Other equations

problem

Hike and Hitch

Age

14 to 16

Challenge level

Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the road (at 4 miles per hour)...
problem

Around and Back

Age

14 to 16

Challenge level

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns around and heads back to the starting point where he meets the runner who is just finishing his first circuit. Find the ratio of their speeds.
problem

Deep Roots

Age

14 to 16

Challenge level

Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$
problem

Not a Polite Question

Age

11 to 14

Challenge level

When asked how old she was, the teacher replied: My age in years is not prime but odd and when reversed and added to my age you have a perfect square...
problem

Plutarch's Boxes

Age

11 to 14

Challenge level

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?
problem

Top-Heavy Pyramids

Age

11 to 14

Challenge level

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
problem

N Is a Number

Age

11 to 14

Challenge level

N people visit their friends staying N kilometres along the coast. Some walk along the cliff path at N km an hour, the rest go by car. How long is the road?
problem

Building Tetrahedra

Age

14 to 16

Challenge level

Can you make a tetrahedron whose faces all have the same perimeter?
problem

Hand Swap

Age

14 to 16

Challenge level

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the minute hand and hour hand had swopped places. What time did the train leave London and how long did the journey take?
problem
Favourite

Two Trees

Age

16 to 18

Challenge level

Can you find the distance between the two trees using the information given?