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 $n \neq 0$ .

Solution

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

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?

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

Problem 7

Let the following infinite summation be shown: $\left \lfloor \cdots \log_2 \sum_{k=2}^\infty \left \lfloor k+ \log_2 \sum_{j=2}^\infty {\left \lfloor {j+\log_{2} \sum_{i=2}^\infty \left \lfloor {\frac{\log i}{\log i+1}} \right \rfloor} \right \rfloor} \right \rfloor \right \rfloor \cdots$ 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 $>4$ ?