2002 Paper 3 Q4

Year: 2002
Paper: 3
Question Number: 4

Course: LFM Stats And Pure
Section: Quadratics & Inequalities

Difficulty: 1700.0 Banger: 1490.1

inequality proof by contradiction cubic quadratic integer solutions Diophantine equation algebraic manipulation number theory

Problem

Show that if \(x\) and \(y\) are positive and \(x^3 + x^2 = y^3 - y^2\) then \(x < y\,\). Show further that if \(0 < x \le y - 1\), then \(x^3 + x^2 < y^3 - y^2\). Prove that there does not exist a pair of {\sl positive} integers such that the difference of their cubes is equal to the sum of their squares. Find all the pairs of integers such that the difference of their cubes is equal to the sum of their squares.

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

Difficulty Comparisons: 0

Banger Rating: 1490.1

Banger Comparisons: 3

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Problem source

Show that if $x$ and $y$ are positive  and $x^3 + x^2 = y^3 - y^2$ then $x < y\,$.
Show further that if $0 < x \le y - 1$, then $x^3 + x^2 < y^3 - y^2$.
Prove that there does not exist a pair of {\sl positive} integers 
such that the difference of their cubes is
 equal to the sum of their squares.
Find all the pairs of integers such that the 
difference of their cubes  is equal to the sum of their squares.

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