#Olympiad problem

29 messages · Page 1 of 1 (latest)

iron cliffBOT
gleaming scaffold
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Consider the area of a potential square with vertices on the lattice points. Can you deduce a contradiction?

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also this is a PROMYS problem not an olympiad problem

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you’re technically supposed to do these on your own :\

glacial marsh
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ik 🥲 im trying man but i gotta get in nd idk quite h ow to prove this

glacial marsh
gleaming scaffold
glacial marsh
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not quite for eu

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but stil like

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i like maths but i found out abt the program rly late

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and its due in 5 days

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for the 8 questions

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nd they r jst hella difficult like for some of them i have no idea where to start dyk any like information or smth i cld read to help cz i gotta prove every step of my working

celest musk
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is this php?

glacial marsh
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whats that?

glacial marsh
celest musk
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pigeonhole principle

glacial marsh
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nah thats q8 which icl is bamboozlingly difficult

glacial marsh
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like u can use them to show the first part but 0.1 and 0.01 part idk

celest musk
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ah

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idk man

glacial marsh
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Anyone know?

celest musk
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u can ping the helpers

glacial marsh
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<@&286206848099549185>

last geyser
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Since it says "at least" so let's consider the cases where the drawn square has it's bottom side horizontal. The minimum vertical distance between any 2 grid points is (½√3)•2=√3, and the minimum horizontal distance between any 2 grid points is 1. Thus for the first question it is not possible. Now for the second question, if we draw a square with horizontal side 1, the vertical side must also be 1, so the remaining vertical distance is √3-1≈0.732. But that is the case when the first picked horizontal side has 2 of it's vertices lie on 2 grid points. To make the distance between the vertices of the square and grid points minimum, we move the square downwards vertically till the vertical distance of each point with the nearest grid point is equal, which is (√3-1)/2≈0.366. The distance is smaller but it is still more than ⅒=0.1. Thus for the second question it is also not possible. Lastly, for the third question, 1/100 < 1/10 so it is obviously not possible either.

glacial marsh
last geyser
glacial marsh