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Problems Methods Extremal principle Extremal principle (other) Discrete Mathematics Set theory and logic Order relations Algebraic methods Proof by exhaustion

13–15 3.0

The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from $n$ books of different heights, if: a) $n = 4$ ; b) $n = 5$ ?

Problem #PRU-98016

Problems Discrete Mathematics Combinatorics Integer lattices Integer lattices (other) Set theory and logic Order relations Methods Pigeonhole principle Pigeonhole principle (other)

10–15 4.0

We are given 101 rectangles with integer-length sides that do not exceed 100.

Prove that amongst them there will be three rectangles $A, B, C$ , which will fit completely inside one another so that $A \subset B \subset C$ .