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

## $show=home 1. Rearrange the following rational numbers in increasing order $$\frac{1005}{2002}, \frac{1007}{2006}, \frac{1009}{2010}, \frac{1011}{2014}$$ 2. In an isosceles triangle$A B C$(at vertex$A$), choose a point$M$such that $$\widehat{M A C}=\widehat{M B A}=\widehat{M C B}.$$ Compare the areas of the triangles$A B M$and$C B M$. 3. Determine the sum of all rational numbers of the form$\dfrac{a}{b}$where$a$,$b$are natural divisors of$27000$and$\gcd(a, b)=1$. 4. Given$a$,$b$,$c$such that$a>0$,$b>c$,$a^{2}=b c$,$a+b+c=a b c$. Prove the inequalities $$a \geq \sqrt{3},\quad b \geq \sqrt{3},\quad 0<c \leq \sqrt{3}.$$ 5. Let$ABCD$be a quadrilateral where$\widehat{A B C}=\widehat{A D C}=90^{\circ}$and$\widehat{B C D}<90^{\circ}$. Choose a point$E$on the opposite ray of$A C$such that$D A$is the angle-bisector of$B D E$. Let$M$be chosen arbitrarily between$D$and$E$, choose another point$N$on the opposite ray of$B E$such that$\widehat{N C B}=\widehat{M C D}$. Prove that$M C$is the angle bisector of$D M N$. 6. Prove that the equation$x^{3}+3 y^{3}=5$has infinitely many rational solutions. 7. If$a$,$b$,$c$are the length of the sides of a triangle, prove that $$\frac{1}{\sqrt{a b+a c}}+\frac{1}{\sqrt{b c+b a}}+\frac{1}{\sqrt{c a+c b}} \geq \frac{1}{\sqrt{a^{2}+b c}}+\frac{1}{\sqrt{b^{2}+a c}}+\frac{1}{\sqrt{c^{2}+a b}}.$$ 8. Let$A B C D . A^{\prime} B^{\prime} C^{\prime} D^{\prime}$be a parallelepiped and let$S_{1}$,$S_{2}$,$S_{3}$denote the areas of the sides$A B C D$,$A B B^{\prime} A^{\prime}$and$A D D^{\prime} A^{\prime}$respectively. Given that the sum of squares of the areas of all sides of the tetrahedron$A B^{\prime} C D^{\prime}$equals$3$, find the smallest possible value of the following expression $$T=2\left(\frac{1}{S_{1}}+\frac{1}{S_{2}}+\frac{1}{S_{3}}\right)+3\left(S_{1}+S_{2}+S_{3}\right)$$ ##$type=three$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: 2009 Issue 381
2009 Issue 381
Mathematics & Youth