Difference between revisions of "2020 CIME I Problems"
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==Problem 8== | ==Problem 8== | ||
A person has been declared the first to inhabit a certain planet on day </math>N=0<math>. For each positive integer </math>N>0<math>, if there is a positive number of people on the planet, then either one of the following three occurs, each with probability </math>\frac{1}{3}<math>: | A person has been declared the first to inhabit a certain planet on day </math>N=0<math>. For each positive integer </math>N>0<math>, if there is a positive number of people on the planet, then either one of the following three occurs, each with probability </math>\frac{1}{3}<math>: | ||
+ | |||
:(i) the population stays the same; | :(i) the population stays the same; | ||
:(ii) the population increases by </math>2^N<math>; or | :(ii) the population increases by </math>2^N<math>; or | ||
:(iii) the population decreases by </math>2^{N-1}<math>. (If there are no greater than </math>2^{N-1}<math> people on the planet, the population drops to zero, and the process terminates). | :(iii) the population decreases by </math>2^{N-1}<math>. (If there are no greater than </math>2^{N-1}<math> people on the planet, the population drops to zero, and the process terminates). | ||
+ | |||
The probability that at some point there are exactly </math>2^{20}+2^{19}+2^{10}+2^9+1<math> people on the planet can be written as </math>\frac{p}{3^q}<math>, where </math>p<math> and </math>q<math> are positive integers such that </math>p<math> is not divisible by </math>3<math>. Find the remainder when </math>p+q<math> is divided by </math>1000$. | The probability that at some point there are exactly </math>2^{20}+2^{19}+2^{10}+2^9+1<math> people on the planet can be written as </math>\frac{p}{3^q}<math>, where </math>p<math> and </math>q<math> are positive integers such that </math>p<math> is not divisible by </math>3<math>. Find the remainder when </math>p+q<math> is divided by </math>1000$. |
Revision as of 14:05, 30 August 2020
2020 CIME I (Answer Key) | AoPS Contest Collections | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Problem 1
A knight begins on the point in the coordinate plane. From any point
the knight moves to either
or
. Find the number of ways the knight can reach
.
Problem 2
At the local Blast Store, there are sufficiently many items with a price of for each nonnegative integer
. A sales tax of
is applied on all items. If the total cost of a purchase, after tax, is an integer number of cents, find the minimum possible number of items in the purchase.
Problem 3
In a math competition, all teams must consist of between and
members,
inclusive. Mr. Beluhov has
students and he realizes that he cannot form
teams so that each of his students is on exactly one team. Find the sum of all
possible values of
.
Problem 4
There exists a unique positive real number satisfying
Given that
can be written in the form
for integers
with
, find
.
Problem 5
Let be a rectangle with sides
and let
be the reflection of
over
. If
and the area of
is
, find the area of
.
Problem 6
Find the number of complex numbers satisfying
and
+z^{350}+1=0$.
[[2020 CIME I Problems/Problem 6 | Solution]]
==Problem 7==
For every positive integer$ (Error compiling LaTeX. Unknown error_msg)nf(1)+f(2)+\cdots+f(2020)
\frac{p}{q}
p
q
p
1000$.
[[2020 CIME I Problems/Problem 7 | Solution]]
==Problem 8==
A person has been declared the first to inhabit a certain planet on day$ (Error compiling LaTeX. Unknown error_msg)N=0N>0
\frac{1}{3}$:
- (i) the population stays the same;
- (ii) the population increases by$ (Error compiling LaTeX. Unknown error_msg)2^N
2^{N-1}
2^{N-1}$people on the planet, the population drops to zero, and the process terminates).
The probability that at some point there are exactly$ (Error compiling LaTeX. Unknown error_msg)2^{20}+2^{19}+2^{10}+2^9+1\frac{p}{3^q}
p
q
p
3
p+q
1000$.