• anonymous
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!
Mathematics

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