Functions

Suppose for all integers $$ x $$, there are two functions $$ f $$ and $$ g $$ such that $$ f(x) + f(x-1) - 1 = 0 $$ ...

Let $$ 0 \leq a \leq x \leq 100 $$ and $$ f(x) = |x - a| + |x - 100| + |x - (a + 50)|. $$ Then the maximum value of...

For any real number $$ x $$, let $$ \lfloor x \rfloor $$ be the largest integer less than or equal to $$ x $$. If $$ ...

The largest real value of $$ a $$ for which the equation $$ |x + a| + |x - 1| = 2 $$ has an infinite number of solution...

If $$ x $$ is a positive real number such that $$ x^8 + \left(\frac{1}{x}\right)^8 = 47 $$, then the value of $$ ...

The sum of all possible values of $$ x $$ satisfying the equation $$ 2^{4x^2} - 2^{2x^2 + x + 16} + 2^{2x + 30} = 0 $...

If $$ f $$ maps the set of natural numbers to whole numbers such that $$ f(xy) = f(x)f(y) + f(x) + f(y) $$ for all $$ x,...

If $$x$$ and $$y$$ satisfy the equations $$|x| + x + y = 15$$ and $$x + |y| - y = 20$$, then $$(x - y)$$ equals ...

For any non-zero real number $$ x $$, let $$ f(x) + 2f\left(\frac{1}{x}\right) = 3x. $$ Then, the sum of all possi...