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Difference between revisions of "2000 SMT/Advanced Topics Problems"

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==Problem 1==
 
==Problem 1==
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How many different ways are there to paint the sides of a tetrahedron with exactly 4 colours? Each side
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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.
 
gets its own colour, and two colourings are the same if one can be rotated to get the other.
  

Latest revision as of 08:43, 22 July 2020

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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