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

## $show=home 1. How many integers$n$are there such that$-1964 \leq n \leq 2011$and the fraction$\dfrac{n^{2}+2}{n+9}$is reducible? 2. Let$a_{1}, a_{2}, a_{3}, \ldots$be a sequence satisfying the conditions $$a_{2}=3,\, a_{50}=300,\quad a_{n}+a_{n+1}=a_{n+2},\quad \forall n \geq 1 .$$ Find the sum of the first$48$terms$S=a_{1}+a_{2}+\ldots+a_{48}$. 3. Let$p$be a prime number. Let$x, y$be nonzero natural numbers such that$\dfrac{x^{2}+p y^{2}}{x y}$is also a natural number. Prove that $$\frac{x^{2}+p y^{2}}{x y}=p+1$$ 4. Solve the system of equations $$\begin{cases}x-2 \sqrt{y+1} &=3 \\ x^{3}-4 x^{2} \sqrt{y+1}-9 x-8 y &=-52-4 x y \end{cases}$$ 5. Let$A B$be a fixed line segment. Point$M$is such that$M A B$is an acute triangle. Let$H$be the orthocenter of$M A B$,$I$is the midpoint of$A B$,$D$is the projection of$H$onto$MI$. Prove that the product$M I$.$D I$does not depend on the position of$M$. 6. Let$a, b, c$be real numbers such that$\sin a+\sin b+\sin c \geq \dfrac{3}{2}$. Prove the inequality $$\sin \left(a-\frac{\pi}{6}\right)+\sin \left(b-\frac{\pi}{6}\right)+\sin \left(c-\frac{\pi}{6}\right) \geq 0$$ 7. Denote by$[x]$the largest integer not exceeding$x$. Solve the equation $$x^{2}-(1+[x]) x+2011=0.$$ 8. Let$E$be the center of the nine-point circle (the Euler's circle) of triangle$A B C$with edge-lengths$B C=a$,$A C=b$,$A B=c$;$E_{1}$,$E_{2}$,$E_{3}$are respectively the projections of$E$onto$B C$,$C A$,$A B$and let$R$be the circumradius of triangle$A B C$. Prove that $$\frac{S_{E_{1} E_{2} E_{3}}}{S_{A B C}}=\frac{a^{2}+b^{2}+c^{2}}{16 R^{2}}-\frac{5}{16}$$ 9. Find the minimum value of the expression $$\tan B+\tan C-\tan A \tan A+\tan C-\tan B \quad \tan A+\tan B-\tan C$$ where$A$,$B$,$C$are three angles of an acute triangle$A B C$and$C \geq A$10. a) Prove that for each positive integer$n,$the equation $$x+x^{2}+x^{3}+\ldots+2011 x^{2 n+1}=2009$$ has a unique real root. b) Let$x_{n}$be denote the real solution in part a). Prove that$0<x_{n}<\dfrac{2010}{2011}$. 11. Let$\left(u_{n}\right)$be a sequence given by $$u_{1}=2011,\quad u_{n+1}=\frac{\pi}{8}\left(\cos u_{n}+\frac{\cos 2 u_{n}}{2}+\frac{\cos 3 u_{n}}{3}\right),\, \forall n \geq 1.$$ Prove that the sequence$\left(u_{n}\right)$has a finite limit. 12. Let$A_{1} B_{1} C_{1} D_{1}$and$A_{2} B_{2} C_{2} D_{2}$be two squares in opposite direction (that is, if the vertices$A_{1}$,$B_{1}$,$C_{1}$,$D_{1}$are in clockwise order, then$A_{2}$,$B_{2}$,$C_{2}$,$D_{2}$are ordered counterclockwise) with centers$O_{1}$,$O_{2}$suppose that$D_{2}$,$D_{1}$are respectively in$A_{1} B_{1}$,$A_{2} B_{2}$. Prove that the lines$B_{1} B_{2}$,$C_{1} C_{2}$and$O_{1} O_{2}$are concurrent. ##$type=three$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 408
2011 Issue 408
Mathematics & Youth