2007 Issue 359

  1. Determine the sum of $2005$ terms $$S=\frac{2^{2}}{1.3}+\frac{3^{2}}{2.4}+\frac{4^{2}}{3.5}+\ldots+\frac{2006^{2}}{2005.2007}$$
  2. Let $ABC$ be an isosceles right triangle with right angle at $A .$ Pick an arbitrary point $M$ on $A C$. The foot of the altitude with $B C$ through $M$ is $H .$ Let $I$ be the midpoint of $BM$. Find the value of the angle $\angle H A I .$
  3. Prove that there exist infinitely many positive integers $n$ such that $n !$ is a multiple of $n^{2}+1$
  4. Let $a, b, c$ be positive numbers such that $a+b+c=a b c .$ Prove that $$\frac{a}{b^{3}}+\frac{b}{c^{3}}+\frac{c}{a^{3}} \geq 1.$$
  5. Solve the equation $$\frac{4}{x}+\sqrt{x-\frac{1}{x}}=x+\sqrt{2 x-\frac{5}{x}}$$
  6. Let $A B C$ be a right triangle at $A$ with $A B<A C$, $B C=2+2 \sqrt{3}$ and its incircle has radius is $1 .$ Determine the value of the angles $\angle B$ and $\angle C$.
  7. Let $A B C$ be an acute triangle with orthocenter $H .$ Let $A G$ be the altitude through $A$ and let $D$ be the midpoint of $B C .$ The circles, whose diameters are $B C$ and $A D$ respectively, meet at $E$ and $F$. Prove that a) The circumcircles of $H, E$ and $G$ and of $H$, $F$ and $G$ are both tangent to the circle whose diameter is the side $B C$. b) $H, E$, and $F$ lie on the same line.
  8. Find the maximum value of the following expression $$x_{1}^{3} x_{2}^{2}+x_{2}^{3} x_{3}^{2}+\ldots+x_{n}^{3} x_{1}^{2}+n^{2(n-1)} x_{1}^{3} x_{2}^{3} \ldots x_{n}^{3}$$ where $x_{1}, x_{2}, \ldots, x_{n}$ are non-negative numbers whose sum is $1$ $(n \geq 2)$.
  9. Solve the system of equations $$\begin{cases}20\left(x+\dfrac{1}{x}\right) &=11\left(y+\dfrac{1}{y}\right) &=2007\left(z+\dfrac{1}{z}\right) \\ x y+y z+z x &=1 \end{cases}$$
  10. Find the limit at $x=0$ of the following function $$\frac{\sqrt{\frac{\cos 2 x+\sqrt[3]{1+3 x}}{2}}-\sqrt[3]{\frac{\cos 3 x+3 \cos x-\ln (1+x)^{4}}{4}}}{x}$$
  11. Let $\left(O_{1}\right)$ be a circle inside a triangle $A B C$ and touches both sides $A B$ and $A C$. Construct a second circle $\left(O_{2}\right)$ through $B$, $C$ and touches outside of $\left(O_{1}\right)$ at $T .$ Prove that the angle bisector of $\widehat{B T C}$ passes through the incenter of the triangle $A B C$.
  12. Let $A_{1} A_{2} A_{3} A_{4}$ be a tetrahedron with perpendicular opposite edges. Denote by $S_{i}$ the area of the opposing faces of the vertices $A_{I}$ $(i=1,2,3,4) .$ Find all possible points $M$ where the sum $$S_{1} \cdot M A_{1}+S_{2} \cdot M A_{2}+S_{3} \cdot M A_{3}+S_{4} \cdot M A_{4}$$ is smallest possible.




Mathematics & Youth: 2007 Issue 359
2007 Issue 359
Mathematics & Youth
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS CONTENT IS PREMIUM Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy