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Difference between revisions of "G285 2021 MC10B"

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Find the smallest <math>n</math> such that: <cmath>n \equiv 3 \pmod{9}</cmath><cmath>2n \equiv 7 \pmod{13}</cmath><cmath>5n \equiv 14 \pmod{17}</cmath>
 
Find the smallest <math>n</math> such that: <cmath>n \equiv 3 \pmod{9}</cmath><cmath>2n \equiv 7 \pmod{13}</cmath><cmath>5n \equiv 14 \pmod{17}</cmath>
  
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<math>\textbf{(A)}\ 1560\qquad\textbf{(B)}\ 1713\qquad\textbf{(C)}\ 2211\qquad\textbf{(D)}\ 3273\qquad\textbf{(E)}\ 3702</math>
  
 
[[G285 MC10B Problems/Problem 4|Solution]]
 
[[G285 MC10B Problems/Problem 4|Solution]]
  
 
==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>.
 +
 +
<math>\textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7</math>
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 +
[[G285 MC10B Problems/Problem 5|Solution]]
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 +
==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]]
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 +
==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>
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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:
 +
 +
<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 13:01, 28 May 2021

Problem 1

Find $\left \lceil {\frac{3!+4!+5!+6!}{2+3+4+5+6}} \right \rceil$

$\textbf{(A)}\ 42\qquad\textbf{(B)}\ 43\qquad\textbf{(C)}\ 44\qquad\textbf{(D)}\ 45\qquad\textbf{(E)}\ 46$

Solution

Problem 2

If $deg(Q(x))=3$, and $deg(K(x))=2$, and $Q(x)=(x-2)K(x)$, what is $deg(Q(x)-2K(x))$?

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$

Solution

Problem 3

A convex hexagon of length $s$ is inscribed in a circle of radius $r$, where $r \neq s$. If $\frac{s}{2r}=\frac{21}{29}$, and $rs=58$, find the area of the hexagon.

$\textbf{(A)}\ 42\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 84\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 120$

Solution

Problem 4

Find the smallest $n$ such that: \[n \equiv 3 \pmod{9}\]\[2n \equiv 7 \pmod{13}\]\[5n \equiv 14 \pmod{17}\]

$\textbf{(A)}\ 1560\qquad\textbf{(B)}\ 1713\qquad\textbf{(C)}\ 2211\qquad\textbf{(D)}\ 3273\qquad\textbf{(E)}\ 3702$

Solution

Problem 5

A principal is pushing out an emergency COVID-19 alert to his school of $40$ teachers and $500$ students. Suppose the announcement is first approved by his $5$ aides. Then, each of the aides share the announcement to $n$ teachers and $t$ students, where $n,t \in \mathbb{Z}$ and for every aide $n \neq t$. Moreover, $n+t = (u+1)^2$, where $u$ is the round number ( for the aides releasing info it is round 1, then round 2....) After every round $u$, some $k$ teachers in the previous round share the announcement to a new group of $n$ teachers and $t$ students, where $k=(u+1)^2$. How many rounds will it take until the entire school is informed? Assume that after all teachers are informed, $n=0$, but $t$ still grows as if $n \neq 0$.

$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$

Solution

Problem 6

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

$\textbf{(A)}\ {1,2}\qquad\textbf{(B)}\ {1,2,3}\qquad\textbf{(C)}\ {1,2,3,4}\qquad\textbf{(D)}\ {2,3}\qquad\textbf{(E)}\ {2,3,4}$

Solution

Problem 7

Let the following infinite summation be shown: \[\left \lfloor \cdots \sum_{k=2}^{11} \left \lfloor -k+ \sum_{j=2}^{10} {\left \lfloor {-j+\sum_{i=2}^\infty \left \lfloor {\frac{10i^2+11i-2}{i^3}} \right \rfloor} \right \rfloor} \right \rfloor \right \rfloor \cdots\] Suppose each individual sum is denoted by a constant $\mu$, where $\mu=1$ is the inner most sum, and $\mu>1$ evaluates sums going outward. For what minimum value of $\mu$ will the expression be $>100$?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{it is never bigger than 100}$

Solution

Problem 8

Find the sum $S$ of all real values $x$ if:

\[y^{\frac{1}{3} \cdot 3^x -1} = \sqrt{3}\] where $y=\log{3}$

Solution

Problem 9

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

$\textbf{(A)}\ 740\qquad\textbf{(B)}\ 820\qquad\textbf{(C)}\ 900\qquad\textbf{(D)}\ 940\qquad\textbf{(E)}\ 1000$

Solution

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