Arguments by Contradiction
You can prove that something
is true
by showing that not something
being true
would lead to a contradictory, absurd conclusion.
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Without appealing to technology,
prove that there do not exist three consecutive natural numbers
such that the cube of the largest is equal to
the sum of the cubes of the other two.
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Twenty five boys and twenty five girls sit around a table.
Prove that it is always possible to find a person
sitting directly between two girls.
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Prove there isn’t
a continuous function \(f\colon \mathbf{R} \to \mathbf{R}\)
such that \(f\big(f(x)\big) = -x.\)
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Putnam and Beyond
Show that no set of nine consecutive integers
can be partitioned into two sets
with the product of the elements of the first set
equal to the product of the elements of the second set.
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German Math Olympiad (1985)
Every point of three-dimensional space
is colored red, green, or blue.
Prove that one of the colors attains all distances.
I.e. for one of the colors,
for any positive real number you can find two points
of that color that are that far apart.
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Romanian Math Olympiad (1986)
Suppose that \(\mathcal{F} = \{E_1, \dotsc, E_s\}\) is a family of subsets
of some universal set \(U\) that each have \(r\) elements.
Show that if the intersection of any \(r+1\)
(not necessarily distinct) sets in \(\mathcal{F}\) is nonempty
then the intersection of all of the sets in \(\mathcal{F}\) is nonempty.
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Putnam and Beyond
Prove that \(\sqrt{2}+\sqrt{3}+\sqrt{5}\)
is an irrational number.
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Putnam and Beyond
Show that there does not exist a function
\(f \colon \mathbf{Z} \to \{1, 2, 3\}\)
satisfying \(f(x) \neq f(y)\)
for all \(x,y \in \mathbf{Z}\)
such that \(|x - y| \in \{2,3, 5\}\).
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Putnam and Beyond
Show that there does not exist
a strictly increasing function
\(f \colon \mathbf{N} \to \mathbf{N} \)
satisfying \(f(2) = 3\)
and \(f(mn) = f(m) f(n)\) for all \(m,n \in \mathbf{N}\).
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Putnam and Beyond
Let \(x \gt 1\) be an arbitrary real number
and let \(n\) be the number of positive prime numbers
less than or equal to \(x\).
Select \(n+1\) positive integers such that
none of them divides the product of all the others.
Prove that there exists a number among the chosen \(n+1\)
that is bigger than \(x\).