Maths item of the month
Cubic with Integer Points
Find a cubic equation, with three distinct roots, such that the roots and the stationary points all lie on points with integer coordinates.
If the coefficient of x3 is equal to 1, what is the cubic with this property where the sum of the absolute value of the roots is as small as possible?
Summer All Over
Take the set of the reciprocals of all the positive integers from 1 to n.
Show that the sum of the products of the possible subsets (including the complete set) is equal to n.
e.g. for n = 3:
An arithmetic progression (AP) is a sequence of numbers with a common difference between them: e.g. 3, 7, 11, 15, …
A geometric progression (GP) is a sequence of numbers with a common ratio between them: e.g. 2, 6, 18, 54, …
If the 1st, 2nd and 6th terms of an AP form a GP what is the common ratio?
If the 1st, 2nd and nth terms of an AP form a GP what is the common ratio?
How many circles?
Circles plotted on coordinate axes can be categorised based on the following criteria (ignoring the scales on the axes):
- Position of the centre
- Positions of the points of intersection with the x-axis
- Positions of the points of intersection with the y-axis
For example the circles above can be categorised as follows:
- Centre in the top-right quadrant
- One intersection with the positive x-axis and one intersection with the negative x-axis
- One intersection with the positive y-axis and one intersection with the negative y-axis
- Centre on the positive x-axis
- Two intersections with the positive x-axis
- No intersections with the y-axis
- Centre in the bottom-right quadrant
- One (repeated) intersection with the positive x-axis
- Two intersections with the negative y-axis
Using these criteria how many different categories of circles are there?
Divisibility of consecutive integers
Let N = (n+1)(n+2)...(2n), i.e. the product of n consecutive integers from n+1 to 2n.
Prove that, for any positive integer n, N is divisible by 2n but not a higher power of 2.
For example, for n = 3:
N = 4×5×6
120 is divisible by 8 but not divisible by 16.
In the diagram various regular polygons, P, have been drawn whose sides are tangents to a circle, C.
Show that for any regular polygon drawn in this way:
Cones from a Circle
An angle θ is cut out of a circle of card to create two sectors: a major sector and a minor sector. The two sectors are then folded to make cones.
What angle θ is required to obtain the largest value for the sum of the volumes of the two cones?
An Unexpected Answer
Mr Student sets his class the following problem:
A committee of 3 students is to be chosen from a group of 13 students of which 8 are girls and 5 are boys. The students are selected at random, without replacement. What is the expected number of girls on the committee?
Anne Student immediately responds that the answer is .
She gives the reason that there are 3 students to be chosen and the proportion of girls is
so she calculated
Is she correct?
If the number of students was different would her method work?