2009 Issue 381

  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 $$T=2\left(\frac{1}{S_{1}}+\frac{1}{S_{2}}+\frac{1}{S_{3}}\right)+3\left(S_{1}+S_{2}+S_{3}\right)$$




Mathematics & Youth: 2009 Issue 381
2009 Issue 381
Mathematics & Youth
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