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Do you have enough information to work out the area of the shaded quadrilateral?

Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

Mind Your Ps and Qs

Age 16 to 18 Short Challenge Level:

Here are 16 propositions involving a real number $x$:
 

 

$x\int^x_0 ydy < 0$ $x> 1$ $0< x< 1 $ $x^2+4x+4 =0$
$x=0 $ $\cos(x/2)> \sin(x/2)$ $x> 2$ $x=1$
$2\int^{x^2}_0ydy> x^2 $ $x< 0 $ $x^2+x-2=0$ $x=-2 $
$x^3> 1$ $|x|> 1$ $x> 4$ $\int^x_0 \cos y dy =0$

[Note: the trig functions are measured in radians]


By choosing $p$ and $q$ from this list, how many correct mathematical statements of the form $p\Rightarrow q$ or $p\Leftrightarrow q$ can you make?


It is possible to rearrange the statements into four statements $p\Rightarrow q$ and four statements $p\Leftrightarrow q$. Can you work out how to do this?
 


NOTES AND BACKGROUND

Logical thinking is at the heart of higher mathematics: In order to construct clear, correct arguments in ever more complicated situations mathematicians rely on clarity of language and logic. Logic is also at the heart of computer programming and circuitry.