Difference between revisions of "2003 Pan African MO Problems/Problem 4"
Rockmanex3 (talk | contribs) (Solution to Problem 4 (credit to akhan98) -- complicated function problem) |
Rockmanex3 (talk | contribs) (Fixed LaTeX) |
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Thus, we can group the values of <math>b</math> into groups of 2004. WLOG, let one group be <math>\{ b_1, b_2, b_3, \cdots b_{2004} \}</math>. We can have | Thus, we can group the values of <math>b</math> into groups of 2004. WLOG, let one group be <math>\{ b_1, b_2, b_3, \cdots b_{2004} \}</math>. We can have | ||
| − | <cmath>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</cmath> | + | <cmath>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.</cmath> |
as the function that satisfies the initial conditions for all <math>b</math> 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 <math>f(x)</math> that satisfies the conditions for all non-negative integers. | as the function that satisfies the initial conditions for all <math>b</math> 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 <math>f(x)</math> that satisfies the conditions for all non-negative integers. | ||
Latest revision as of 12:59, 27 January 2020
Problem
Let 
. Does there exist a function 
such that:
![\[f^{2003}(n)=5n, \forall n \in \mathbb{N}_0\]](http://latex.artofproblemsolving.com/2/1/4/2146fb4becc1f80f67da479190d7459f6d0017c2.png)
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
![\[f(5^a \cdot b_n) = \left \{ \begin{array}{l} 5^a \cdot b_{n+1}, n < 2004 \\ 5^{a+1} \cdot b_1, n = 2004\end{array} \right.\]](http://latex.artofproblemsolving.com/4/1/0/4108935c6619c4d13f8e27a216aa8dd163204826.png)
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 | ||
