angle wanted, 3 squares related

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Source: 2022 European Math Tournament - Senior Grand League - Math Battle 4.8

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#1 h Dec 19, 2022, 1:33 PM

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The figure shows three squares. Find the marked angle.

https://cdn.artofproblemsolving.com/attachments/b/4/291ac111faa614ca2eac350806c03e1a670acf.png

This post has been edited 2 times. Last edited by parmenides51, Dec 19, 2022, 1:35 PM

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#2 h Dec 19, 2022, 1:34 PM

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posted for the image links

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#3 h Dec 19, 2022, 1:43 PM

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are there information about ratio of sides of squares? I think angle is depended on this ratio

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#4 h Dec 19, 2022, 1:49 PM • 1 Y

Y by ETS1331

No other data was given, original wording source

My comment: The two small squares seem to be equal.

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#5 h Dec 19, 2022, 1:52 PM

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RagvaloD wrote:

are there information about ratio of sides of squares? I think angle is depended on this ratio

surprisingly, it's not

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#6 h Dec 19, 2022, 5:22 PM • 2 Y

Y by rayfish, cosmicgenius

Spiral sim at $Q$ maps $AB$ to $CD$ $\implies (ABPQ), (CDPQ)$ cyclic $\implies \boxed{45^\circ}$

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#7 h Dec 19, 2022, 5:45 PM

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alternatively, assume small squares have side length 0

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#8 h Dec 19, 2022, 5:52 PM • 1 Y

Y by Mango247

AoPS111 wrote:

alternatively, assume small squares have side length 0

Yes, if there isn't anything restraining the size of the small squares, and we can find the angle doubtless. It doesn't matter what the size of the square is. The angle is always the same no matter the size.

We can still find that the angle is 45 degrees.

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#9 h Dec 25, 2022, 10:38 PM

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sol

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#10 h Dec 26, 2022, 1:02 AM

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ryanbear wrote:

whats wrong with this solution?
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if all squares are 1x1 then c is not symmetric to D so theres 2 different lengths

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