this month, Stage 3 & 4

May 1999

 

Problems

problem Icon

Symmetricality

Stage:3 Challenge Level:Challenge Level:1

Add up all 5 equations given below. What do you notice? Solve the system and find the values of a, b, c , d and e. b + c + d + e = 4 a + c + d + e = 5 a + b + d + e = 1 a + b + c + e = 2 a + b. . . .

problem Icon

Why 8?

Stage:3 Challenge Level:Challenge Level:1

Choose any four consecutive even numbers. Multiply the two middle numbers together. Multiply the first and last numbers. Now subtract your second answer from the first. Try it with your own. . . .

problem Icon

Flagging

Stage:3 Challenge Level:Challenge Level:2Challenge Level:2

How many tricolour flags are possible with 5 available colours such that two adjacent stripes must NOT be the same colour. What about 256 colours?

problem Icon

Tied up

Stage:3 Challenge Level:Challenge Level:2Challenge Level:2

In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . .

problem Icon

The Pillar of Chios

Stage:3 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

problem Icon

Rotating Triangle

Stage:3 and 4 Challenge Level:Challenge Level:1

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

problem Icon

Coke Machine

Stage:4 Challenge Level:Challenge Level:1

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly. . . .

problem Icon

Overturning Fracsum

Stage:4 Challenge Level:Challenge Level:2Challenge Level:2

Solve the system of equations to find the values of x, y and z: xy/(x+y)=1/2, yz/(y+z)=1/3, zx/(z+x)=1/7

problem Icon

One and Three

Stage:4 Challenge Level:Challenge Level:2Challenge Level:2

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

problem Icon

Grid Lockout

Stage:4 Challenge Level:Challenge Level:2Challenge Level:2

What remainders do you get when square numbers are divided by 4?