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Difference between revisions of "2023 WSMO Speed Round Problems/Problem 1"

(Created page with "==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...")
 
 
(2 intermediate revisions by the same user not shown)
Line 25: Line 25:
  
 
==Solution==
 
==Solution==
 +
We divide the shaded region into three sections as follows
 +
 +
<asy>
 +
size(4cm);
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for (int i=0; i<5; ++i)
 +
{
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for (int j=0; j<5; ++j)
 +
{
 +
  dot((i,j));
 +
}
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}
 +
pair a = (0,2);
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pair b = (1,4);
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pair c = (3,2);
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pair d = (4,4);
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pair e = (3,0);
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pair f = (1,2);
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path x = a--b--c--d--e--f--cycle;
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fill(x, cyan);
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draw(x, linewidth(2));
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 +
draw(b--f,red+linewidth(2));
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draw(c--e,red+linewidth(2));
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</asy>
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 +
The left triangle has an area of <math>\tfrac{1}{2}\cdot1\cdot2 = 1</math>, the center parallelogram has an area of <math>2\cdot2 = 4</math>, and the right triangle has an area of <math>\tfrac{1}{2}\cdot1\cdot2 = 1</math>. Thus, our answer is <math>1+4+1 = \boxed{6}.</math>
 +
 +
~pinkpig

Latest revision as of 11:05, 12 September 2025

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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