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

## $show=home 1. Given the sum $$S=\frac{5}{1.2 .3}+\frac{8}{2.3 .4}+\frac{11}{3.4 .5}+\ldots+\frac{6026}{2008.2009 .2010}.$$ Compare$S$with$2$. 2. Let$A B C$be a triangle with$\widehat{B A C}=50^{\circ}$,$\widehat{A B C}=72^{\circ}$. Outside of the triangle$A B C,$draw a triangle$B D C$such that$\widehat{C B D}=28^{\circ}$;$\widehat{B C D}=22^{\circ}$. Find the measure of the angle$A D B$. 3. Find all possible pair of integers$x$,$y$satisfying the following condition $$x^{2}+y^{2}=(x-y)(x y+2)+9$$ 4. Given$a, b, c, d \in[0 ; 1]$satisfies the following condition $$a+b+c+d=x+y+z+t=1.$$ Prove the inequality $$a x+b y+c z+d t \geq 54 a b c d.$$ 5. Let$A B C$be a triangle with$\widehat{B A C}=45^{\circ}$. The attitudes$B D$and$C E$intersect at$H .$Let$I$be a midpoint of$D E .$Prove that the line$H I$goes through the centroid of the triangle$A B C$. 6. Solve the equation $$\sqrt{x}+\sqrt{x}+4 \sqrt{17-x}+8 \sqrt{17-x}=34.$$ 7. A number is said to be an interesting number if it has$10$digits, all are distinct, and is a multiple of$11111$. How many interesting numbers are there? 8. Let$A_{1} A_{2} A_{3} \ldots A_{n}$be a convex polygon$(n \geq 3)$on the plane$(P)$and let$S$be a point outside$(P)$. Another plane$(\alpha)$intersects the sides$S A_{1}, S A_{2}, \ldots, S A_{n}$at$B_{1}, B_{2},\ldots, B_{n}$respectively such that $$\frac{S A_{1}}{S B_{1}}+\frac{S A_{2}}{S B_{2}}+\ldots+\frac{S A_{n}}{S B_{n}}=a$$ where$a$is a given positive number. Prove that such a plane$(\alpha)$always contains a fixed point. 9. Two circles$\omega_{1}$,$\omega_{2}$intersect at points$A$,$B$.$C D$is a common tangent line of$\omega_{1}$,$\omega_{2}(C \in \omega_{1}, D \in \omega_{2}$) where point$B$is closer to$C D$than point$A$.$C B$cuts$A D$at$E$,$D B$cuts$CA$at$F$and$E F$cuts$A B$at$N$.$K$is the orthogonal projection of$N$onto$C D$. a) Prove that$\widehat{C A B}=\widehat{D A K}$. b) Let$O$be the circumcenter of the triangle$A C D$and$H$is the orthocenter of the triangle$K E F .$Prove that$O$,$B$,$H$are collinear 10. Let$\left(x_{n}\right)$be the sequence where $$x_{1}=5,\quad x_{n+1}=\frac{x_{n}^{2010}+3 x_{n}+16}{x_{n}^{2009}-x_{n}+11},\,n=1,2, \ldots$$ For each positive number$n,$put$\displaystyle y_{n}=\sum_{i=1}^{n} \frac{1}{x_{i}^{2009}+7}$. Determine$\displaystyle \lim_{n\to\infty}y_{n}$. 11. Find all functions$f: \mathbb{R} \rightarrow \mathbb{R}$satisfy the following equation $$f(f(x-y))=f(x) \cdot f(y)+f(x)-f(y)-x y,\,\forall x, y \in \mathbb{R}.$$ 12. For each$n \in \mathbb{N}$, let$a_{n}$be a number of bijections$f:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, n\}$such that$f(f(k))=k$for all$k \in\{1,2, \ldots, n\}$. a) Prove that$a_{n}$is an even number for every$n \geq 2$. b) Prove that for every$n \geq 10$and$n$is divisible by$3$then$a_{n}-a_{n-9}$is divisible by$3$. ##$type=three$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

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Mathematics & Youth: 2011 Issue 403
2011 Issue 403
Mathematics & Youth