MathDB

2

Part of 2018 Chile National Olympiad

Problems(1)

area of a ... square inside a square wanted (Chile 2018 L2 P1)

Source:

10/22/2022
Consider ABCDABCD a square of side 11. Points P,Q,R,SP,Q,R,S are chosen on sides ABAB, BCBC, CDCD and DADA respectively such that ∣AP∣=∣BQ∣=∣CR∣=∣DS∣=a|AP| = |BQ| =|CR| =|DS| = a, with a<1a < 1. The segments AQAQ, BRBR, CSCS and DPDP are drawn. Calculate the area of the quadrilateral that is formed in the center of the figure.
[asy] unitsize(1 cm);
pair A, B, C, D, P, Q, R, S;
A = (0,3); B = (0,0); C = (3,0); D = (3,3); P = (0,2); Q = (1,0); R = (3,1); S = (2,3);
draw(A--B--C--D--cycle); draw(A--Q); draw(B--R); draw(C--S); draw(D--P);
label("AA", A, NW); label("BB", B, SW); label("CC", C, SE); label("DD", D, NE); label("PP", P, W); label("QQ", Q, dir(270)); label("RR", R, E); label("SS", S, N); label("aa", (A + P)/2, W); label("aa", (B + Q)/2, dir(270)); label("aa", (C + R)/2, E); label("aa", (D + S)/2, N); [/asy]
geometrysquareareas