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

## $show=home 1. Let $$A=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots+\frac{1}{2007}+\frac{1}{2008},$$ $$B=\frac{2007}{1}+\frac{2006}{2}+\frac{2005}{3}+\ldots+\frac{2}{2006}+\frac{1}{2007}.$$ Determine$\dfrac{B}{A}$. 2. Let$A B C$be a right isosceles triangle with right angle at$A$.$M$is an arbitrary point on the side$B C$(M differs from$B$,$C$as well as the midpoint of$B C$). The altitudes from$B$and$C$onto$A M$meet$A M$at$H$and$K$, respectively. The line through$C$and parallel to$A M$meets$B H$at$N$,$A N$meets$C K$at$P$,$B P$intersects with$A M$at$I$Prove that$I B=I P$. 3. Let$a, b, c, d$and$e$be natural numbers such that $$a^{4}+b^{4}+c^{4}+d^{4}+e^{4}=2009^{2008}.$$ Prove that abcde is a multiple of$10^{4}$. 4. Let$a, b, c$be positive real numbers such that$a \geq b \geq c .$Prove the inequality $$a^{2} b(a-b)+b^{2} c(b-c)+c^{2} a(c-a) \geq 0.$$ 5. Let$A M$and$B N$be two tangent lines from two points$A$,$B$($A$differs from$B$) outside the circle$(O)$($M$,$N$are on the circle,$M$and$N$are different). Prove that if$A M=B N$, then the line$M N$is either parallel to$A B$or passes through its midpoint. 6. A number is said to be a beautiful number if it is a composite number and but it is not a multiple of either$2$,$3$or$5$(for example, the three smallest beautiful numbers are$49,77$and$91$). How many beautiful numbers which are less than$1000 ?$7. Let$D$and$E$be two points on the side$B C$of a triangle$A B C$such that$\dfrac{B D}{C D}=2\dfrac{C E}{B E}$. The circumcircle of$A D E$meets$A B$and$A C$at$M$and$N,$respectively. Prove that regardless of the positions of the points$D$and$E$on$B C,$the centroid of the triangle$A M N$lies on a fixed line. 8. Find the smallest real number$k$such that the following inequality holds for all nonnegative real numbers$a, b, c$$$\frac{a+b+c}{3} \leq \sqrt{a b c}+k \cdot \max \{|a-b|, b-c|,| c-a \mid\}.$$ 9. Determine all triple of real numbers$x, y, z$such that $$x^{6}+y^{6}+z^{6}-6\left(x^{4}+y^{4}+z^{4}\right)+10\left(x^{2}+y^{2}+z^{2}\right) - 2\left(x^{3} y+y^{3} z+z^{3} x\right)+6(x y+y z+z x)=0.$$ 10. Find all functions$f: \mathbb{R} \rightarrow \mathbb{R}$such that $$f\left(x^{3}-y\right)+2 y\left(3 f^{2}(x)+y^{2}\right)=f(y+f(x)),\, \forall x, y \in \mathbb{R}.$$ 11. Consider the sequence$\left(u_{n}\right)(n=1,2,\ldots)$given by the following recursive formula $$u_{1}=u_{2}=1,\quad u_{n+1}=4 u_{n}-5 u_{n-1},\,\forall n \geq 2.$$ Prove that$\displaystyle \lim_{n \rightarrow+\infty}\left(\frac{u_{n}}{a^{n}}\right)=0$for all real number$a>\sqrt{5}$. 12. Choose three points$A_{1}$,$B_{1}$,$C_{1}$on the sides of a triangle$A B C$,$A_{1} \in B C$,$B_{1} \in A C$,$C_{1} \in A B$such that$A A_{1}$,$B B_{1}$,$C C_{1}$meet at a common point. Again, choose three points$A_{2}$,$B_{2}$,$C_{2}$on the sides of the triangle$A_{1} B_{1} C_{1}$,$A_{2} \in B_{1} C_{1}$,$B_{2} \in A_{1} C_{1}$,$C_{2} \in A_{1} B_{1}$. Prove that the three lines$A A_{2}$,$B B_{2}$,$C C_{2}$meet at a common point if and only if so do$A_{1} A_{2}$,$B_{1} B_{2}$,$C_{1} C_{2}$. ##$type=three$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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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,
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Mathematics & Youth: 2008 Issue 377
2008 Issue 377
Mathematics & Youth