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The numerical function $f$ is such that for any $x$ and $y$, the equality $f(x+y)=f(x)+f(y)+80xy$ holds. Find $f(1)$, if $f(0.25)=2$.

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$f(0,5)=f(0,25+0,25)=f(0,25)+f(0,25)+80 \cdot 0,25 \cdot 0,25=2+2+5=9$. Similarly $f(1)=f(0,5+0,5)=f(0,5)+$ $f(0,5)+80 \cdot 0,5 \cdot 0,5=9+9+20=38$.

## Answer

38. 

## [ Rational and irrational numbers ] Problem 116563 Topics: [Identities (trigonometry).] [ Proof by contradiction

Author: Senderov V...

Does there exist a real $\alpha$ such that the number $\cos \alpha$ is irrational, while all the numbers $\cos 2 \alpha, \cos 3 \alpha, \cos 4 \alpha, \cos 5 \alpha$ are rational?

## Solution

Assume the opposite. Then the number $A=\cos \alpha+\cos 5 \alpha$ is irrational as the sum of a rational and an irrational number. On the other hand,

$A=2 \cos 2 \alpha \cos 3 \alpha$ is rational as the product of three rational numbers. Contradiction.

## Answer

No.