1

Part of 2006 IMC

Problems(2)

IMC 2006, problem 1, day 1

Source: Easy

f: \mathbb{R}\to \mathbb{R}

be a real function. Prove or disprove each of the following statements. (a) If f is continuous and range(f)=

\mathbb{R}

then f is monotonic (b) If f is monotonic and range(f)=

\mathbb{R}

then f is continuous (c) If f is monotonic and f is continuous then range(f)=

\mathbb{R}

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IMC 2006, problem 1, day 2

Source: Easy

V

be a convex polygon. (a) Show that if

V

3k

V

can be triangulated such that each vertex is in an odd number of triangles. (b) Show that if the number of vertices is not divisible with 3, then

V

can be triangulated such that exactly 2 vertices have an even number of triangles.

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