Proof Tutorial Class, Week 21

Question One §

Question §

Use mathematical induction to prove that $2^n>100n$ for any $n\ge 10$.

Solution §

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Question Two §

Question §

Where is the mistake in the following “proof” that all triangles have the same area?

Obviously, it is sufficient to prove that, given any $n$ triangles, they all have the same area. We use induction on $n$. Indeed,

1. If we have one triangle, the assertion is of course true.
2. Suppose the assertion is true for $n=k$. Consider any $k+1$ triangles $T_1,\dots ,T_{k+1}$. Apply the induction hypothesis to the first $k$ of the triangles: $T_1,\dots ,T_{k+1}$ all have the same area. Then apply the induction hypothesis to the last $k$ of them: $T_2,\dots ,T_{k+1}$ all have the same area. Then obviously all $k+1$ triangles $T_1,\dots ,T_{k+1}$ have the same area: area of $T_1=\text{area of }{T}_2=\cdots =\text{area of }{T}_k=\text{area of }{T}_{k+1}$. Thus, by the Axiom of Mathematical Induction, the assertion is true for all $n$, and therefore all triangles have the same area.

Solution §

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Question Three §

Question §

How does one tie a goat so it can eat grass within exactly a semicircle? It can be leashed multiple times, with several pegs. For example, one peg and a leash mean that the goat eats within a circle. It is also allowed to string one rope tightly between two pegs, and to tie the goat with a leash to a small ring sliding on the first rope.

Hint: use intersections of sets.

Solution §

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