Other equations

There are 22 NRICH Mathematical resources connected to Other equations
Two Trees
problem
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Two trees

Age
16 to 18
Challenge level
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Two trees 20 metres and 30 metres long, lean across a passageway between two vertical walls. They cross at a point 8 metres above the ground. What is the distance between the foot of the trees?
Polynomial Relations
problem
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Polynomial relations

Age
16 to 18
Challenge level
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Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.
Root to Poly
problem
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Root to poly

Age
14 to 16
Challenge level
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Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.
One and three
problem
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One and three

Age
14 to 16
Challenge level
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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 metres from B. How long is the lake?
Three four five
problem

Three four five

Age
14 to 16
Challenge level
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Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
Plutarch's Boxes
problem

Plutarch's boxes

Age
11 to 14
Challenge level
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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?
Real(ly) numbers
problem

Real(ly) numbers

Age
16 to 18
Challenge level
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If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?
Around and Back
problem

Around and back

Age
14 to 16
Challenge level
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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.
Building Tetrahedra
problem

Building tetrahedra

Age
14 to 16
Challenge level
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Can you make a tetrahedron whose faces all have the same perimeter?
Deep Roots
problem

Deep roots

Age
14 to 16
Challenge level
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Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$