Inequalities
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
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.
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
Thousand words
Age
16 to 18
Challenge level
Here the diagram says it all. Can you find the diagram?
problem
Powerful order
Age
14 to 16
Challenge level
Powers of numbers might look large, but which of these is the largest...
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
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.
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?