MathDB

4

Part of 1996 May Olympiad

Problems(2)

angle wanted, square related

Source: II May Olympiad (Olimpiada de Mayo) 1996 L2 P4

9/17/2022
Let ABCDABCD be a square and let point FF be any point on side BCBC. Let the line perpendicular to DFDF, that passes through BB, intersect line DCDC at QQ. What is value of ∠FQC\angle FQC?
geometryanglessquare
numbers on drawing

Source: II May Olympiad (Olimpiada de Mayo) 1996 L1 P4

9/17/2022
(a) In this drawing, there are three squares on each side of the square. Place a natural number in each of the boxes so that the sum of the numbers of two adjacent boxes is always odd. https://cdn.artofproblemsolving.com/attachments/e/6/75517b7d49857abd3f8f0430a70ae5b0eb1554.gif
(b) In this drawing, there are now four squares on each side of the triangle. Justify why a natural number cannot be placed in each box so that the sum of the numbers in two adjacent boxes is always odd. https://cdn.artofproblemsolving.com/attachments/c/8/061895b9c1cdcb132f7d37087873b7de3fb5f3.gif
(c) If you now draw a polygon with51 51 sides and on each side you place 5050 boxes, taking care that there is a box at each vertex. Can you place a natural number in each box so that the sum of the numbers in two adjacent boxes is always odd? Why?
combinatorics