2007 Issue 356

  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.




Mathematics & Youth: 2007 Issue 356
2007 Issue 356
Mathematics & Youth
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