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

## $show=home 1. Prove that a number of the form$(\overline{33 \ldots 3})^{2}$with$k(k>0)$digits$3$can always be written as the difference between of the number whose digits are 1 and the one whose digits are$2$. 2. Find all real numbers$a$such that$|3 a-2| \leq 1$and$A=\dfrac{3 a-1}{4 a^{4}+a^{2}}$is an integer. 3. Find the greatest value of the expression $$P=3 \sqrt{x}+8 \sqrt{y}$$ where$x, y$are two non-negative numbers satisfying$17 x^{2}-72 x y+90 y^{2}-9=0$4. Solve the equation $$\sqrt{x-2}+\sqrt{4-x}+\sqrt{2 x-5}=2 x^{2}-5 x$$ 5. From a point$A$outside a given circle witlı center$O$, construct two tangent lines$A B$and$A C$with$B, C$are the tangency points. On$O B$, choose$N$such that$B N=2 O N$. The perpendicular bisector of the line segment$C N$meets$O A$at$M .$Determine the ratio$\dfrac{A M}{A O}$6. Find the least value of the expression $$A=\frac{1}{a^{4}(b+1)(c+1)}+\frac{1}{b^{4}(c+1)(a+1)}+\frac{1}{c^{4}(a+1)(b+1)}$$ where$a, b, c$are positive real numbers satisfying$a b c=1$. 7. Let$a, b$be two real numbers in the interval$(0 ; 1) .$A sequence$\left(u_{n}\right)(n=0,1,2, \ldots)$is given by $$u_{0}=a, u_{1}=b,\quad u_{n+2}=\frac{1}{2010} \cdot u_{n+1}^{4}+\frac{2009}{2010} \sqrt{u_{n}},\,\forall n \in \mathbb{N}.$$ Prove that$\left(u_{n}\right)$has a finite limit, and find that limit. 8. Let$l_a$,$l_b$,$l_c$be respectively the lengths of the interior angle-bisectors from the three vertices$A$,$B$,$C$of a triangle$A B C$. Let$R$be its circumradius. Prove that $$\frac{l_{a}+l_{b}+l_{c}}{R} \leq 2\left(\cos \frac{A}{2} \cos \frac{B}{2}+\cos \frac{B}{2} \cos \frac{C}{2}+\cos \frac{C}{2} \cos \frac{A}{2}\right).$$ When does the equality occur? 9. Let$A B$be a fixed chord on a given circle$(O) .$A point$C$moves on the circle and let$M$be a point on the zigzag$A C B$(consisting of two line segments$A C$and$C B)$such that it divides this zigzag into two parts of equal length. Find the locus of$M$when$C$. moves around the circle$(O)$. 10. Let$m, n, d$be positive integers such that$m<n$and$\gcd(d, m)=1$,$\gcd(d, n)=1$. (Here,$\gcd(a, b)$denotes the greatest common divisor of$a$and$b$.) Prove that in an arithmetic progression of$n$terms with common difference$d$, there always exist two distinct terms whose product is a multiple of$m n$. 11. A function$f: \mathbb{R} \rightarrow \mathbb{R}$is given with the following properties •$f(0)=1$•$f(x) \leq 1,\, \forall x \in \mathbb{R}$•$\displaystyle f\left(x+\frac{11}{24}\right)+f(x)=f\left(x+\frac{1}{8}\right)+f\left(x+\frac{1}{3}\right)$Put$\displaystyle F(x)=\sum_{n=0}^{2009} f(x+n)$. Find$F(2009)$12. Given two positive degrees polynomials with real coefficients $$P(x)=x^{n}+a_{1} x^{n-1}+\ldots+a_{n-1} x+a_{n}$$ and $$Q(x)=x^{m}+b_{1} x^{m-1}+\ldots+b_{m-1} x+b_{m}$$ where$Q(x)$has exactly$m$real roots and$P(x)$is divisible by$Q(x)$. Prove that if there exists a$k \in\{1,2, \ldots, m\}$such that$\left|b_{k}\right|>C_{m}^{k} .2009^{k},$then there also exists an$i \in\{1,2, \ldots, n\}$such that$|a,|>2008$##$type=three$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 388
2009 Issue 388
Mathematics & Youth