Difference between revisions of "1981 IMO Problems/Problem 6"

Revision as of 16:39, 29 October 2006

Problem

The function $\displaystyle f(x,y)$ satisfies

(1) $\displaystyle f(0,y)=y+1,$

(2) $\displaystyle f(x+1,0)=f(x,1),$

(3) $\displaystyle f(x+1,y+1)=f(x,f(x+1,y)),$

for all non-negative integers $\displaystyle x,y$ . Determine $\displaystyle f(4,1981)$ .

Solution

We observe that $\displaystyle f(1,0) = f(0,1) = 2$ and that $\displaystyle f(1, y+1) = f(1, f(1,y)) = f(1,y) + 1$ , so by induction, $\displaystyle f(1,y) = y+2$ . Similarly, $\displaystyle f(2,0) = f(1,1) = 3$ and $\displaystyle f(2, y+1) = f(2,y) + 2$ , yielding $\displaystyle f(2,y) = 2y + 3$ .

We continue with $\displaystyle f(3,0) + 3 = 8$ ; $\displaystyle f(3, y+1) + 3 = 2(f(3,y) + 3)$ ; $\displaystyle f(3,y) + 3 = 2^{y+3}$ ; and $\displaystyle f(4,0) + 3 = 2^{2^2}$ ; $\displaystyle f(4,y) + 3 = 2^{f(4,y) + 3}$ .

It follows that $\displaystyle f(4,1981) = 2^{2\cdot ^{ . \cdot 2}} - 3$ when there are 1984 2s, Q.E.D.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

Resources

@@ Line 17: / Line 17: @@
 We continue with <math>\displaystyle f(3,0) + 3 = 8 </math>; <math>\displaystyle f(3, y+1) + 3 = 2(f(3,y) + 3)</math>; <math>\displaystyle f(3,y) + 3 = 2^{y+3}</math>; and <math>\displaystyle f(4,0) + 3 = 2^{2^2}</math>; <math>\displaystyle f(4,y) + 3 = 2^{f(4,y) + 3}</math>.
-It follows that <math>\displaystyle f(4,1981) = 2^{2\cdot ^{ . \cdot 2}}</math> when there are 1984 2s, Q.E.D.
+It follows that <math>\displaystyle f(4,1981) = 2^{2\cdot ^{ . \cdot 2}} - 3 </math> when there are 1984 2s, Q.E.D.
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Difference between revisions of "1981 IMO Problems/Problem 6"

Revision as of 16:39, 29 October 2006

Problem

Solution

Resources