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Cosines Rule

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

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DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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Root to Poly

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

Alternating Sum

Stage: 4 Short Challenge Level: Challenge Level:1

$n$ is clearly odd, so we can write the series as:
$1+(-2+3)+(-4+5)+...+(-(n-1)+n)$. So the sum is the same as $1+1+...+1=(n+1)/2$.

So $(n+1)=4016$, so $n=4015$


If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.
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