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Broad Topics > Algebra > Inequality/inequalities

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Random Inequalities

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Can you build a distribution with the maximum theoretical spread?

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Squareness

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

The family of graphs of x^n + y^n =1 (for even n) includes the circle. Why do the graphs look more and more square as n increases?

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Discrete Trends

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Find the maximum value of n to the power 1/n and prove that it is a maximum.

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In Between

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Can you find the solution to this algebraic inequality?

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Eyes Down

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

The symbol [ ] means 'the integer part of'. Can the numbers [2x]; 2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three different values?

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Proofs with Pictures

Stage: 5

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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Approximating Pi

Stage: 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?

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Unit Interval

Stage: 4 and 5 Challenge Level: Challenge Level:1

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

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Giants

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Which is the bigger, 9^10 or 10^9 ? Which is the bigger, 99^100 or 100^99 ?

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Tet-trouble

Stage: 4 Challenge Level: Challenge Level:1

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

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Quadratic Harmony

Stage: 5 Challenge Level: Challenge Level:1

Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

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Square Mean

Stage: 4 Challenge Level: Challenge Level:1

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

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Weekly Challenge 1: Inner Equality

Stage: 4 and 5 Short Challenge Level: Challenge Level:1

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

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Erratic Quadratic

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Farey Neighbours

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Classical Means

Stage: 5 Short Challenge Level: Challenge Level:1

Use the diagram to investigate the classical Pythagorean means.

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All-variables Sudoku

Stage: 3, 4 and 5 Challenge Level: Challenge Level:1

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

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Almost Total Inequality

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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' Tis Whole

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Number Chains

Stage: 5 Challenge Level: Challenge Level:1

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

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Integral Inequality

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

An inequality involving integrals of squares of functions.

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Integral Sandwich

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Generalise this inequality involving integrals.

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Balance Point

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Inside Outside

Stage: 4 Challenge Level: Challenge Level:1

Balance the bar with the three weight on the inside.

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Spread

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Fracmax

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

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Thousand Words

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Climbing

Stage: 5 Challenge Level: Challenge Level:1

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

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Max Box

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Mediant

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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Plutarch's Boxes

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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. . . .

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Comparing Continued Fractions

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Which of these continued fractions is bigger and why?

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Inequalities

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

A bag contains 12 marbles. There are more red than green but green and blue together exceed the reds. The total of yellow and green marbles is more than the total of red and blue. How many of. . . .

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Christmas Trees

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Christmas trees are planted in a rectangular array of 10 rows and 12 columns. The farmer chooses the shortest tree in each of the columns... the tallest tree from each of the rows ... Which is. . . .

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Not Continued Fractions

Stage: 4 and 5 Challenge Level: Challenge Level:1

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

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Tetra Inequalities

Stage: 5 Challenge Level: Challenge Level:1

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.

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Diverging

Stage: 5 Challenge Level: Challenge Level:1

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.

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Reciprocals

Stage: 5 Challenge Level: Challenge Level:1

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

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Power Up

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Big, Bigger, Biggest

Stage: 5 Challenge Level: Challenge Level:1

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

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Without Calculus

Stage: 5 Challenge Level: Challenge Level:1

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.

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After Thought

Stage: 5 Challenge Level: Challenge Level:1

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

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Rationals Between

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Code to Zero

Stage: 5 Challenge Level: Challenge Level:1

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.

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Exhaustion

Stage: 5 Challenge Level: Challenge Level:1

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

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Two Cubes

Stage: 4 Challenge Level: Challenge Level:1

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. . . .

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Shades of Fermat's Last Theorem

Stage: 5 Challenge Level: Challenge Level:1

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?