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There are **45** NRICH Mathematical resources connected to **Inequalities**, you may find related items under Algebraic expressions, equations and formulae.

Problem
Primary curriculum
Secondary curriculum
### Which Is Bigger?

Which is bigger, n+10 or 2n+3? Can you find a good method of answering similar questions?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Which Is Cheaper?

When I park my car in Mathstown, there are two car parks to choose from. Can you help me to decide which one to use?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Random Inequalities

Can you build a distribution with the maximum theoretical spread?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Squareness

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Discrete Trends

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Eyes Down

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?

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Proofs with Pictures

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Age 14 to 18

Problem
Primary curriculum
Secondary curriculum
### Approximating Pi

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?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Unit Interval

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

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Giants

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tet-trouble

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Quadratic Harmony

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square Mean

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Unit Interval

Can you prove our inequality holds for all values of x and y between 0 and 1?

Age 16 to 18

ShortChallenge Level

Problem
Primary curriculum
Secondary curriculum
### Erratic Quadratic

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Farey Neighbours

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Classical Means

Use the diagram to investigate the classical Pythagorean means.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### All-variables Sudoku

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

Age 11 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### ' Tis Whole

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?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Integral Inequality

An inequality involving integrals of squares of functions.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fracmax

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Climbing

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Biggest Enclosure

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Mediant Madness

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Plutarch's Boxes

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 their surface areas equal to their volumes?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Comparing Continued Fractions

Which of these continued fractions is bigger and why?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Inequalities

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 each colour there are in the bag?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Not Continued Fractions

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

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tetra Inequalities

Can you prove that in every tetrahedron there is a vertex where the three edges meeting at that vertex have lengths which could be the sides of a triangle?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Diverging

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Reciprocals

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Power Up

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Big, Bigger, Biggest

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Without Calculus

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Making Waves

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rationals Between...

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Code to Zero

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Exhaustion

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Two Cubes

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 prove you have found all possible solutions.]

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shades of Fermat's Last Theorem

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?

Age 16 to 18

Challenge Level