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Difference between revisions of "G285 2021 MC10A"

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==Problem 7==
 
==Problem 7==
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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>.
+
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 10:24, 12 May 2021

Problem 1

What is the smallest value of $x$ that minimizes $|||2^{|x^2|} - 4|-4|-8|$?

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

Solution

Problem 2

Suppose the set $S$ denotes $S = \{1,2,3 \cdots n\}$. Then, a subset of length $1<k<n$ is chosen. All even digits in the subset $k$ are then are put into group $k_1$, and the odd digits are put in $k_2$. Then, one number is selected at random from either $k_1$ or $k_2$ with equal chances. What is the probability that the number selected is a perfect square, given $n=4$?

$\textbf{(A)}\ \frac{1}{2}\qquad\textbf{(B)}\ \frac{3}{11}\qquad\textbf{(C)}\ \frac{6}{11}\qquad\textbf{(D)}\ \frac{7}{13}\qquad\textbf{(E)}\ \frac{3}{5}$

Solution

Problem 3

Let $ABCD$ be a unit square. If points $E$ and $F$ are chosen on $AB$ and $CD$ respectively such that the area of $\triangle AEF = \frac{3}{2} \triangle CFE$. What is $EF^2$?

$\textbf{(A)}\ \frac{13}{9}\qquad\textbf{(B)}\ \frac{8}{9}\qquad\textbf{(C)}\ \frac{37}{36}\qquad\textbf{(D)}\ \frac{5}{4}\qquad\textbf{(E)}\ \frac{13}{36}$

Solution

Problem 4

What is the smallest value of $k$ for which \[2^{18k} \equiv 76 \mod 100\]

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 20$

Solution

Problem 5

Let a recursive sequence be denoted by $a_n$ such that $a_0 = 1$ and $a_1 = k$. Suppose $a_{n-1} = n+a_n$ for $n>1$. Let an infinite arithmetic sequence $P$ be such that $P=\{k+1, k-p+1, k-2p+1 \cdots\}$. If $k$ is prime, for what value of $p$ will $k_{2021} = k-2022p+1$?

$\textbf{(A)}\ 1011\qquad\textbf{(B)}\ \frac{1011}{2}\qquad\textbf{(C)}\ 2021\qquad\textbf{(D)}\ \frac{2021}{2}\qquad\textbf{(E)}\ 4042$

Solution

Problem 6

Find \[\sum_{j=1}^{50} s^3 \sum_{h=3}^{10} {4h+5}\]

$\textbf{(A)}\ 323400\qquad\textbf{(B)}\ 336600\qquad\textbf{(C)}\ 673200\qquad\textbf{(D)}\ 646800\qquad\textbf{(E)}\ 2124150$

Solution

Problem 7

A regular tetrahedron has length $4$. Suppose on the center of each surface, a hemisphere of diameter $2$ 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 $\frac{m\sqrt{n}}{r\sqrt{n}-e}$, where $n$ is square free, and $gcd(m,e,r) = 1$. Find $m+n+r+e$.

$\textbf{(A)}\ 19\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 22\qquad\textbf{(D)}\ 23\qquad\textbf{(E)}\ 25$

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