Two international triangle counting problems

With two problems counting triangles in the international exam for 6th and 7th graders, how long can you solve them?

Lesson 1 (Excerpt from APMOPS 2005 Math Olympiad exam):

Given 12 points on 2 lines (a), (b) as shown below. Ask how many triangles have 3 vertices taken from 3 of the 12 given points.

Lesson 2 (Excerpt from BIMC 2013 Math Olympiad):

Given 21 points on 3 lines (a), (b), (c) as shown below. Ask how many triangles with 3 vertices taken from 3 of the 21 given points are there at most.

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Tran Phuong

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