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Farey Neighbours

Age 16 to 18 Challenge Level:

Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?

Inner Equality

Age 16 to 18 Short Challenge Level:

Can you solve this inequalities challenge?

Rationals Between...

Age 14 to 16 Challenge Level:

What fractions can you find between the square roots of 65 and 67?

Two Cubes

Age 14 to 16 Challenge Level:

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to. . . .

Erratic Quadratic

Age 16 to 18 Challenge Level:

Can you find a quadratic equation which passes close to these points?

Unit Interval

Age 16 to 18 Short Challenge Level:

Can you prove our inequality holds for all values of x and y between 0 and 1?

Balance Point

Age 14 to 16 Challenge Level:

Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. Move the bar from side to side until you find a balance point. Is it possible to predict that position?

Almost Total Inequality

Age 14 to 16 Challenge Level:

Fracmax

Age 14 to 16 Challenge Level:

Find the maximum value of 1/p + 1/q + 1/r where this sum is less than 1 and p, q, and r are positive integers.

Big, Bigger, Biggest

Age 16 to 18 Challenge Level:

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

' Tis Whole

Age 14 to 18 Challenge Level:

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

Shades of Fermat's Last Theorem

Age 16 to 18 Challenge Level:

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?

Biggest Enclosure

Age 14 to 16 Challenge Level:

Three fences of different lengths form three sides of an enclosure. What arrangement maximises the area?

Code to Zero

Age 16 to 18 Challenge Level:

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

Comparing Continued Fractions

Age 16 to 18 Challenge Level:

Which of these continued fractions is bigger and why?

Mediant Madness

Age 14 to 16 Challenge Level:

Kyle and his teacher disagree about his test score - who is right?

Exhaustion

Age 16 to 18 Challenge Level:

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

Classical Means

Age 16 to 18 Challenge Level:

Use the diagram to investigate the classical Pythagorean means.

Power Up

Age 16 to 18 Challenge Level:

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

Reciprocals

Age 16 to 18 Challenge Level:

Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.

Diverging

Age 16 to 18 Challenge Level:

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

All-variables Sudoku

Age 11 to 18 Challenge Level:

The challenge is to find the values of the variables if you are to solve this Sudoku.

Thousand Words

Age 16 to 18 Challenge Level:

Here the diagram says it all. Can you find the diagram?

Not Continued Fractions

Age 14 to 18 Challenge Level:

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

Climbing

Age 16 to 18 Challenge Level:

Sketch the graphs of y = sin x and y = tan x and some straight lines. Prove some inequalities.

Integral Inequality

Age 16 to 18 Challenge Level:

An inequality involving integrals of squares of functions.

Tetra Inequalities

Age 16 to 18 Challenge Level:

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

Integral Sandwich

Age 16 to 18 Challenge Level:

Generalise this inequality involving integrals.

Inside Outside

Age 14 to 16 Challenge Level:

Balance the bar with the three weight on the inside.

After Thought

Age 16 to 18 Challenge Level:

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

Without Calculus

Age 16 to 18 Challenge Level:

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.