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Part of 2001 Pan African

Problems(2)

find positive integer n

Source: Pan African 2001

10/4/2005
Find all positive integers nn such that: n3+3n2+7 \dfrac{n^3+3}{n^2+7} is a positive integer.
searchquadratics
Equation

Source: Pan African 2001

10/4/2005
Let nn be a positive integer, and let a>0a>0 be a real number. Consider the equation: ∑i=1n(xi2+(a−xi)2)=na2 \sum_{i=1}^{n}(x_i^2+(a-x_i)^2)= na^2 How many solutions (x1,x2⋯ ,xnx_1, x_2 \cdots , x_n) does this equation have, such that: 0≤xi≤a,i∈N+ 0 \leq x_i \leq a, i \in N^+