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Difference between revisions of "2004 Indonesia MO Problems/Problem 8"

(Created page with "==Problem 8== A floor with an area of <math>3 \text{ m}^2</math> will be covered by <math>5</math> rugs with various shapes, each having an area of <math>1 \text{ m}^2</math>...")
 
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A floor with an area of <math>3 \text{ m}^2</math> will be covered by <math>5</math> rugs with various shapes, each having an area of <math>1 \text{ m}^2</math>. Show that there exist <math>2</math> overlapping rugs with the overlapped area at least <math>1/5 \text{ m}^2</math>.
 
A floor with an area of <math>3 \text{ m}^2</math> will be covered by <math>5</math> rugs with various shapes, each having an area of <math>1 \text{ m}^2</math>. Show that there exist <math>2</math> overlapping rugs with the overlapped area at least <math>1/5 \text{ m}^2</math>.
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==Solution==
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Let the first 3 rugs occupy the entire floor, then the next rug that you add in, by the pigeon hole principle, it must overlap with another rug. Let a, b and c be the overlapping region with rug 1, rug 2 and rug 3 respectively, a+b+c = 1, thus at least one of a, b and c must be greater than 0.2.
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==Solution==
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Suppose that a, b, c, d, e are the area each rag covers. Then, by principle of inclusion-exclusion,
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<math>(a+b+c+d+e) - (a\cap b + a\cap c + a\cap d + a\cap e + b\cap c + b\cap d + b\cap e + c\cap d + c\cap e + d\cap e)</math>
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<math>+(a\cap b\cap c + ...+c\cap d\cap e) - (a\cap b\cap c\cap d + ...+ b\cap c\cap d\cap e)</math>
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<math>+(a\cap b\cap c\cap d\cap e) = 3</math>
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Let p, q, r, s indicate the area overlapped by at least 2 rags, 3 rags, 4 rags, and 5 rags, respectively, then the above equation can be replaced as <math>5+p-q+r-s=3</math>
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We will use contradiction
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Suppose it is indeed possible for an arrangement such that each overlap of 2 or more rags covers less than 0.2 area and the total covered area is equal to 3.
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Then, <math>0\le s < 0.2</math>.
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Since r represents the sum of 5 intersecting rags, with each overlapped area < 0.2, <math>s\le r < 1</math>.
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Similarly, <math>r\le q < 2</math>, since q is the sum of <math>{5 \choose 3} = 10</math> overlapped area, where each area < 0.2.
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<math>q\le p < 2</math>, since p is the sum of <math>{5\choose 2} = 10</math> areas.
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From the restrictions, we know that the total area = <math>5-p+q-r+s \ge 5-p+r-r+s > 5-2+s = 3+s \ge 3</math>. Thus, Total Covered Area <math>= 5-p+q-r+s > 3</math>, which is a contradiction.

Latest revision as of 20:57, 13 September 2024

Problem 8

A floor with an area of $3 \text{ m}^2$ will be covered by $5$ rugs with various shapes, each having an area of $1 \text{ m}^2$. Show that there exist $2$ overlapping rugs with the overlapped area at least $1/5 \text{ m}^2$.

Solution

Let the first 3 rugs occupy the entire floor, then the next rug that you add in, by the pigeon hole principle, it must overlap with another rug. Let a, b and c be the overlapping region with rug 1, rug 2 and rug 3 respectively, a+b+c = 1, thus at least one of a, b and c must be greater than 0.2.

Solution

Suppose that a, b, c, d, e are the area each rag covers. Then, by principle of inclusion-exclusion,

$(a+b+c+d+e) - (a\cap b + a\cap c + a\cap d + a\cap e + b\cap c + b\cap d + b\cap e + c\cap d + c\cap e + d\cap e)$ $+(a\cap b\cap c + ...+c\cap d\cap e) - (a\cap b\cap c\cap d + ...+ b\cap c\cap d\cap e)$ $+(a\cap b\cap c\cap d\cap e) = 3$

Let p, q, r, s indicate the area overlapped by at least 2 rags, 3 rags, 4 rags, and 5 rags, respectively, then the above equation can be replaced as $5+p-q+r-s=3$


We will use contradiction


Suppose it is indeed possible for an arrangement such that each overlap of 2 or more rags covers less than 0.2 area and the total covered area is equal to 3.

Then, $0\le s < 0.2$.

Since r represents the sum of 5 intersecting rags, with each overlapped area < 0.2, $s\le r < 1$.

Similarly, $r\le q < 2$, since q is the sum of ${5 \choose 3} = 10$ overlapped area, where each area < 0.2.

$q\le p < 2$, since p is the sum of ${5\choose 2} = 10$ areas.

From the restrictions, we know that the total area = $5-p+q-r+s \ge 5-p+r-r+s > 5-2+s = 3+s \ge 3$. Thus, Total Covered Area $= 5-p+q-r+s > 3$, which is a contradiction.

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