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Difference between revisions of "2016 AIME II Problems/Problem 7"

m (→‎Solution)
(→‎Solution)
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Solution by Shaddoll (edited by ppiittaattoo)
 
Solution by Shaddoll (edited by ppiittaattoo)
 +
 +
<asy>
 +
pair A,B,C,D,E,F,G,H,I,J,K,L;
 +
A=(0,0);
 +
B=(2016,0);
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C=(2016,2016);
 +
D=(0,2016);
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I=(1008,0);
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J=(2016,1008);
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K=(1008,2016);
 +
L=(0,1008);
 +
E=(504,504);
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F=(1512,504);
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G=(1512,1512);
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H=(504,1512);
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draw(A--B--C--D--A);
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draw(I--J--K--L--I);
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draw(E--F--G--H--E);
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label("$A$",A,SW);
 +
</asy>
 +
*Drawing not finished, someone please finish it.
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2016|n=II|num-b=6|num-a=8}}
 
{{AIME box|year=2016|n=II|num-b=6|num-a=8}}

Revision as of 10:40, 21 February 2019

Problem

Squares $ABCD$ and $EFGH$ have a common center and $\overline{AB} || \overline{EF}$. The area of $ABCD$ is 2016, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on a side of $IJKL$. Find the difference between the largest and smallest positive integer values for the area of $IJKL$.

Solution

Letting $AI=a$ and $IB=b$, we have $IJ^{2}=a^{2}+b^{2} \geq 1008$ by Cauchy-Schwarz inequality. Also, since $EFGH||ABCD$, the angles that each square cuts another are equal, so all the triangles are formed by a vertex of a larger square and $2$ adjacent vertices of a smaller square are similar. Therefore, the areas form a geometric progression, so since $2016=12^{2} \cdot 14$, we have the maximum area is $2016 \cdot \dfrac{11}{12} = 1848$ (the areas of the squares from largest to smallest are $12^{2} \cdot 14, 11 \cdot 12 \cdot 14, 11^{2} \cdot 14$ forming a geometric progression).


The minimum area is $1008$ (every square is half the area of the square whose sides its vertices touch), so the desired answer is $1848-1008=\boxed{840}$.

Solution by Shaddoll (edited by ppiittaattoo)

[asy] pair A,B,C,D,E,F,G,H,I,J,K,L; A=(0,0); B=(2016,0); C=(2016,2016); D=(0,2016); I=(1008,0); J=(2016,1008); K=(1008,2016); L=(0,1008); E=(504,504); F=(1512,504); G=(1512,1512); H=(504,1512); draw(A--B--C--D--A); draw(I--J--K--L--I); draw(E--F--G--H--E); label("$A$",A,SW); [/asy]

  • Drawing not finished, someone please finish it.

See also

2016 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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