Resources tagged with: Inequalities

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There are 41 results

Broad Topics > Algebraic expressions, equations and formulae > Inequalities

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?

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.

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.

' 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?

Proofs with Pictures

Age 14 to 18

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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.

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.

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

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. . . .

Thousand Words

Age 16 to 18
Challenge Level

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

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?

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 Inequality

Age 16 to 18
Challenge Level

An inequality involving integrals of squares of functions.

Integral Sandwich

Age 16 to 18
Challenge Level

Generalise this inequality involving integrals.

Big, Bigger, Biggest

Age 16 to 18
Challenge Level

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

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?

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.

In Between

Age 16 to 18
Challenge Level

Can you find the solution to this algebraic inequality?

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?

Rationals Between...

Age 14 to 16
Challenge Level

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

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.

Biggest Enclosure

Age 14 to 16
Challenge Level

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

Classical Means

Age 16 to 18
Challenge Level

Use the diagram to investigate the classical Pythagorean means.

Mediant Madness

Age 14 to 16
Challenge Level

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

Comparing Continued Fractions

Age 16 to 18
Challenge Level

Which of these continued fractions is bigger and why?

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?

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?

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.

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.

Almost Total Inequality

Age 14 to 16
Challenge Level

Erratic Quadratic

Age 16 to 18
Challenge Level

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

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?

Random Inequalities

Age 16 to 18
Challenge Level

Can you build a distribution with the maximum theoretical spread?

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.

After Thought

Age 16 to 18
Challenge Level

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

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?

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 ?

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

Inner Equality

Age 16 to 18 Short
Challenge Level

Can you solve this inequalities challenge?

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?

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?