2020 CIME I Problems/Problem 13
Problem 13
Chris writes on a piece of paper the positive integers from to
in that order. Then, he randomly writes either
or
between every two adjacent numbers, each with equal probability. The expected value of the expression he writes can be expressed as
for relatively prime positive integers
and
. Find the remainder when
is divided by
.
First we thought we should make two variables for two different types of games, (away games)
(home games). We knew
We also knew that
, which means
. So we replaced
in our first equation with
, so now it is:
. Solving this we get:
solving this further, we get
. Solving this we get
and going back to
we replace
with
and because
.
Solution
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See also
2020 CIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All CIME Problems and Solutions |
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