From the 2004 Tournament of Towns:

*Each day, the price of the shares of the corporation “Soap Bubble, Limited” either increases or decreases by n%, where n is an integer such that 0<n<100. The price is calculated with unlimited precision. Does there exist an n for which the price can take the same value twice?*

The Russians are at it again with their well conceived, intriguing maths problems.

What is really being asked is can we find a solution to

?

This leads to

.

Now the right hand side is divisible by the primes 2 and 5 only. This severely restricts the possible values of ; in fact, the only value of in the given range that is even worth trying is . That is,

.

Alas, this fails: 2 appears to a higher power than 5 in both 160 and 40, but 2 and 5 must appear equally often on the right hand side. Thus there is **no value of n **for which the price can take the same value twice.

It is another fantastic problem from that part of the world that produces more interesting problems than anywhere else. I just wish there had been a solution – then the problem would have been that much sweeter…

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