Resources tagged with: Inequalities

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Broad Topics > Algebraic expressions, equations and formulae > Inequalities

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Rationals Between...

Age 14 to 16 Challenge Level:

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

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Giants

Age 16 to 18 Challenge Level:

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

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In Between

Age 16 to 18 Challenge Level:

Can you find the solution to this algebraic inequality?

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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.

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Integral Sandwich

Age 16 to 18 Challenge Level:

Generalise this inequality involving integrals.

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Eyes Down

Age 16 to 18 Challenge Level:

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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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. . . .

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Big, Bigger, Biggest

Age 16 to 18 Challenge Level:

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

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Square Mean

Age 14 to 16 Challenge Level:

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

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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.

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Discrete Trends

Age 16 to 18 Challenge Level:

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

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Almost Total Inequality

Age 14 to 16 Challenge Level:

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Thousand Words

Age 16 to 18 Challenge Level:

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

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Unit Interval

Age 14 to 18 Challenge Level:

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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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?

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Proofs with Pictures

Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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Quadratic Harmony

Age 16 to 18 Challenge Level:

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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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?

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Erratic Quadratic

Age 16 to 18 Challenge Level:

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

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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.

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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.

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Inner Equality

Age 16 to 18 Short Challenge Level:

Can you solve this inequalities challenge?

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Integral Inequality

Age 16 to 18 Challenge Level:

An inequality involving integrals of squares of functions.

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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

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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.

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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.

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Mediant Madness

Age 14 to 16 Challenge Level:

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

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Comparing Continued Fractions

Age 16 to 18 Challenge Level:

Which of these continued fractions is bigger and why?

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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?

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Biggest Enclosure

Age 14 to 16 Challenge Level:

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

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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?

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Squareness

Age 16 to 18 Challenge Level:

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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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.

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Approximating Pi

Age 14 to 18 Challenge Level:

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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Inside Outside

Age 14 to 16 Challenge Level:

Balance the bar with the three weight on the inside.

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After Thought

Age 16 to 18 Challenge Level:

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

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Random Inequalities

Age 16 to 18 Challenge Level:

Can you build a distribution with the maximum theoretical spread?

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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.

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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?

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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

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Tet-trouble

Age 14 to 16 Challenge Level:

Is it possible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units?

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Classical Means

Age 16 to 18 Challenge Level:

Use the diagram to investigate the classical Pythagorean means.

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' 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?