Other tags that relate to Thousand Words
Visualising. Generalising. Mathematical reasoning & proof. Powers & roots. Creating and manipulating expressions and formulae. Complex numbers. Indices. Quadratic equations. Making and proving conjectures. Inequalities.There are 41 results
Broad Topics > Algebraic expressions, equations and formulae > Inequalities
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.
Integral Inequality
Age 16 to 18
Challenge Level
An inequality involving integrals of squares of functions.
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.
Proofs with Pictures
Age 14 to 18
Some diagrammatic 'proofs' of algebraic identities and inequalities.
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.
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.
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 ?
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?
Big, Bigger, Biggest
Age 16 to 18
Challenge Level
Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?
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?
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?
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. . . .
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.
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
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?
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.
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?
Rationals Between...
Age 14 to 16
Challenge Level
What fractions can you find between the square roots of 65 and 67?
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?
Comparing Continued Fractions
Age 16 to 18
Challenge Level
Which of these continued fractions is bigger and why?
Biggest Enclosure
Age 14 to 16
Challenge Level
Three fences of different lengths form three sides of an enclosure. What arrangement maximises the area?
Mediant Madness
Age 14 to 16
Challenge Level
Kyle and his teacher disagree about his test score - who is right?
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.
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
Classical Means
Age 16 to 18
Challenge Level
Use the diagram to investigate the classical Pythagorean means.
' 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?
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?
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.
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.
Random Inequalities
Age 16 to 18
Challenge Level
Can you build a distribution with the maximum theoretical spread?
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?
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.
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?
After Thought
Age 16 to 18
Challenge Level
Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?
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?
NRICH is part of the family of activities in the Millennium Mathematics Project.