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As you might know, the 2010 Euclid Contest was officially taken yesterday. So lets discuss!

I thought it wasnt too bad. #10 was hard though (the triangle one). Here was the question:

For each positive integer n, let T(n) be the number of triangles with integer side lengths, positive area, and perimeter n. For example, T(6) = 1 since only such triangle with a perimeter of 6 has side lengths 2, 2 and 2.

(b) If m is a positive integer with m >=(greater than or equal to) 3, prove that T(2m) = T(2m-3).

(c) Determine the smallest positive integer n such that T(n) > 2010.

Also, #9 was hard:

(b) In triangle ABC, BC = a, AC = b, AB = c, and a < .5(b+c).

Prove that angle BAC < .5 (angle ABC + angle ACB).

Can you please help me with those problems?

Also, if anyone wants to share how they solved #7 and 8, that would be appreciated :)

I thought it wasnt too bad. #10 was hard though (the triangle one). Here was the question:

For each positive integer n, let T(n) be the number of triangles with integer side lengths, positive area, and perimeter n. For example, T(6) = 1 since only such triangle with a perimeter of 6 has side lengths 2, 2 and 2.

(b) If m is a positive integer with m >=(greater than or equal to) 3, prove that T(2m) = T(2m-3).

(c) Determine the smallest positive integer n such that T(n) > 2010.

Also, #9 was hard:

(b) In triangle ABC, BC = a, AC = b, AB = c, and a < .5(b+c).

Prove that angle BAC < .5 (angle ABC + angle ACB).

Can you please help me with those problems?

Also, if anyone wants to share how they solved #7 and 8, that would be appreciated :)

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