G285 2021 MC10B

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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