# 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

### May 2021 Square pegs in a semi-round hole

Two non-overlapping squares sit inside a semi-circle of radius r. What is the the maximum possible total area of the two squares?

### April 2021 Angling for an answer

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.

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.

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

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.

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?

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

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).

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.

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.