Difference between revisions of "1992 IMO Problems/Problem 2"
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==Solution== | ==Solution== | ||
− | + | [quote=probability1.01]Set x = 0 to get <math>f(f(y)) = y+f(0)^{2}</math>. We'll let <math>c = f(0)^{2}</math>, so <math>f(f(y)) = y+c</math>. Then | |
− | + | <math>f(a^{2}+f(f(b))) = f(b)+f(a)^{2}</math> | |
+ | <math>f(a^{2}+b+c) = f(b)+f(a)^{2}</math> | ||
+ | <math>f(f(a^{2}+b+c)) = f(f(b)+f(a)^{2}) = b+f(f(a))^{2}</math> | ||
+ | <math>a^{2}+b+2c = b+(a+c)^{2}</math> | ||
+ | <math>2c = c^{2}+2ac</math> | ||
− | Since <math>f(y)=y</math>, | + | Since this holds for all <math>a</math>, it follows that <math>c = 0</math>. Now we have <math>f(0) = 0 \implies f(f(y)) = y</math>. Note that f must be surjective since we may let y vary among all reals, and f must be injective since if <math>f(a) = f(b)</math>, then <math>a+f(x)^{2}= f(x^{2}+f(a)) = f(x^{2}+f(b)) = b+f(x)^{2}</math>. Finally, if <math>u > v</math>, then there is some <math>t</math> s.t. <math>u = t^{2}+v</math>, and so <math>f(u) = f(t^{2}+v) = f^{-1}(v)+f(t)^{2}> f^{-1}(v) = f(f(f^{-1}(v))) = f(v)</math>. Hence f is strictly increasing. It is now clear that since <math>f(f(y)) = y</math>, we must have <math>f(x) = x</math> for all x.[/quote] |
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==See Also== | ==See Also== |
Latest revision as of 14:53, 12 March 2024
Problem
Let denote the set of all real numbers. Find all functions
such that
Solution
[quote=probability1.01]Set x = 0 to get . We'll let
, so
. Then
Since this holds for all , it follows that
. Now we have
. Note that f must be surjective since we may let y vary among all reals, and f must be injective since if
, then
. Finally, if
, then there is some
s.t.
, and so
. Hence f is strictly increasing. It is now clear that since
, we must have
for all x.[/quote]
See Also
1992 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |