HMMT November 2021 #0kk4

HMMT November 2021Hard

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Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Problem

Problem:

$O$ is the center of square $ABCD$ , and $M$ and $N$ are the midpoints of $\overline{BC}$ and $\overline{AD}$ , respectively. Points $A'$ , $B'$ , $C'$ , $D'$ are chosen on $\overline{AO}$ , $\overline{BO}$ , $\overline{CO}$ , $\overline{DO}$ , respectively, so that $A'B'MC'D'N$ is an equiangular hexagon. The ratio $\frac{[A'B'MC'D'N]}{[ABCD]}$ can be written as $\frac{a+b\sqrt{c}}{d}$ , where $a, b, c, d$ are integers, $d$ is positive, $c$ is square-free, and $\operatorname{gcd}(a, b, d)=1$ . Find $1000a+100b+10c+d$ .