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Maths item of the month

Curriculum mapping

A list of Maths Items of the Month categorised by GCSE/A level topics can be seen at: Maths Items of the Month Curriculum mapping.

Recent Maths Items of the Month

April 2021
Angling for an answer

isosceles triangle with missing angle

Find the size of the angle α in this isosceles triangle.

March 2021
Colouring the plane with three colours

Every point in the plane is coloured either red, blue or yellow.

all the points in the plane coloured red, blue or yellow

Prove that there must be two points of the same colour that are exactly one unit apart.

Solution

February 2021
It's odd but is it rational?

The graph with equation y = 3x2 + 7x − 5 and the roots of the equation
3x2 + 7x − 5= 0 are shown in the image below.

roots of a quadratic equation

If the values of a, b ánd c are all odd integers will the equation ax2 + bx + c = 0 ever have rational roots?

If the values of a, b ánd c are integers there are 8 possible combinations for these to be odd or even. For which of these combinations is it possible for the equation ax2 + bx + c = 0 to have rational roots?

Solution

January 2021
Happy 2021

What is the longest string of consecutive positive integers that adds to 2021?

Solution

December 2020
Santa's score draw

20 presents to 4 houses

Santa has four houses to go and notices he has 20 identical stocking fillers in his sleigh, in addition to the presents the children in those houses have asked for.

In how many distinct ways can he distribute these between the four houses if each house has to get at least one?

November 2020
Fault-free tilings

A fault-free tiling is an arrangement of tiles in an m×n grid such that there are no vertical or horizontal lines that can be placed on the grid without crossing one of the tiles.

fault-free tilings

In the image above a 6×4 and a 6×8 grid have been tiled with 2×1 dominoes. The 6×4 tiling has 2 vertical fault lines and is therefore not fault free. The 6×8 tiling is fault-free.

Is it possible, using 2×1 dominoes, to find fault-free tilings of:

  • a 6×5 grid
  • a 6×6 grid
  • a 6×7 grid

Solution

October 2020
Ritangle 2020

A rectangle has sides of length 12 and 8 units. A square of side c is drawn in one corner, creating the rectangular areas P, Q, R and S as in the diagram. What is the minimum value that (Q + R)/(P + S) can take?

12 by 8 rectangle with a square of side c

This problem was taken from Ritangle 2017. Ritangle is a competition for teams of students aged 16 - 18 of A level Mathematics, the IB and Scottish Highers. Registration for Ritangle 2020 starts opens on 5th October 2020: integralmaths.org/ritangle.

Solution

September 2020
Path of the midpoint

Intersection of y=1/x and y=2x+c

M is the midpoint of the points of intersection of y=1/x and y=2x+c. What path does M trace as c varies?

Solution

July 2020
Function of a function

The function f(x)=(x+1)/(x+2).

f(x)=(x+1)/(x+2)

Find f2(x), f3(x), f4(x)..., where f2(x)=f(f(x)).

June 2020
Area of an arbelos

An arbelos is the shape created inside a semi-circle with diameter AB when two further semi-circles are drawn to a point C on AB so that their diameters are AC and BC. The arbelos is the area inside the larger semi-circle but outside the two smaller semi-circles: shown as the blue region below.

An arbelos

Show that the area of an arbelos is the same as the area of a circle with diameter CD where D is the point on the larger semi-circle directly above C: shown as the red circle in the picture above.

May 2020
Walk on by

Two old women started at sunrise and each walked at a constant velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. At what time was the sunrise on that day?