Odd "How many triangles are there in the picture?" Question From the First Stage of Israel's International Mathematical Olympiad Selection Process
You can follow this picture, the number of triangles are written inside of each drawing:
A triangle is determined by three lines. There are $12$ lines in the picture, so as a first approximation we have $\binom{12}3=220$ possible triangles. But not every triple of lines makes a triangle, so we have to count the bad triples and subtract. There are three types of bad triples.
Type I. Three lines meeting at one point. This happens $\boxed{20}$ times; the point of concurrency can only be a corner of the big square or the midpoint of a side of the big square.
Type II. Two of the three lines are parallel. There are $6$ pairs of parallel lines, so we get a bad triple of this kind by taking two parallel lines and a random third line, in $6\cdot10=\boxed{60}$ ways.
Type III. None of the three lines are parallel, but at least two of them fail to meet within the picture. In this case one of the lines, call it $S$, must be a side of the big square, and another is one of the two lines, call them $A$ and $B$, which are not parallel to $S$ but do not meet $S$ within the picture. (If $S$ is the top side, then $A$ is the line from the lower left corner of the big square to the middle of the right side, and $B$ is its mirror image.) There are $8$ type III triples containing $S$ and $A$, as the third line can be any of the $8$ lines not parallel to $S$ or $A$. Likewise there are $8$ type III triples containing $S$ and $B$. Since the triple $\{S,A,B\}$ has been counted twice, the total number of type III triples containing $S$ and $A$ or $B$ is $8+8-1=15$. Since there are $4$ choices for $S$ (and no overlaps between different choices for $S$), the total number of bad triples of type III is $4\cdot15=\boxed{60}$.
The number of triangles in the picture is $$220-20-60-60=\boxed{80}.$$
There are several types of triangles. Two types are symmetric (the first two in the following figure), they occur $4$ times each. All other types occur $8$ times each. I found $9$ of them, so that we have $80$ triangles in all.
