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My Favorite Math Problems
Here are some great math problems for high school students. To
make this list, a problem must be accessible and delightful, conducive
to further exploration, but non-standard. Hints, solutions, and
extensions are provided for each of them. However, no peeking at
the hint until you've made an honest effort to solve the problem.
Have fun with the mathematics!
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Note:
the phrase no three collinear means
that any line in the plane will pass through at most two
of the points at a time. We need this condition, for otherwise
we could place all 101 points in a row, with point V
on one end, to obtain a counterexample to the statement. |
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Note:
a unit disc consists of all points inside or on
a circle with radius one. Also, each segment must have positive
length; no points are allowed. As usual, a tiling
implies that none of the segments intersect and that every
point is covered by exactly one segment. |
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| Note:
the three expressions appearing here are commonly called binomial
coefficients. The first one on the left is usually read
“n choose 3” and is often written as
C(n,3) in text. It represents the number
of ways to choose 3 objects from among a collection of n
different objects. For example, C(4,3)=4. (Just count
the number of ways there are to select three of the four fingers
on your left hand.) The same sort of interpretation applies
for C(n,4) and C(n+1,4).
The goal here is to show that the two sides of the above equation
are equal based solely on what they mean, without using any
algebra. |
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I will gradually be expanding this
list. The most recent addition was posted on 11/24/05. Feel free
to Contact Us if you would like to
suggest your own favorite math problem, but be forewarned that the
editor is somewhat choosy about which questions to include.
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