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

## $show=home 1. Let be given two bottles, the first bottle contains 1 liter of water, the second bottle is empty. One pours$\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 $$1+\sqrt{1+\frac{4}{3 !}}+\sqrt{1+\frac{9}{4 !}}+\ldots+\sqrt[n]{1+\frac{n^{2}}{(n+1) !}}<n+\frac{1}{2}$$ 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 $$\frac{M B_{1}^{2}}{h_{1}}+\frac{M B_{2}^{2}}{h_{2}}+\frac{M B_{3}^{2}}{h_{3}}+\frac{M B_{4}^{2}}{h_{4}} \geq r.$$ ##$type=three$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: 2007 Issue 357
2007 Issue 357
Mathematics & Youth