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Problem #PRU-100267

Problems Methods Mathematical induction Mathematical induction in geometry

Illustrate with a picture

(a) $(a - b)^{2} = a^{2} - 2 a b + b^{2}$ ,

(b) $a^{2} - b^{2} = (a - b) (a + b)$ ,

(c) $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 a c + 2 b c$ .

Problem #PRU-100271

Problems Methods Mathematical induction Mathematical induction in geometry

Circles and lines are drawn on the plane. They divide the plane into non-intersecting regions, see the picture below.

Show that it is possible to colour the regions with two colours in such a way that no two regions sharing some length of border are the same colour.

Problem #PRU-78680

Problems Geometry Plane geometry Circles Central angle. Arc length and circumference Methods Mathematical induction Mathematical induction in geometry Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–15 3.0

On a circle of radius 1, the point $O$ is marked and from this point, to the right, a notch is marked using a compass of radius $l$ . From the obtained notch $O_{1}$ , a new notch is marked, in the same direction with the same radius and this is process is repeated 1968 times. After this, the circle is cut at all 1968 notches, and we get 1968 arcs. How many different lengths of arcs can this result in?