Question

Fool's problems of the day,

[** EDIT: Complimentary problem added **]

On the April fools' day I give you three cute problems on number theory/Combinatorics:

\((1) \) Find the number of non-negative integral solutions of \(3x+4y=120 \)
[Solved: @2bornot2b ]

\( (2) \) If \(81x + 64y = n\) find the greatest possible of \( n \) such that both \( (x, y) \) are not positive.

\( (3) \) Let \(n\) and \( m\) be positive integers. An \( n \times  m \) rectangle is tiled with unit squares. Let \(r(n,m) \) denote the number of rectangles formed by the edge of these unit squares. Thus, for example, \(r(2, 1) = 3\). Can you find \( r(11, 12) \)?.
[ Solved by @satellite73]

PS: The problems are arranged in an increasing order of difficulty. (However, (2) could seem very very easy to a number theorist!)

** Complimentary for those who feels these are very hard:**

If the tangent the to the curve \(\sqrt{x} + \sqrt{y} = \sqrt{a} \) at any point on cuts x-axis and y-axis at two distinct points. Can you find the sum of the intercepts ?

Good luck!