Substitutions/solving equations

**usagi_killer** · November 12th 2009, 05:35 AM

1. Suppose that $a,b,c,d$ are real numbers such that

$a^2 + b^2 = 1$

$c^2 + d^2 = 1$

$ac+bd=0$

Show that

$a^2 + c^2 = 1$

$b^2 + d^2 = 1$

$ab + cd = 0$

(Problem can get messy but there is an elegant and complete solution)

2. Find all integer solutions $(n,m)$ to $n^4+2n^3+2n^2+2n+1=m^2$

3. Find the smallest positive integer whose cube ends in 888.

**Drexel28** · November 12th 2009, 06:52 AM

Originally Posted by usagi_killer

1. Suppose that $a,b,c,d$ are real numbers such that

$a^2 + b^2 = 1$

$c^2 + d^2 = 1$

$ac+bd=0$

Show that

$a^2 + c^2 = 1$

$b^2 + d^2 = 1$

$ab + cd = 0$

(Problem can get messy but there is an elegant and complete solution)

2. Find all integer solutions $(n,m)$ to $n^4+2n^3+2n^2+2n+1=m^2$

3. Find the smallest positive integer whose cube ends in 888.

Are these challenge questions?

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