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

## $show=home 1. For each integer$n$greater than$6,$denote by$A_{n}$the collection of integers which are less than$n$and not less than$\dfrac{n}{2} .$Find$n$such that there are no perfect square in$A_{n}$. 2. Let$A B C$be an acute triangle with$\widehat{B A C}=60^{\circ} . E$and$F$are two points on$A C$and$A B$respectively such that$\widehat{E B C}=\widehat{F C B}=30^{\circ}$. Prove that $$B F=F E=E C \geq \dfrac{B C}{2}.$$ 3. Find four distinct integers$a, b, c, d$in the set$\{10 ; 21 ; 37 ; 51\}$such that $$a b+b c-a d=637.$$ 4. Solve for$x$$$(x+3) \sqrt{(4-x)(12+x)}=28-x.$$ 5. Let$A B C$be a triangle with$\widehat{B A C} \neq 45^{\circ}$and$\widehat{A I O}=90^{\circ}$where$(O)$and$(I)$are its circumcircle and incircle, respectively. Choose a point$D$on the ray$B C$such that$B D=A B+A C$. The tangent line through$D$touches$(O)$at$E$. The tangent line through$B$of$(O)$meets$D E$at$F$;$C F$meets$(O)$at another point denoted by$K$. Let$G$be the centroid of the triangle$A B C$. Prove that$I G$is parallel to$E K$. 6. Let$a_{1}, a_{2}, \ldots, a_{n}$be positive integers such that the following two properties hold •$a_{i}<2008$for all$i=1,2, \ldots, n$• The greatest common divisor of any pair of numbers is greater than$2008$. Prove that$\displaystyle\sum_{i=1}^{n} \frac{1}{a_{i}}<2$. 7. Prove the following inequality $$(3 a+2 b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \leq \frac{45}{2}$$ where$a, b,$and$c$are in the interval$[1 ; 2]$When does equality occur? 8. Given a triangle$A B C$with circumcircle$(O)$and three sides$B C=a$,$C A=b$,$A B^{\prime}=c .$Denote by$A_{1}$,$B_{1},$and$C_{1}$the midpoints of$B C$,$C A,$and$A B$respectively; and$A_{2}$,$B_{2}$,$C_{2}$are the midpoint of the arcs$\widehat{B C}$(which does not contain$A$),$\widehat{C A}$(which does not contains$B$), and$\widehat{A B}$(which does not contain$C$). Draw the circles$\left(Q_{1}\right),\left(O_{2}\right),$and$\left(O_{3}\right)$whose diameters are$A_{1} A_{2}$,$B_{1} B_{2},$and$C_{1} C_{2}$respectively. Prove the inequality $$\mathcal{P}_{A/\left(O_{1}\right)}+\mathcal{P}_{B /\left(O_{2}\right)}+\mathcal{P}_{C /\left(O_{3}\right)} \geq \frac{(a+b+c)^{2}}{3}.$$ When does equality occur? 9. Find all positive integers$x, y, z, n$such that $$x !+y !+z !=5 . n !$$ where$k !=1 \times 2 \times \ldots \times k$. 10. Let$a$and$b$be two real numbers in the open interval$(0 ; 4) .$A sequence$\left(a_{n}\right),(n=0,1, \ldots)$is constructed by the following recursive formula $$a_{0}=a,\, a_{1}=b,\quad a_{n+2}=\frac{2\left(a_{n+1}+a_{n}\right)}{\sqrt{a_{n+1}}+\sqrt{a_{n}}}.$$ Prove that the sequence$\left(a_{n}\right)$(n=0,1, \ldots)$ converges and find its limit.
11. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$f(f(x))+f(x)=\left(26^{3^{2008}}+\left(26^{32008}\right)^{2}\right) x.$$
12. Let $S$ denote the surface area of a given tetrahedron. Prove that the sum of areas of the six angle-bisectors of this tetrahedron does not exceed $\dfrac{\sqrt{6}}{2} S$.

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Mathematics & Youth: 2008 Issue 369
2008 Issue 369
Mathematics & Youth