Other tags that relate to Thousand Words
Making and proving conjectures. Mathematical reasoning & proof. Inequalities. Indices. Complex numbers. Powers & roots. Quadratic equations. Creating and manipulating expressions and formulae. Visualising. Generalising.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.
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?
Proofs with Pictures
Age 14 to 18
Some diagrammatic 'proofs' of algebraic identities and inequalities.
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.
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.
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.
Big, Bigger, Biggest
Age 16 to 18 Challenge Level:
Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?
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?
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.
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 ?
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
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?
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.
Erratic Quadratic
Age 16 to 18 Challenge Level:
Can you find a quadratic equation which passes close to these points?
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?
Classical Means
Age 16 to 18 Challenge Level:
Use the diagram to investigate the classical Pythagorean means.
Biggest Enclosure
Age 14 to 16 Challenge Level:
Three fences of different lengths form three sides of an enclosure. What arrangement maximises the area?
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?
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?
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. . . .
' 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?
Rationals Between...
Age 14 to 16 Challenge Level:
What fractions can you find between the square roots of 65 and 67?
Comparing Continued Fractions
Age 16 to 18 Challenge Level:
Which of these continued fractions is bigger and why?
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?
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.
Random Inequalities
Age 16 to 18 Challenge Level:
Can you build a distribution with the maximum theoretical spread?
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.
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.
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
After Thought
Age 16 to 18 Challenge Level:
Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?
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.
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?
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.