A basketball player is practising free throws. Their current success rate (ratio of successful throws to total) is exactly 70% (or 0.7 in terms of ratio). After a few more throws that success rate is 90%.
Show that at some point the success rate was exactly 80%.
Can the product of three positive consecutive integers be a perfect square?
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Assume the three integers are $n-1, n, n+1$ and that $(n-1)n(n+1) = n(n^2-1)$ is a perfect square.
Since $n$ is relatively prime to both $n-1$ and $n+1$ (and hence their product $n^2-1$), we must have that $n^2-1$ is a perfect square too.
In a triangle $ABC$ (sides $a,b,c$ opposite $A,B,C$), angle $A$ is twice $B$.
Show that $$a^2 = b(b+c)$$
Try not to use trigonometry if possible.
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Let AD be the angular bisector of A, D lying on BC (might help to draw a figure).
Then by angular bisector theorem
$$BD = \frac{ac}{b+c}, DC = \frac{ab}{b+c}$$
BAD is isosceles, with $AD = BD$. Also triangle $ADC$ is similar to triangle $BAC$.
$AD/AB = DC/AC$ gives the result.
There are non-trigonometric proofs of the angular bisector theorem. For eg, prove for right angled triangles and use affine transform etc.
