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

## $show=home 1. Consider the following sequence $$a_{1}=3,\, a_{2}=4,\, a_{3}=6, \ldots, a_{n+1}=a_{n}+n, \ldots$$ a) Is$2006$a number in the above sequence? b) What is the$2007^{\text {th }}$number in this sequence? c) Calculate the sum of the first$100$numbers in the above sequence? 2. Let$A B C$be a right triangle with right angle at$A,$and$\widehat{A C B}=54^{\circ} .$Choose a point$E$on the open ray in opposite direction to$C A$such that$\widehat{A B E}=54^{\circ} .$Prove that$B C < A E$3. Consider the following sum of$2006$terms $$S=\sqrt{\frac{2+1}{2}}+\sqrt{\frac{3+1}{3}}+\sqrt{\frac{4+1}{4}}+\ldots+200 \sqrt{\frac{2007+1}{2007}}.$$ Find$[S]$. (Here$[a]$denote the largest integer which does not exceed$a$.) 4. Solve the following system of equations $$\begin{cases} x-\dfrac{4}{x} &=2 y-\dfrac{2}{y} \\ 2 x &=y^{3}+3\end{cases}$$ 5. Let$a, b, c$be numbers, all greater than or equal$-\dfrac{3}{2},$such that $$a b c+a b+b c+c a+a+b+c \geq 0.$$ Prove that$a+b+c \geq 0$. 6. Let$M$,$N$and$P$be three points outside a given triangle$A B C$such that $$\angle C A N=\angle C B M=30^{\circ},\,\angle A C N=\angle B C M=20^{\circ},\,\angle P A B=\angle P B A=40^{\circ} .$$ Show that$M N P$is an equilateral triangle. 7. Let$A B C$be a right triangle with right angle at$A$and let$A H$be the altitute through$A$. Denote by$I$,$I_{1}$,$I_{2}$the incenters of the triangles$A B C$,$A H B$and$A H C$. respectively. Prove that the circumcircle of the triangle$I I_{1} I_{2}$is exactly the incircle of the triangle$A B C$. 8. Find all functions$f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$which satisfies the following identity $$f(x) \cdot f(y)=f(x+y f(x)), \forall x, y \in \mathbb{R}^{+}$$ 9. Find the largest real number$m$such that there exists a number$k$in the interval$[1 ; 2]$such that the following inequality $$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)^{2 k} \geq(k+1)(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)+m$$ is satisfied for all positive real numbers$a, b, c$. 10. Let$\left(u_{k}\right)(k=1,2, \ldots, n,)$be an increasing sequence and let$A_{n}$be the set of all positive numbers of the form$u_{i}-u_{j}(1 \leq j<i \leq n)$. Prove that if$A_{n}$has fewer that$n$elements, then$\left(u_{k}\right)(k=1,2, \ldots, n)$forms an arithmetic sequence. 11. Let$A B C D$be a quadrilateral with nonparallel opposite sides and let$O$denote the intersection of the diagonals$A C$and$B D .$The circumcircles of the triangles$O A B$and$O C D$meet at$X$and$O$. The circumcircles of the triangles$O A B$and$O C B$meet at$Y$and$O .$The circles with diameters$A C$and$B D$intersect at$Z$and$T$. Prove that either$X$,$Y$,$Z$,$T$are colinear or they lie in a circle. 12. Consider a closed polygon$A_{1} A_{2} \ldots A_{n}$where each of its$n$sides is tangent to a sphere$\mathscr{C}$at center$O$and radius$R .$Let$G$be the centroid of the system of points$A_{i}$,$i=1,2, \ldots, n$(that is,$\overline{G A_{1}}+\overline{G A_{2}}+\ldots+\overline{G A_{n}}=\overrightarrow{0}$) and denote$\angle A_{i}=\angle A_{i-1} A_{i} A_{i+1}$(with the convention that$A_{0} \equiv A_{n}$and$A_{n+1} \equiv A_{1}$.) Prove the inequality $$\sum_{i=1}^{n} \frac{G A_{i}}{\sin \frac{A_{i}}{2}} \geq \frac{1}{n R} \sum_{1 \leq i<j \leq n} A_{i} A_{j}^{2}$$ ##$type=three$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: 2007 Issue 360
2007 Issue 360
Mathematics & Youth