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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?
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.
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NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.