Other equations

problem
Are you kidding

Age

14 to 16

Challenge level

If the altitude of an isosceles triangle is 8 units and the perimeter of the triangle is 32 units.... What is the area of the triangle?
problem
Root to poly

Age

14 to 16

Challenge level

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.
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?