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

## $show=home 1. Let$a, b$be two natural numbers satisfying $$2006 a^{2}+a=2007 b^{2}+b.$$ Prove that$a-b$is a perfect square. 2. Let$A B C$be a triangle with$\widehat{B A C}=90^{\circ}, \widehat{A B C}=60^{\circ} .$Take the point$M$on the side$B C$such that$A B+B M=A C+C M$. Caculate the measure of$\widehat{C A M}$3. Find all positive integers$x, y$greater than 1 so that$2 x y-1$divisible by$(x-1)(y-1)$. 4. Prove that $$\frac{a^{4} b}{2 a+b}+\frac{b^{4} c}{2 b+c}+\frac{c^{4}}{2 c+a} \geq 1$$ where$a, b, c$are positive numbers satisfying the condition$a b+b c+c a \leq 3 a b c$. When does equality occur? 5. Let be given two circles$\left(O_{1}\right)$,$\left(O_{2}\right)$with centers$O_{1}$,$O_{2}$with distinct radii, externally touching each other at a point$T$. Let$O_{1} A$be a tangent to$\left(O_{2}\right)$at a point$A,$let$O_{2} B$be a tangent to$\left(O_{1}\right)$at a point$B$so that the points$A, B$are on the same side with respect to the line$O_{1} O_{2} .$Let$H$be the point on$O_{1} A, K$be the point on$O_{2} B$so that the lines$B H, A K$are perpendicular to$O_{1} O_{2}$. The line$T H$cuts$\left(O_{1}\right)$again at$E,$the line$T K$cuts$\left(O_{2}\right)$againt at$F$. The line$E F$cuts$A B$at$S$. Prove that the lines$O_{1} A$,$O_{2} B$and$T S$are concurrent. 6. Let$S$be a set consisting of 43 distinct positive integers not exceeding$100 .$For each subset$X$of$S$let$t_{X}$be the product of its elements. Prove that there exist two disjoint substs$A$and$B$of$S$such that$t_{A} t_{B}^{2}$is the cube of a natural numbers. 7. Find the greast value of the expression $$\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}-\frac{a b c d}{(a b+c d)^{2}}$$ where$a, b, c d$are distinct real numbers satisfying the conditions$\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}=4$and$a c=b d$8. a) Let$f(x)$be a polynomial of degree$n$with leading coefficient$a$. Suppose that$f(x)$has$n$distinct roots$x_{1}, x_{2}, \ldots, x_{n}$all not equal to zero. Prove that $$\frac{(-1)^{n-1}}{a x_{1} x_{2} \ldots x_{n}} \sum_{k=1}^{n} \frac{1}{x_{k}}=\sum_{k=1}^{n} \frac{1}{x_{k}^{2} f^{\prime}\left(x_{k}\right)}.$$ b) Does there exist a polynomial$f(x)$of degree$n,$with leading coefficient$a=1,$such that$f(x)$has$n$distinct roots$x_{1}, x_{2}, \ldots, x_{n},$all not equal to zero, satisfying the condition $$\frac{1}{x_{1} f^{\prime}\left(x_{1}\right)}+\frac{1}{x_{2} f^{\prime}\left(x_{2}\right)}+\ldots+\frac{1}{x_{n} f^{\prime}\left(x_{n}\right)}+\frac{1}{x_{1} x_{2} \ldots x_{n}}=0 ?$$ 9. Let$A D$,$B E$,$C F$be the altitudes and$H$be the orthocenter of an acute triangle$A B C .$Let$M$,$N$be respectively the points of intersection of$D E$and$C F$and of$D E$and$B E$. Prove that the line passing through$A$perpendicular to the line$M N$passes through the circumcenter of triangle$B H C$. ##$type=three$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 355
2007 Issue 355
Mathematics & Youth