2003 Pan African MO Problems/Problem 4
Revision as of 13:56, 27 January 2020 by Rockmanex3 (talk | contribs) (Solution to Problem 4 (credit to akhan98) -- complicated function problem)
Problem
Let . Does there exist a function such that: where we define: and , ?
Solution (credit to akhan98)
Let , where are integers and is relatively prime to 5. Notice that there are an infinite number of values that can be.
Thus, we can group the values of into groups of 2004. WLOG, let one group be . We can have
\[f(5^a \cdot b_n) = \left\{\begin{array}{l} 5^a \cdot b_{n+1}, n < 2004 \\ 5^{a+1} \cdot b_1, n = 2004\end{array}\right\] (Error compiling LaTeX. ! Missing delimiter (. inserted).)
as the function that satisfies the initial conditions for all in the given set. Since we can apply a similar function from the other groups of 2004, we can demonstrate that there is a function that satisfies the conditions for all non-negative integers.
See Also
2003 Pan African MO (Problems) | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 5 |
All Pan African MO Problems and Solutions |