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2023 WSMO Speed Round Problems/Problem 1

Problem

Find the number of square units in the area of the shaded region. [asy] size(4cm); for (int i=0; i<5; ++i) { for (int j=0; j<5; ++j) { dot((i,j)); } } pair a = (0,2); pair b = (1,4); pair c = (3,2); pair d = (4,4); pair e = (3,0); pair f = (1,2); path x = a--b--c--d--e--f--cycle; fill(x, cyan); draw(x, linewidth(2)); [/asy]

Solution

We divide the shaded region into three sections as follows

[asy] size(4cm); for (int i=0; i<5; ++i) { for (int j=0; j<5; ++j) { dot((i,j)); } } pair a = (0,2); pair b = (1,4); pair c = (3,2); pair d = (4,4); pair e = (3,0); pair f = (1,2); path x = a--b--c--d--e--f--cycle; fill(x, cyan); draw(x, linewidth(2)); draw(b--f,red+linewidth(2)); draw(c--e,red+linewidth(2)); [/asy]

The left triangle has an area of $\tfrac{1}{2}\cdot1\cdot2 = 1$, the center parallelogram has an area of $2\cdot2 = 4$, and the right triangle has an area of $\tfrac{1}{2}\cdot1\cdot2 = 1$. Thus, our answer is $1+4+1 = \boxed{6}.$

~pinkpig

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