Tough Problem,Lattice,Circles

Sorry for the typesetting

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just a little correction of mistake: 1. sort by slope in DESCENDING order (but the order does not really matter

because of symmetric properties) 2. sort by slope

the slope of the line passing thru. O and the point

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Andrew: you can wer it. it is good if this problem can be solved. but it could be solved

i would still award you best wer

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1. Pick"s theorem? Brilliant. You want to pare the area of triangles P(i) O P(i+1) using 1/2absin theta and Pick"s? (<--a guess) 2. Actually this is only a part of a big problem. The big problem itself is a minmax problem. and now we are at the max stage.

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3. Yes

it breaks. But what is your idea? perhaps there exists a line that is suitable to bound the points without changing the no. of point.

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(supp.) 3. because the quarter circle gives the points to be considered

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Please recall that Pm is the m-th point (sort by slope in ascending or descending order) in the set of points. And we"d like to find the largest angle that the line P_m O makes with P_m+1 O

Working on it. Couple hints for others: 1.) Consider Pick"s theorem - I think it is relevant here. 2.) The GCD constraint is really simply removing lines with exactly same slope. 2010-06-24 13:25:20 补充： We can think of this problem as a maximin problem - for a particular point

minmize by finding a point nearby that minimize the angle

and maximize by finding the point that maximize the quantity above. 2010-06-24 13:25:29 补充： From this perspective - we can approach the problem this way. From a point

try to find an upper bound of the minimized angle. If this upper bound is less than the one beeen P1 and P2

we are done. 2010-06-24 13:25:39 补充： By Pick"s Theorem - we have the area of the triangle = 0.5

I wonder how can we make use of this fact. It sounds like a constrainted maximization problem - find a triangle with with integer coordinates

area = 0.5

with maximum angle. 2010-06-24 13:25:46 补充： kwanhimshek - if you have a simulation program

can you remove the constraint that the points are in the quarter circle - does that break the conjecture? 2010-06-29 11:06:52 补充： I don"t have a plete wer yet

but I want to share what I think about the problem and hopefully we can wer that together. First of all

we can leverage Pick"s theorem. For all the triangles that are formed by three lattice point on the vertexs and no lattice point on the edge in interior

the area must be 0.5. We can also calculate the area in a different way using vector cross product

suppose the points are (a

b) and (c

d) respectively

we have 2A = |ad - bc| = 1 We can also calculate the area in a third way

in particular

we have 2A = |(a

b)| |(c

d)| sin theta = 1 Now since sine is an increasing function

therefore the larger the angle

the larger the sine

therefore

if we want big angles

we will have *** all length products. Unfortunately

Pick"s theorem does not help us to eliminate all wrong triangles

in particular

we can have triangles with no lattice points in the interior and boundary but yet not a valid triangle

(0

1)-(0

0)-(1

0) is a counter example. In fact

it gives the largest possible angle = 90 degree. 2010-06-30 12:16:45 补充： I think we need more explanation here: If we join P(m)

O

P(m + 1) into a triangle

it will be a triangle that 1) Have no points in the interior (for if there is one

then there is a line with *** aller angle and P(m) and P(m+1) could not be next to each other)

and 2010-06-30 12:16:49 补充： 2) Have no point in the edge (or otherwise GCD(a

b) > 1 or GCD(c

d) > 1) Therefore it will satisfy the Pick"s theorem

let us call these right triangles. There are wrong triangles that also satisfy Pick"s theorem

as I mention in the last paragraph. 2010-07-02 10:36:30 补充： I have more findings here. For ad-bc must equals to 1

and gcd(a

b) = 1. We can guarentee the existence of (c

d) by the extended euclidean algorithm. Suppose we get some (c

d) here such that ad-bc=1

another pair of (c"

d") that also satisfy the relation:: ad" - bc" = 1 ad - bc = 1 a(d"-d)-b(c"-c)=0

参考: Pick"s theorem

i"ll try to understand your wer..but i"m afraid i can"t- -