Problem

Source: 2nd National Women's Contest of Mexican Mathematics Olympiad 2023 , level 1+2 p2

Tags: Mexico, algebra



Matilda drew $12$ quadrilaterals. The first quadrilateral is an rectangle of integer sides and $7$ 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 $1$. What is the maximum area possible for the first quadrilateral? [asy][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); // Paralelogramo dot(M); dot(N); dot(P); dot(Q); [/asy][/asy] $\textbf{Note:}$ The above figure illustrates the first two quadrilaterals that Matilda drew.