I have written about the Soifer Mathematical Olympiad before. Go here for the first in a mini-series of two blogs that will give you a bit of the flavor. We try to make the first problem really easy, but we don’t always succeed. This year’s problem 1 was so easy we almost didn’t use it, yet students had trouble with it: only 40 of 200 solved it.

Triangular Question (A. Soifer).Each side of triangle T is longer than each side of triangle R. Does that mean that the area of T is greater than the area of R?

If you think about if for a few seconds, you know without any complicated mathematics that the answer is NO. Here’s the figure accompanying Dr. Soifer’s solution:

The shortest side of DEF (triangle T) is longer than the longest side of ABC (triangle R) and DEF’s area is half that of ABC. In my solution, I simply compared an equilateral triangle of side 2 (area square root of 3) with a triangle with sides 3, 4, 6.9999 (area virtually 0). As I say, a very simple problem.

The simplicity didn’t prevent students from answering incorrectly. Some, because they were very inexperienced at math and didn’t know what they were doing.

Others, however, way more experienced in mathematics, used formulas I hadn’t thought of in decades. One student based his “proof” on the formula Area = .5 x a x b x sin(C), where a and b are two sides of the triangle and sin(C) is the sine of the angle between them. Very sophisticated, and the student wrote two or three pages “proving” that the area of triangle T is always larger than the area of triangle R. Beautifully written, completely wrong.

Of course, it’s not just in math…

Certain persons…have wandered away into vain discussion…without understanding either what they are saying or the things about which they make confident assertions. (1 Timothy 1.6, 7, ESV)

If anyone teaches a different doctrine and does not agree with the sound words of our Lord Jesus Christ and the teaching that accords with godliness, he is puffed up with conceit and understands nothing. He has an unhealthy craving for controversy and for quarrels about words, which produce envy, dissension, slander, evil suspicions… (1 Timothy 6.3, 4, ESV)

And Jesus answered them, “See that no one leads you astray…And many false prophets will arise and lead many astray.” (Matthew 24.4, 11, ESV)

PS The formula the student attempted to use could be used to arrive at the correct answer. The closer angle C gets to a straight line, the smaller the triangle’s area is. And as angle C gets larger (approaching a straight line) its sine approaches 0, and the formula will tell you that the triangle’s area approaches 0 no matter how long the sides are.