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

## $show=home 1. Caculate the following sum$S$of 1002 terms $$S=\frac{1.3}{3.5}+\frac{2.4}{5.7}+\ldots+\frac{(n-1)(n+1)}{(2 n-1)(2 n+1)}+\ldots+\frac{1002.1004}{2005.2007}.$$ 2. Let$B E$and$C F$be two altitudes of a triangle$A B C .$Prove that$A B=A C$when and only when$A B+B E=A C+C F$. 3. Let$A$be a natural number greater than$9$, written in decimal system with digits$1,3,7,9$. Prove that$A$has at least a prime divisor not less than$11 .$4. Find the least value of the expression $$P=\frac{b_{1}+b_{2}+b_{3}+b_{4}+b_{5}}{a_{1}+a_{2}+a_{3}+a_{4}+a_{5}}$$ where$a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, b_{1}, b_{2}, b_{3}, b_{4}, b_{5}$are non negative real numbers satisfying the conditions$a_{i}^{2}+b_{i}^{2}=1(i=1,2,3,4,5)$and$a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}=1$5. Let be given a nonright-angled triangle$A B C$with its altitudes$A A^{\prime}$,$B B^{\prime}$,$C C^{\prime}$. Let$D$,$E$,$F$be the centers of the escribed circles of the trangles$A B^{\prime} C^{\prime}$,$B C^{\prime} A^{\prime}$,$C A^{\prime} B^{\prime}$opposite$A$,$A^{\prime}$,$A^{\prime}$respectively. The escribed circle of triangle$A B C$opposite$A$touches the lines$B C$,$C A$,$A B$at$M$,$N$,$P$respectively. Prove that the circumcenter of triangle$D E F$is the orthocenter of triangle$M N P$. 6. Ten teams participated in a football competition where each team played against every other team exactly once. When the competition was over, it turned out that for every three teams$A$,$B$,$C$if$A$defeated$B$and$B$defeated$C$then$A$defeated$C$. Prove that there were four teams$A$,$B$,$C$,$D$such that$A$defeated$B$,$B$defeated$C$,$C$defeated$D$or such that each match between them was a draw. 7. Prove that for abitrary positive numbers$a, b, c$such that$a b c \geq 1,$we have $$a+b+c \geq \frac{1+a}{1+b}+\frac{1+b}{1+c}+\frac{1+c}{1+a}.$$ When does equality occur? 8. The sequence of numbers$\left(u_{n}\right) ; n=1$,$2, \ldots,$is defined by :$u_{0}=a$and $$u_{n+1}=\sin ^{2}\left(u_{n}+11\right)-2007$$ for all natural number$n,$where$a$is a given real number. Prove that a) The equation$\sin ^{2}(x+11)-x=2007$has a unique solution. Denote it by$b$. b)$\lim u_{n}=b$9. Let$A B C$be a nonregular triangle. Take three points$A_{1}$,$B_{1}$,$C_{1}$lying on the sides$B C$.$C A$,$A B$respectively such that$\dfrac{B A_{1}}{B C}=\dfrac{C B_{1}}{C A}=\dfrac{A C_{1}}{A B}$. Prove that if the triangles$A B_{1} C_{1}$,$B C_{1} A_{1}$,$C A_{1} B_{1}$have equal circumradii then$A_{1}$,$B_{1}$,$C_{1}$are the midpoints of the sides$B C$,$C A$,$A B$respectively. ##$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 356
2007 Issue 356
Mathematics & Youth