# 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>. | ||

+ | |||

+ | <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]] | ||

− | <math>\textbf{(A)}\ 3\qquad\textbf{(B)}\ | + | ==Problem 6== |

+ | Let <math>k</math> planes parallel to the horizontal slice a sphere with radius <math>r</math> at not necessarily distinct random locations to create <math>k</math> cross sections, and <math>k+2</math> partial spheres. What range of values for <math>k</math> will the cumulative area of the cross-sections never be able to exceed the sum of the outer surface areas of the partial spheres? | ||

+ | |||

+ | <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> | ||

+ | |||

+ | [[G285 MC10B Problems/Problem 6|Solution]] | ||

+ | |||

+ | ==Problem 7== | ||

+ | Let the following infinite summation be shown: <cmath>\left \lfloor \cdots \sum_{k=2}^{11} \left \lfloor -k+ \sum_{j=2}^{10} {\left \lfloor {-j+\sum_{i=2}^\infty \left \lfloor {\frac{10i^2+11i-2}{i^3}} \right \rfloor} \right \rfloor} \right \rfloor \right \rfloor \cdots</cmath> | ||

+ | Suppose each individual sum is denoted by a constant <math>\mu</math>, where <math>\mu=1</math> is the inner most sum, and <math>\mu>1</math> evaluates sums going outward. For what minimum value of <math>\mu</math> will the expression be <math>>100</math>? | ||

+ | |||

+ | <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{it is never bigger than 100}</math> | ||

+ | |||

+ | [[G285 MC10B Problems/Problem 7|Solution]] | ||

+ | |||

+ | ==Problem 8== | ||

+ | Find the sum <math>S</math> of all real values <math>x</math> if: | ||

+ | |||

+ | <cmath>y^{\frac{1}{3} \cdot 3^x -1} = \sqrt{3}</cmath> | ||

+ | where <math>y=\log{3}</math> | ||

+ | |||

+ | [[G285 MC10B Problems/Problem 8|Solution]] | ||

+ | |||

+ | ==Problem 9== | ||

+ | Call a 3-digit positive integer <math>palindromic</math> if it can be represented as the difference of two distinct palindromes, and the number itself is NOT a palindrome. Find the number of <math>palindromic</math> <math>numbers</math> | ||

− | + | <math>\textbf{(A)}\ 740\qquad\textbf{(B)}\ 820\qquad\textbf{(C)}\ 900\qquad\textbf{(D)}\ 940\qquad\textbf{(E)}\ 1000</math> | |

− | |||

− | + | [[G285 MC10B Problems/Problem 9|Solution]] |

## Latest revision as of 14:01, 28 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

## Problem 9

Call a 3-digit positive integer if it can be represented as the difference of two distinct palindromes, and the number itself is NOT a palindrome. Find the number of