2007 Issue 357

  1. Let be given two bottles, the first bottle contains 1 liter of water, the second bottle is empty. One pours $\frac{1}{2}$ of the quantity of water contained in the first bottle into the second one, then one pours $\frac{1}{3}$ of the quantity of water contained in the second bottle into the first one, then one pours $\frac{1}{4}$ of the quantity of water contained in the first bottle into the second one, and so on, one pours $\frac{1}{5},$ then $\frac{1}{6},$ then $\frac{1}{7}, \ldots$ After the $2007^{\mathrm{th}}$ turn of such pouring, what are the quantities of water left in each bottle?
  2. Let $A B C$ be a right-angled triangle with right angle at $A$ and let $I$ be the point of intersection of its innner angled-bisectors. Take the orthogonal projection $E$ of $A$ on the line $B I$ then the orthogonal projection $F$ of $A$ on the line $C E .$ Prove that $2 E F^{2}=A I^{2}$
  3. Find all finite subset $A \subset \mathbb{N}^{*}$ such that there exists a finite subset $B \subset \mathbb N^{*}$ containning $A$ so that the sum of the numbers in $B$ is equal to the sum of the squares of the numbers in $A$.
  4. Prove that for every natural number $n \geq 2,$ we have $$1+\sqrt{1+\frac{4}{3 !}}+\sqrt[3]{1+\frac{9}{4 !}}+\ldots+\sqrt[n]{1+\frac{n^{2}}{(n+1) !}}<n+\frac{1}{2}$$ where $n !$ denotes $1.2 .3 \ldots n$
  5. Let $A B C D$ be a square inscribed in the circle $(O) .$ On the minor arc $\widehat{B C},$ take an arbitrary point $M$ distinct from $B, C .$ The lines $C M$ and $D B$ intersect at a point $E,$ the lines $D M$ and $A B$ intersect at a point $F$. Prove that the triangles $A B E$ and $D O F$ have equal areas.
  6. Prove that for every couple of positive integers $n, k$ the number $(\sqrt{n}-1)^{k}$ can be written in the form $\sqrt{a_{k}}-\sqrt{a_{k}-(n-1)^{k}}$ with $a_{k} \in \mathbb{N}^{*}$
  7. Prove that for every triangle $A B C$, we have $$\frac{3 \sqrt{3}}{2 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}}+8 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \geq 5.$$ When does equality occur?
  8. Consider the sequence of numbers $\left(x_{n}\right)(n=0,1,2, \ldots)$ defined as follows $x_{0}$, $x_{1}$, $x_{2}$ are given positive numbers; $$x_{n+2}=\sqrt{x_{n+1}}+\sqrt{x_{n}}+\sqrt{x_{n-1}},\, \forall n \geq 1 .$$ Prove that the sequence $\left(x_{n}\right)(n=0,1,2, \ldots)$ has a finite limit and find its limit.
  9. Let be given a tetrahedron $A_{1} A_{2} A_{3} A_{4}$ Its inscribed sphere has center $I$, has radius $r$ and touches the faces opposite the vertices $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$ at $B_{1}$, $B_{2}$, $B_{3}$, $B_{4}$ respectively. Let $h_{1}$, $h_{2}$, $h_{3}$, $h_{4}$ be the measures of the altitudes of $A_{1} A_{2} A_{3} A_{4}$ issued from $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$ respectively. Prove that for every point $M$ in space, we have $$\frac{M B_{1}^{2}}{h_{1}}+\frac{M B_{2}^{2}}{h_{2}}+\frac{M B_{3}^{2}}{h_{3}}+\frac{M B_{4}^{2}}{h_{4}} \geq r.$$




Mathematics & Youth: 2007 Issue 357
2007 Issue 357
Mathematics & Youth
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