# Resources tagged with: Inequalities

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Broad Topics > Algebraic expressions, equations and formulae > Inequalities ### Thousand Words

##### Age 16 to 18Challenge Level

Here the diagram says it all. Can you find the diagram? ### Without Calculus

##### Age 16 to 18Challenge 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

##### Age 16 to 18Challenge Level

An inequality involving integrals of squares of functions. ### Tetra Inequalities

##### Age 16 to 18Challenge 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. ### Proofs with Pictures

##### Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities. ### In Between

##### Age 16 to 18Challenge Level

Can you find the solution to this algebraic inequality? ### Reciprocals

##### Age 16 to 18Challenge 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 18Challenge 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. ### Giants

##### Age 16 to 18Challenge Level

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

##### Age 16 to 18 ShortChallenge Level

Can you prove our inequality holds for all values of x and y between 0 and 1? ### Big, Bigger, Biggest

##### Age 16 to 18Challenge Level

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

##### Age 16 to 18Challenge Level

Generalise this inequality involving integrals. ### Square Mean

##### Age 14 to 16Challenge Level

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

##### Age 14 to 18Challenge Level

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

##### Age 14 to 16Challenge 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. . . . ### Quadratic Harmony

##### Age 16 to 18Challenge 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. ### Exhaustion

##### Age 16 to 18Challenge Level

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

##### Age 16 to 18Challenge 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? ### All-variables Sudoku

##### Age 11 to 18Challenge Level

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

##### Age 16 to 18Challenge Level

Can you find a quadratic equation which passes close to these points? ### Almost Total Inequality

##### Age 14 to 16Challenge Level ### Eyes Down

##### Age 16 to 18Challenge 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? ### Inner Equality

##### Age 16 to 18 ShortChallenge Level

Can you solve this inequalities challenge? ### Rationals Between...

##### Age 14 to 16Challenge Level

What fractions can you find between the square roots of 65 and 67? ### Not Continued Fractions

##### Age 14 to 18Challenge Level

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

##### Age 16 to 18Challenge Level

Which of these continued fractions is bigger and why? ### Biggest Enclosure

##### Age 14 to 16Challenge Level

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

##### Age 14 to 16Challenge Level

Kyle and his teacher disagree about his test score - who is right? ### Code to Zero

##### Age 16 to 18Challenge 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. ### Power Up

##### Age 16 to 18Challenge 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 ### Classical Means

##### Age 16 to 18Challenge Level

Use the diagram to investigate the classical Pythagorean means. ### ' Tis Whole

##### Age 14 to 18Challenge 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 18Challenge 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? ### Fracmax

##### Age 14 to 16Challenge 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. ### Discrete Trends

##### Age 16 to 18Challenge Level

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

##### Age 16 to 18Challenge Level

Can you build a distribution with the maximum theoretical spread? ### Squareness

##### Age 16 to 18Challenge 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? ### Climbing

##### Age 16 to 18Challenge Level

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

##### Age 14 to 16Challenge Level

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

##### Age 16 to 18Challenge Level

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

##### Age 14 to 18Challenge 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?