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

## $show=home 1. Given $$A=\frac{1}{4} \cdot \frac{3}{6} \cdot \frac{5}{8} \cdot . . \frac{995}{998} \cdot \frac{997}{1000}$$ and $$B=\frac{2}{5} \cdot \frac{4}{7} \cdot \frac{6}{9} \ldots \frac{996}{999} \cdot \frac{998}{1001}.$$ a) Compare$A$and$B$. b) Prove that$A<\dfrac{1}{12900}$. 2. Let$A B C$be a triangle, the median$B M$and the angle bisector$C D$meets at$J$and$J B=J C .$From$A$draw$A H$perpendicular to$B C$. Prove that$J M=J H$. 3. Assume that$n \in \mathbb{N}$,$n \geq 2$. Consider all natural numbers$a_{n}=\overline{11 \ldots 1}$consisting of exactly$n$digits$1 .$Prove that if$a_{n}$is a prime number then$n$is a divisor of$a_{n}-1$. 4. Let$a, b, c, d$be real numbers in the half-open interval$\left(0 ; \frac{1}{2}\right] .$Prove that $$\left(\frac{a+b+c+d}{4-a-b-c-d}\right)^{4} \geq \frac{a b c d}{(1-a)(1-b)(1-c)(1-d)}$$ 5. Let$A B C D$be a square whose side length is$a, M$is an arbitrary point on$A B(M \neq A, M \neq B)$.$M C$meets$B D$at$P$,$M D$cuts$A C$at$Q$. Find the maximum value of the area of triangle$M P Q$and the minimum value of the area of the quadrilateral$C P Q D$. 6. Solve the equation $$25 x+9 \sqrt{9 x^{2}-4}=\frac{2}{x}+\frac{18 x}{x^{2}+1}$$ 7. Let$A B C$be a triangle with incenter$I$and centroid$G$. Let$R_{1}$,$R_{2}$,$R_{3}$be the circumradii of the triangles$I B C$,$I C A$and$I A B$respectively. Let$R_{1}^{\prime}$,$R_{2}^{\prime}$,$R_{3}^{\prime}$be the of circumradii of the triangles$G B C$,$G C A$and$G A B$respectively. Prove that $$R_{1}^{\prime}+R_{2}^{\prime}+R_{3}^{\prime} \geq R_{1}+R_{2}+R_{3}$$ 8. Let$f:|a ; b| \rightarrow \mathbb{R}(0<a<b)$be a continuous function on$|a ; b|$and differentiable on$(a ; b)$,$f(x) \neq 0$for all$x \in(a ; b)$. Prove that there exists$c \in(a ; b)$so that $$\frac{2}{a-c}<\frac{f^{\prime}(c)}{f(c)}<\frac{2}{b-c}$$ 9. Let$\left(a_{n}\right)(n=1,2, \ldots)$be a sequence given by $$a_{1}=1,\quad a_{n+1}=1+\frac{1}{a_{n}+1},\,\forall n \in \mathbb{N}^{\circ} .$$ Prove that $$a_{1}^{2}+a_{2}^{2}+\ldots+a_{2010}^{2}<4020$$ 10. Given any set$A \subset \mathbb{R},$let$A+1$be the set$A+1=\{a+1 \mid a \in A\} .$How many subsets$A$of set$\{1,2, \ldots, n\}(n \geq 1, n \in \mathbb{N})$are there such that$A \cup(A+1)=\{1,2, \ldots, n\}$? 11. Find all the functions$f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$so that $$f(x) f(y)=\beta f(x+y f(x)),\,\forall x, y \in \mathbb{R}^{+}$$ (for a given$\beta \in \mathbb{R}$,$\beta>1$.) 12. Let$A B C$be a triangle and$M$be the midpoint of the arc$B C$of its circumcircle. Let$I$,$J$,$K$be the projections of$M$onto the lines$A B$,$B C$,$C A$respectively;$X$is the intersection of$B K$and$A J$;$L$is the intersection of$C X$and$1 J$. The ray$J y$perpendicular to$M K$cuts$A L$at$T$. Prove that$C T$is perpendicular to$I M$. ##$type=three$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 Issue 400
2010 Issue 400
Mathematics & Youth