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

## $show=home 1. Let$n$be a natural number greater than$11$. Does there exist a natural number$x$so that$n^{2010}<x<n^{2011}$and the last 2011 digits of$x$are$0 ?$2. Let$p$be a prime number,$a$and$b$are natural numbers$(a<b)$such that the sum of all irreducible fractions with denominator$p$which lies between$a$and$b$is equal to$2011 .$Find the values of$p$,$a$,$b$. 3. Let$a$be an$n$-digits natural number (in decimal system) and$a^{3}$has$m$digits. Can$n+m$be equal to$2011 ?$Why? 4. Solve the equation $$x+y+z+\sqrt{x y z}=2(\sqrt{x y}+\sqrt{y z}+\sqrt{z x}-2).$$ 5. Let$A B C D$be a parallelogram. The angle-bisector of$B A D$meets$B C$,$D C$at$M$,$N$respectively. Let$E$be the other intersection point of the circumcircles of the triangles$B C D$and$C M N .$Find the measure of angle$A E C$. 6. Find the least value of $$A=\frac{2}{|a-b|}+\frac{2}{|b-c|}+\frac{2}{|c-a|}+\frac{5}{\sqrt{a b+b c+c a}}$$ where$a$,$b$,$c$are real numbers satisfying$a+b+c=1$and$a b+b c+c a>0$7. Solve the system of equations $$\begin{cases}4+9.3^{x^{2}-2 y} &= \left(4+9^{x^{2}-2 y}\right) \cdot 7^{2 y-x^{2}+2} \\ 4^{x}+4 &= 4 x+4 \sqrt{2 y-2 x+4}\end{cases}$$ 8. Let$A B C D$be a square of side$4 a$.$M$,$N$are points on the spheres$S(D ; a)$,$S(C ; 2 a)$Determine the minimum value of the sum$M A+2 N B+4 M N$. 9. Let$A B C$be an acute triangle with$A B \neq A C$. Let$P$be a point inside the triangle so that$\widehat{P B A}=\widehat{P C A},$draw lines$P M$and$P N$perpendicular to$A B$and$A C$respectively.$O$is the midpoint of$B C$. The angle-bisectors of$B A C$and$M O N$intersects at$R .$Prove that the circumcircles of the triangles$B M R$and$C N R$meet at another point on the line segment$B C$. 10. Let$a$be a positive real number. Let$\left(x_{n}\right)(n=1,2, \ldots)$be a sequence defined by $$x_{1}=a,\quad x_{n+1}=\frac{x_{n} \sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}}{x_{n}+1},\,\forall n=1,2, \ldots$$ (there are exactly$n$numbers$2$in the numerator). Prove that the sequence$\left(x_{n}\right)(n=1,2, \ldots)$has a finite limit and find this limit. 11. Find all functions$f: \mathbb{R} \rightarrow \mathbb{R}$satisfying the condition $$f(x+f(y))=f^{4}(y)+4 x^{3} f(y)+6 x^{2} f^{2}(y)+4 x f^{3}(y)+f(-x)$$ for all$x$,$y$in$\mathbb{R}$. 12. Let$A B C$be an equilateral triangle with circumradius$R$and let$P$be a point inside the triangle. Prove that $$P A \cdot P B \cdot P C \leq \frac{9}{8} R^{3}.$$ ##$type=three$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

Name

2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
ltr
item
Mathematics & Youth: 2010 Isue 401
2010 Isue 401
Mathematics & Youth