BdMO National Higher Secondary 2010/10
Posted: Mon Feb 07, 2011 12:11 am
Problem 10:
$a_1, a_2,\cdots , a_k, \cdots , a_n$ is a sequence of distinct positive real numbers such that $a_1 < a_2 < \cdots <a_k$ and $a_k > a_{k+1} > \cdots > a_n$. A Grasshopper is to jump along the real axis, starting at the point $O$ and making $n$ jumps to the right of lengths $a_1, a_2, \cdots , a_n$ respectively. Prove that, once he reaches the rightmost point, he can come back to point $O$ by making $n$ jumps to the left of lengths $a_1, a_2, \cdots , a_n$ in some order such that he never lands on a point which he already visited while jumping to the right. (The only exceptions are point $O$ and the rightmost point)
$a_1, a_2,\cdots , a_k, \cdots , a_n$ is a sequence of distinct positive real numbers such that $a_1 < a_2 < \cdots <a_k$ and $a_k > a_{k+1} > \cdots > a_n$. A Grasshopper is to jump along the real axis, starting at the point $O$ and making $n$ jumps to the right of lengths $a_1, a_2, \cdots , a_n$ respectively. Prove that, once he reaches the rightmost point, he can come back to point $O$ by making $n$ jumps to the left of lengths $a_1, a_2, \cdots , a_n$ in some order such that he never lands on a point which he already visited while jumping to the right. (The only exceptions are point $O$ and the rightmost point)