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2000 SMT/Advanced Topics Problems

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Problem 1

How many different ways are there to paint the sides of a tetrahedron with exactly 4 colors? Each side gets its own colour, and two colourings are the same if one can be rotated to get the other.

Solution

Problem 2

Simplify $\left(\frac{-1+i\sqrt{3}}{2}\right)^{6}+\left(\frac{-1-i\sqrt{3}}{2}\right)^{6}$ to the form $a+bi$

Solution

Problem 3

Evaluate $\sum_{n=1}^{\infty} \frac{1}{n^2+2n}$

Solution

Problem 4

Five positive integers from 1 to 15 are chosen without replacement. What is the probability that their sum is divisible by 3?

Solution

Problem 5

Find all 3-digit numbers which are the sums of the cubes of their digits

Solution

Problem 6

6 people each have a hat. If they shuffle their hats and redistribute them, what is the probability that exactly one person gets their own hat back?

Solution

Problem 7

Assume that $a,b,c,d$ are positive intergers and $\frac{a}{c}=\frac{b}{d} = \frac{3}{4}, \sqrt{a^2+b^2}-\sqrt{b^2+d^2} = 15$. Find $ac+bd-ad-bc$.

Solution

Problem 8

How many non-isomorphic graphs with 9 vertices, with each vertex connected to exactly 6 other vertices, are there? (Two graphs are isomorphic if one can relabel the vertices of one graph to make all edges be exactly the same.)

Solution

Problem 9

The Cincinnati Reals are playing the Houston Alphas in the last game of the Swirled Series. The Alphas are leading by 1 run in the bottom of the 9th (last) inning, and the Reals are at bat. Each batter has a 1/3 chance of hitting a single and a 2/3 chance of making an out. If the Reals hit 5 or more singles before they make 3 outs, they will win. If the Reals hit exactly 4 singles before making 3 outs, they will tie the game and send it into extra innings, and they will have a 3/5 chance of eventually winning the game (since they have the added momentum of coming from behind). If the Reals hit fewer than 4 singles, they will LOSE! What is the probability that the Alphas hold off the Reals and win, sending the packed Alphadome into a frenzy? Express the answer as a fraction.

Solution

Problem 10

I call two people A and B and think of a natural number $n$. Then I give the number $n$ to A and the number $n + 1$ to B. I tell them that they have both been given natural numbers, and further that they are consecutive natural numbers. However, I don’t tell A what B’s number is and vice versa. I start by asking A if he knows B’s number. He says “no”. Then I ask B if he knows A’s number, and he says “no” too. I go back to A and ask, and so on. A and B can both hear each other’s responses. Do I ever get a “yes” in response? If so, who responds first with “yes” and how many times does he say “no” before this? Assume that both A and B are very intelligent and logical. You may need to consider multiple cases.

Solution

See Also

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