygon$(n \geq 3)$circumscribed around the circle centered at$J$. Prove that for any point$M$, $$\sum_{i=1}^{n} \cos \frac{A_{i}}{2}\left(M A_{i}-J A_{i}\right) \geq 0$$ 8. Find all functions$f: \mathbb{R} \rightarrow \mathbb{R}$satifying the condition $$f(x y+f(z))=\frac{x f(y)+y f(x)}{2}+z$$ for all$x, y, z$in$\mathbb{R}$. 9. Let$a, b, c$be positive numbers. Prove that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{a} \geq \sqrt{a^{2}-a b+b^{2}}+\sqrt{b^{2}-b c+c^{2}}+\sqrt{c^{2}-c a+a^{2}}.$$ 10. Let$A B C$be an acute triangle and the altitudes$A D, B E, C F$meet at$H .$On$D E$choose a point$K$such that$D K=D H$. On$D H,$choose a point$I$such that$\widehat{I K D}=90^{\circ} .$Prove that the circle centered at$I$with radius$I K$touches the circle whose diameter is$B C$. 11. Let$\left(u_{n}\right)$be a sequence of positive numbers. Put$S_{n}=u_{1}^{3}+u_{2}^{3}+u_{3}^{3}+\ldots+u_{n}^{3}$for$n=1,2, \ldots$. Assume that $$u_{n+1} \leq\left(\left(S_{n}-1\right) u_{n}+u_{n-1}\right) \frac{1}{S_{n+1}},$$ for any$n=2,3, \ldots$Find$\displaystyle\lim_{n\to\infty} u_{n}$12. Let$p$be a prime number and$m, n, q$be natural numbers satisfying$2 \leq n \leq m$and$(p, q)=1 .$Prove that$C_{q p^{n}}^{n}$is divisible by$p^{n+n+1}$(The binomial coefficient$C_{n}^{k}$is the number of ways of picking$k$unordered elements from a set of$n$elements). ##$type=three$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: 2009 Issue 389
2009 Issue 389
Mathematics & Youth