BdMO H.Sec 2010. Problem 10

Zzzz
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BdMO H.Sec 2010. Problem 10

আচ্ছা, আমি কখনোই ১০ নম্বর সমস্যাটার সমাধান কোথাও দেখি নাই। নিজেও করতে পারি নাই অবশ্যই। কেউ কি দিতে পারেন?

সমস্যাটা হইলঃ

Let $a_1,a_2,…,a_k,…,a_n$ is a sequence of distinct positive real numbers such that $a_1<a_2<…<a_k$ and $a_k>a_{k+1}>…>a_n$ . A grasshopper is to jump along the real axis, starting at the point $O$ and making $n$ jumps to right of lengths $a_1,a_2,…,a_n$ respectively. Prove that, once he reaches the rightmost point, he can come back to point $O$ by making $n$ jumps to left of of lengths $a_1,a_2,…,a_n$ 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).
Every logical solution to a problem has its own beauty.

tanvirab
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Re: BdMO H.Sec 2010. Problem 10

The case $n=1$ is trivial. Case $n=2$ is easy. Do induction with base case $n=2$.

Always try induction if the problem is concerning some finite set.

sourav das
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Re: BdMO H.Sec 2010. Problem 10

Solution:
Please check my solution. Am I missing any point?
You spin my head right round right round,
When you go down, when you go down down......
(-$from$ "$THE$ $UGLY$ $TRUTH$" )