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.

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Recent Maths Items of the Month

June 2021
Rounded square?

The value of the nearest integer to n + √n is shown in the table below for the first few positive integers, n.

n	1	2	3	4	5
n + √n	2	3.41...	4.73...	6	7.23...
Nearest integer	2	3	5	6	7

Will there ever be a square number in the bottom row for any positive integer n?

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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?

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April 2021
Angling for an answer

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

Solution

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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

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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

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January 2021
Happy 2021

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

Solution

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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?

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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

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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

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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

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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)).