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(PQRS)<= 1/2 (ABCD), quadr. inside a square

Source: 2018 Ecuador Juniors (OMEC) L2 p3

October 24, 2022
geometrysquaregeometric inequalityareas

Problem Statement

Let ABCDABCD be a square. Point P,Q,R,SP, Q, R, S are chosen on the sides ABAB, BCBC, CDCD, DADA, respectively, such that AP+CR≥AB≥BQ+DSAP + CR \ge AB \ge BQ + DS. Prove that area  (PQRS)≤12  area  (ABCD)area \,\, (PQRS) \le \frac12 \,\, area \,\, (ABCD) and determine all cases when equality holds.
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