Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "G285 2021 MC10B"

m
m
Line 36: Line 36:
  
 
==Problem 6==
 
==Problem 6==
−
Let a plane parallel to the horizontal slice a sphere with radius <math>r</math> at a random location to create a cross section, and two partial spheres. For what minimum radius <math>r</math> will the area of the cross-section never be able to exceed the sum of the outer surface areas of the partial spheres?
+
Let <math>k</math> planes parallel to the horizontal slice a sphere with radius <math>r</math> at not necessarily distinct random location 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)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</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>

Revision as of 11:21, 15 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 $n \neq 0$.

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 location 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?

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=G285_2021_MC10B&oldid=153762"