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

## $show=home 1. The first$100$positive integer numbers are written consecutively in a certain order. Call the resulting number$A$. Is$A$a multiple of$2007$? 2. Let$ABC$be a nonisosceles triangle, where$AB$is the shortest side. Choose a point$D$in the opposite ray of$BA$such that$BD=BC$. Prove that$\angle ACD<90^\circ$. 3. Let$a$,$b$,$c$be positive reals such that$a+b+c=1$. Prove that $$\left(a+\dfrac{1}{b}\right)\left(b+\dfrac{1}{c}\right)\left(c+\dfrac{1}{a}\right)\geq \left(\dfrac{10}{3}\right)^3.$$ 4. Solve the equation in$\mathbb{R}$$$(x^4+5x^3+8x^2+7x+5)^4+(x^4+5x^3+8x^2+7x+3)^4=16$$ 5. Let$AH$denote the altitude of a right triangle$ABC$, right angle at$A$and suppose that$AH^2=4AM\cdot AN$where$M$,$N$are the feet of the altitude from$H$to$AB$and$AC$, respectively. Find the measures of the angles of triangle$ABC$. 6. Find all$(x,y)\in\mathbb{Z}^2$such that $$x^{2007}=y^{2007}-y^{1338}-y^{669}+2.$$ 7. Let$(x_n)$be a sequence given by $$x_1=5,\quad x_{n+1}=x_n^2-2,\,\forall n\geq 1.$$ Calculate$\displaystyle\lim_{n\to\infty}\dfrac{x_{n+1}}{x_1x_2\cdots x_n}.$8. Let$a$,$b$,$c$and denote the three sides of a triangle$ABC$. Its altitudes are$h_a$,$h_b$,$h_c$and the radius of its three escribed circles are$r_a$,$r_b$,$r_c$. Prove that $$\dfrac{a}{h_a+r_a}+\dfrac{b}{h_b+r_b}+\dfrac{c}{h_c+r_c}\geq \sqrt{3}.$$ 9. In a quadrilateral$ABCD$, where$AD=BC$meets at$O$, and the angle bisector of the angles$DAB$,$CBA$meets at$I$. Prove that the midpoints of$AB$,$CD$,$OI$are colinear. 10. Prove that for all$a,b,c\in [1,+\infty)$we have $$\left(a-\dfrac{1}{b}\right)\left(b-\dfrac{1}{c}\right)\left(c-\dfrac{1}{a}\right)\geq \left(a-\dfrac{1}{a}\right)\left(b-\dfrac{1}{b}\right)\left(c-\dfrac{1}{c}\right).$$ 11. Find the number of binary strings of length$n(n>3)$in which the substring$01$occurs exactly twice. 12. Let$f:\mathbb{N}\to\mathbb{R}$be a function such that $$f(1)=\dfrac{2007}{6},\quad \dfrac{f(1)}{1}+\dfrac{f(2)}{2}+\cdots+\dfrac{f(n)}{n}=\dfrac{n+1}{2}\cdot f(n)\forall n\in\mathbb{N}.$$ Find the limit$\displaystyle\lim_{n\to\infty} (2008+n)f(n)$. ##$type=three$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: 2007 Issue 363
2007 Issue 363
Mathematics & Youth