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Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

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In y = ax +b when are a, -b/a, b in arithmetic progression. The polynomial y = ax^2 + bx + c has roots r1 and r2. Can a, r1, b, r2 and c be in arithmetic progression?

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Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.

Polynomial Relations

Age 16 to 18 Challenge Level:

Good solutions to this problem were received from Tyrone of Cyfarthfa High School in Merthyr Tydfil, and Koopa of Boston College in the USA.

Tyrone solved the problem by relating both polynomials to $(x+1)^2$ :

$$ \eqalign { p(x)=x^2 + 2x \Rightarrow &p &=&(x+1)^2 - 1 \\ &p+1 &=& (x+1)^2 \\ } $$
(1)
$$ \eqalign { q(x)=x^2 + x + 1 \Rightarrow &q &=&(x+1)^2 - x \\ &q+x &=& (x+1)^2 \\ } $$
(2)

$ \Rightarrow p+1=q+x $ (combining eqns (1) and (2)).


But $x=(p+1)^{1/2}-1$ (from eqn (1)). So

$$ \eqalign { \Rightarrow p+1&=&q+ ((p+1)^{1/2}-1) \\
&=&q+ (p+1)^{1/2}-1 }$$
$$ \eqalign { p-q+2&=&(p+1)^{1/2} \\
(p-q+2)^2&=&p+1}$$.

Squaring the bracket,

$$ \eqalign { &p^2-pq+2p-pq+q^2-2q+2p-2q+4=p+1 \\ &p^2 -2pq+q^2+4p-4q+4=p+1 \\ &p^2-2pq+q^2+3p-4q+3=0 } $$.