quadrilateral with midpoints
Source: 2nd National Women's Contest of Mexican Mathematics Olympiad 2023 , level 1+2 p2
July 22, 2023
Mexicoalgebra
Problem Statement
Matilda drew quadrilaterals. The first quadrilateral is an rectangle of integer sides and times more width than long. Every time she drew a quadrilateral she joined the midpoints of each pair of consecutive sides with a segment. It´s is known that the last quadrilateral Matilda drew was the first with area less than . What is the maximum area possible for the first quadrilateral?
[asy]size(200);
pair A, B, C, D, M, N, P, Q;
real base = 7;
real altura = 1;A = (0, 0);
B = (base, 0);
C = (base, altura);
D = (0, altura);
M = (0.5*base, 0*altura);
N = (0.5*base, 1*altura);
P = (base, 0.5*altura);
Q = (0, 0.5*altura);draw(A--B--C--D--cycle); // Rectángulo
draw(M--P--N--Q--cycle); // Paralelogramodot(M);
dot(N);
dot(P);
dot(Q);
[/asy]
The above figure illustrates the first two quadrilaterals that Matilda drew.