Other tags that relate to Thousand Words
Indices. Visualising. Quadratic equations. Mathematical reasoning & proof. Inequalities. Powers & roots. Creating and manipulating expressions and formulae. Generalising. Complex numbers. Making and proving conjectures.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.
Big, Bigger, Biggest
Age 16 to 18 Challenge Level:
Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?
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.
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.
Proofs with Pictures
Age 14 to 18
Some diagrammatic 'proofs' of algebraic identities and 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.
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?
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 ?
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?
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
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.
Classical Means
Age 16 to 18 Challenge Level:
Use the diagram to investigate the classical Pythagorean means.
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?
Mediant Madness
Age 14 to 16 Challenge Level:
Kyle and his teacher disagree about his test score - who is right?
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. . . .
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?
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?
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.
Biggest Enclosure
Age 14 to 16 Challenge Level:
Three fences of different lengths form three sides of an enclosure. What arrangement maximises the area?
Comparing Continued Fractions
Age 16 to 18 Challenge Level:
Which of these continued fractions is bigger and why?
Rationals Between...
Age 14 to 16 Challenge Level:
What fractions can you find between the square roots of 65 and 67?
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
' 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?
Random Inequalities
Age 16 to 18 Challenge Level:
Can you build a distribution with the maximum theoretical spread?
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.
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?
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?
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 ?
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.