Difference between revisions of "G285 2021 MC10A"
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==Problem 7== | ==Problem 7== | ||
− | A regular tetrahedron has length <math>4</math>. Suppose on the center of each surface, a hemisphere of diameter <math>2</math> is constructed such that the hemisphere falls inside the volume of the figure. If the ratio between the radius of the largest sphere that can be inscribed inside the old tetrahedron and new tetrahedron <math>\frac{m\sqrt{n}}{r\sqrt{n}-e}</math>, where <math>n</math> is square free and <math>m,r | + | A regular tetrahedron has length <math>4</math>. Suppose on the center of each surface, a hemisphere of diameter <math>2</math> is constructed such that the hemisphere falls inside the volume of the figure. If the ratio between the radius of the largest sphere that can be inscribed inside the old tetrahedron and new tetrahedron <math>\frac{m\sqrt{n}}{r\sqrt{n}-e}</math>, where <math>n</math> is square free, and <math>\gcd(m,e,r) = 1</math>. Find <math>m+n+r+e</math>. |
<math>\textbf{(A)}\ 19\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 22\qquad\textbf{(D)}\ 23\qquad\textbf{(E)}\ 25</math> | <math>\textbf{(A)}\ 19\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 22\qquad\textbf{(D)}\ 23\qquad\textbf{(E)}\ 25</math> |
Revision as of 11:24, 12 May 2021
Problem 1
What is the smallest value of that minimizes
?
Problem 2
Suppose the set denotes
. Then, a subset of length
is chosen. All even digits in the subset
are then are put into group
, and the odd digits are put in
. Then, one number is selected at random from either
or
with equal chances. What is the probability that the number selected is a perfect square, given
?
Problem 3
Let be a unit square. If points
and
are chosen on
and
respectively such that the area of
. What is
?
Problem 4
What is the smallest value of for which
Problem 5
Let a recursive sequence be denoted by such that
and
. Suppose
for
. Let an infinite arithmetic sequence
be such that
. If
is prime, for what value of
will
?
Problem 6
Find
Problem 7
A regular tetrahedron has length . Suppose on the center of each surface, a hemisphere of diameter
is constructed such that the hemisphere falls inside the volume of the figure. If the ratio between the radius of the largest sphere that can be inscribed inside the old tetrahedron and new tetrahedron
, where
is square free, and
. Find
.