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Farey Neighbours

Stage: 5 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?

Integral Sandwich

Stage: 5 Challenge Level:

Generalise this inequality involving integrals.

Not Continued Fractions

Stage: 4 and 5 Challenge Level:

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

Big, Bigger, Biggest

Stage: 5 Challenge Level:

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

Tetra Inequalities

Stage: 5 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.

Spread

Stage: 5 Challenge Level:

Given the mean and standard deviation of a set of marks, what is the greatest number of candidates who could have scored 100%?

Balance Point

Stage: 4 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?

Weekly Challenge 1: Inner Equality

Stage: 4 and 5 Short Challenge Level:

Our first weekly challenge. We kick off with a challenge concerning inequalities.

Almost Total Inequality

Stage: 4 Challenge Level:

Thousand Words

Stage: 5 Challenge Level:

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

Without Calculus

Stage: 5 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.

Integral Inequality

Stage: 5 Challenge Level:

An inequality involving integrals of squares of functions.

Mediant

Stage: 4 Challenge Level:

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

Power Up

Stage: 5 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

Diverging

Stage: 5 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.

Comparing Continued Fractions

Stage: 5 Challenge Level:

Which of these continued fractions is bigger and why?

Code to Zero

Stage: 5 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.

Inside Outside

Stage: 4 Challenge Level:

Balance the bar with the three weight on the inside.

Exhaustion

Stage: 5 Challenge Level:

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

Shades of Fermat's Last Theorem

Stage: 5 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?

' Tis Whole

Stage: 4 and 5 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?

Erratic Quadratic

Stage: 5 Challenge Level:

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

Classical Means

Stage: 5 Short Challenge Level:

Use the diagram to investigate the classical Pythagorean means.

All-variables Sudoku

Stage: 3, 4 and 5 Challenge Level:

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

Fracmax

Stage: 4 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.

Number Chains

Stage: 5 Challenge Level:

Find all the periodic cycles and fixed points in this number sequence using any whole number as a starting point.

Two Cubes

Stage: 4 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. . . .

After Thought

Stage: 5 Challenge Level:

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

Max Box

Stage: 4 Challenge Level:

Three rods of different lengths form three sides of an enclosure with right angles between them. What arrangement maximises the area

Rationals Between

Stage: 4 Challenge Level:

What fractions can you find between the square roots of 56 and 58?

Climbing

Stage: 5 Challenge Level:

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

Reciprocals

Stage: 5 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.