Difference between revisions of "G285 2021 MC10B"
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==Problem 5== | ==Problem 5== | ||
A principal is pushing out an emergency COVID-19 alert to his school of <math>40</math> teachers and <math>500</math> students. Suppose the announcement is first approved by his <math>5</math> aides. Then, each of the aides share the announcement to <math>n</math> teachers and <math>t</math> students, where <math>n,t \in \mathbb{Z}</math> and for every aide <math>n \neq t</math>. Moreover, <math>n+t = (u+1)^2</math>, where <math>u</math> is the round number ( for the aides releasing info it is round 1, then round 2....) After every round <math>u</math>, some <math>k</math> teachers in the previous round share the announcement to a new group of <math>n</math> teachers and <math>t</math> students, where <math>k=(u+1)^2</math>. How many rounds will it take until the entire school is informed? Assume that after all teachers are informed, <math>n=0</math>, but <math>t</math> still grows as if <math>n \neq 0</math>. | A principal is pushing out an emergency COVID-19 alert to his school of <math>40</math> teachers and <math>500</math> students. Suppose the announcement is first approved by his <math>5</math> aides. Then, each of the aides share the announcement to <math>n</math> teachers and <math>t</math> students, where <math>n,t \in \mathbb{Z}</math> and for every aide <math>n \neq t</math>. Moreover, <math>n+t = (u+1)^2</math>, where <math>u</math> is the round number ( for the aides releasing info it is round 1, then round 2....) After every round <math>u</math>, some <math>k</math> teachers in the previous round share the announcement to a new group of <math>n</math> teachers and <math>t</math> students, where <math>k=(u+1)^2</math>. How many rounds will it take until the entire school is informed? Assume that after all teachers are informed, <math>n=0</math>, but <math>t</math> still grows as if <math>n \neq 0</math>. | ||
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+ | <math>\textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7</math> | ||
[[G285 MC10B Problems/Problem 5|Solution]] | [[G285 MC10B Problems/Problem 5|Solution]] | ||
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==Problem 6== | ==Problem 6== | ||
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<math>\textbf{(A)}\ {1,2}\qquad\textbf{(B)}\ {1,2,3}\qquad\textbf{(C)}\ {1,2,3,4}\qquad\textbf{(D)}\ {2,3}\qquad\textbf{(E)}\ {2,3,4}</math> | <math>\textbf{(A)}\ {1,2}\qquad\textbf{(B)}\ {1,2,3}\qquad\textbf{(C)}\ {1,2,3,4}\qquad\textbf{(D)}\ {2,3}\qquad\textbf{(E)}\ {2,3,4}</math> | ||
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+ | [[G285 MC10B Problems/Problem 6|Solution]] | ||
==Problem 7== | ==Problem 7== | ||
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<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{never bigger than 4}</math> | <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{never bigger than 4}</math> | ||
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+ | [[G285 MC10B Problems/Problem 7|Solution]] | ||
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+ | ==Problem 8== | ||
+ | Find the sum <math>S</math> of all real values <math>x</math> if: | ||
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+ | <cmath>y^{\frac{1}{3} \cdot 3^x -1} = \sqrt{3}</cmath> | ||
+ | where <math>y=\log{3}</math> | ||
+ | |||
+ | [[G285 MC10B Problems/Problem 8|Solution]] |
Revision as of 20:01, 19 May 2021
Contents
Problem 1
Find
Problem 2
If , and
, and
, what is
?
Problem 3
A convex hexagon of length is inscribed in a circle of radius
, where
. If
, and
, find the area of the hexagon.
Problem 4
Find the smallest such that:
Problem 5
A principal is pushing out an emergency COVID-19 alert to his school of teachers and
students. Suppose the announcement is first approved by his
aides. Then, each of the aides share the announcement to
teachers and
students, where
and for every aide
. Moreover,
, where
is the round number ( for the aides releasing info it is round 1, then round 2....) After every round
, some
teachers in the previous round share the announcement to a new group of
teachers and
students, where
. How many rounds will it take until the entire school is informed? Assume that after all teachers are informed,
, but
still grows as if
.
Problem 6
Let planes parallel to the horizontal slice a sphere with radius
at not necessarily distinct random locations to create
cross sections, and
partial spheres. What range of values for
will the cumulative area of the cross-sections never be able to exceed the sum of the outer surface areas of the partial spheres?
Problem 7
Let the following infinite summation be shown:
Suppose each individual sum is denoted by a constant
, where
is the inner most sum, and
evaluates sums going outward. For what minimum value of
will the expression be
?
Problem 8
Find the sum of all real values
if:
where