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1972 IMO Problems/Problem 5: Difference between revisions

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Let <math>u>0</math> be the least upper bound for <math>|f(x)|</math> for all <math>x</math>. So, <math>|f(x)| \leq u</math> for all <math>x</math>. Then, for all <math>x,y</math>,
Let <math>u>0</math> be the least upper bound for <math>|f(x)|</math> for all <math>x</math>. So, <math>|f(x)| \leq u</math> for all <math>x</math>. Then, for all <math>x,y</math>,


<math>2u \geq |f(x+y)|+|f(x-y)| \geq |f(x+y)+f(x-y)| = |2f(x)g(y)|=2|f(x)||g(y)|</math>
<math>2u \geq |f(x+y)+f(x-y)| = |2f(x)g(y)|=2|f(x)||g(y)|</math>


Therefore, <math>u \geq |f(x)||g(y)|</math>, so <math>|f(x)| \leq u/|g(y)|</math>.
Therefore, <math>u \geq |f(x)||g(y)|</math>, so <math>|f(x)| \leq u/|g(y)|</math>.

Revision as of 10:04, 11 June 2020

Let $f$ and $g$ be real-valued functions defined for all real values of $x$ and $y$, and satisfying the equation \[f(x + y) + f(x - y) = 2f(x)g(y)\] for all $x, y$. Prove that if $f(x)$ is not identically zero, and if $|f(x)| \leq 1$ for all $x$, then $|g(y)| \leq 1$ for all $y$.

Solution

Let $u>0$ be the least upper bound for $|f(x)|$ for all $x$. So, $|f(x)| \leq u$ for all $x$. Then, for all $x,y$,

$2u \geq |f(x+y)+f(x-y)| = |2f(x)g(y)|=2|f(x)||g(y)|$

Therefore, $u \geq |f(x)||g(y)|$, so $|f(x)| \leq u/|g(y)|$.

Since $u$ is the least upper bound for $|f(x)|$, $u/|g(y)| \geq u$. Therefore, $|g(y)| \leq 1$.

Borrowed from http://www.cs.cornell.edu/~asdas/imo/imo/isoln/isoln725.html

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