Difference between revisions of "2003 Pan African MO Problems/Problem 6"
Rockmanex3 (talk | contribs) (Solution to Problem 6 -- epic functional equation problem) |
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By letting <math>x = 0, y = n</math>, we have <math>f(0) - f(n^2) = n (f(0) - f(n))</math>, and by letting <math>x = 0, y = -n</math>, we have <math>f(0) - f(n^2) = -n (f(0) - f(-n))</math>. Substitution and simplification results in | By letting <math>x = 0, y = n</math>, we have <math>f(0) - f(n^2) = n (f(0) - f(n))</math>, and by letting <math>x = 0, y = -n</math>, we have <math>f(0) - f(n^2) = -n (f(0) - f(-n))</math>. Substitution and simplification results in | ||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
− | n (f(0) - f(n)) &= -n (f(0) - f(n)) \ | + | n (f(0) - f(n)) &= -n (f(0) - f(-n)) \ |
f(0) - f(n) &= -f(0) + f(-n) \ | f(0) - f(n) &= -f(0) + f(-n) \ | ||
2 f(0) &= f(n) + f(-n). | 2 f(0) &= f(n) + f(-n). |
Latest revision as of 12:01, 25 January 2020
Problem
Find all functions such that: for .
Solution
By letting , we have , and by letting , we have . Substitution and simplification results in Therefore, if , where is a real number, then . Thus, the function has rotational symmetry about .
Additionally, multiplying both sides by results in , and rearranging results in . The equation seems to resemble a rearranged slope formula, and the rotational symmetry seems to hint that is a linear function.
To prove that must be a linear function, we need to prove that the slope is the same from to all the other points of the function. By letting , we have . Additionally, by letting , we have . By rearranging the prior equation, we have
The slope from points and is . By substitution,
The slope from point to is the same as the slope from point to any other point on the function, so must be a linear function.
Let . Using the function on the original equation results in
Thus, can be any linear function, so , where are real numbers.
See Also
2003 Pan African MO (Problems) | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Last Problem |
All Pan African MO Problems and Solutions |