Inequalities

problem
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?
problem
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?
problem
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.
problem
Tetra inequalities

Age

16 to 18

Challenge level

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?
problem
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.
problem
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.
problem
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
problem
Big, bigger, biggest

Age

16 to 18

Challenge level

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