UKMT JMC 2024

When the five expressions below are simplified, how many different values are obtained? \( 2+2 \quad 2\times 2 \quad 2-2 \quad 2 \div 2 \quad 2^2 \)

Which of the following could have a capacity of 10 litres?

Gill is 36 this year. In which year will her age next be a square?

A drink is made by mixing one part of cordial with four parts of water. What percentage of the drink is cordial?

What is the value of \(1+2-3\times4\div5\)?

Which of the following has the same remainder when divided by 3 as it does when divided by 4?

The diagram shows a large square which has been divided into four smaller squares. It also shows both diagonals of the large square and two diagonals of smaller squares. What fraction of the area of the large square has been shaded?

Skye has half as many pens as Ishaa. Ana has twice as many pens as Skye. What fraction of all their pens does Skye have?

The diagram shows the regular hexagon \(PQRSTU\). What is the size of angle \(UPT\)?

In Fred's field there are some humans and some horses. There are 25 heads in total and 60 legs in total. What is the difference between the number of humans and the number of horses?

A hexagon is formed by arranging three equilateral triangles, as shown in the diagram. The side-length of the largest equilateral triangle is 10 cm. What is the perimeter, in cm, of the hexagon?

In the multiplication sum shown, \(y\) represents the same digit each time. What is the value of \(y\)?

In the triangle \(PQR\), the point \(S\) is on the edge \(QR\). \(\\angle QPS = \\angle SPR\), \(PQ = PS = SR\) and \(\\angle PRQ = x°\). What is the value of \(x\)?

The digits 1, 2, 3, 5 and 8 are to be placed in the grid above, one to a cell, to make a correct mathematical statement. Which number should come immediately after the division sign?

In the diagram shown, \(PQRS\) is a rhombus and \(PQT\) is an isosceles triangle in which \(PT = QT\). Angle \(PSR = 110°\). What is the size of angle \(SQT\)?

The world's smallest vertebrate is much shorter than its name! Discovered in 2022, the frog paedophryne amauensis is only 7.7 mm long. Approximately how many of these frogs, placed end to end, would be needed to make a line 1 metre long?

\(PQRS\) is a square with area \(100 \,\mathrm{cm}^2\). The point \(T\) is inside the square. \(QRT\) is a triangle with area \(24 \text{ cm}^2\). What is the area, in \(\mathrm{cm}^2\), of the triangle \(PTS\)?

Goldilocks eats three equal-sized bowls of porridge, one after the other. When she has eaten \(\dfrac{3}{7}\) of the total amount of porridge, what fraction of the porridge in the second bowl has she eaten?

Jokers always lie. Clowns always tell the truth. A group of four, each of whom is a Joker or a Clown, make the following statements about each other: P says, "Q always lies"; Q says, "R always lies"; R says, "P always tells the truth"; S says, "Exactly two of P, Q and R are Jokers". How many of P, Q, R and S are Clowns?

When you cut a regular hexagon into two pieces with a single straight cut, you get two polygons. Which of these shapes cannot be obtained?

The rectangle \(PQRS\) is divided into eight squares: one large unshaded square, four small unshaded squares and three shaded squares, as shown in the diagram. What fraction of the area of rectangle \(PQRS\) is shaded?

Forty furry ferrets weigh the same as fifty fit ferrets. Forty-five fit ferrets weigh the same as fifty-four friendly ferrets. How many friendly ferrets weigh the same as fifty furry ferrets?

The area of a square is six times the area of a rectangle with a length half that of the square and a width 6 cm less than the width of the square. What is the perimeter of the square?

Rovers, United, City and Wanderers played against each other once in a hockey tournament. The results table is shown on the right:

Team	Win	Draw	Loss	Goals for	Goals against
Rovers	3	0	0	5	0
United	0	2	1	3	6
City	1	1	1	4	4
Wanderers	0	1	2	0	2

What was the score in the match between Rovers and United?

In a school, one fifth of the students have blue eyes. One tenth of the left-handed students have blue eyes. One quarter of the right-handed students have blue eyes. What fraction of the students are left-handed?