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Difference between revisions of "G285 2021 MC10A"

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==Problem 9==
 
==Problem 9==
−
Suppose if a real number <math>k</math> is  <math>happy</math> , <math>k^3+5k-3 \ge (k-1)^4</math>. If a real number <math>l</math>  is  <math>unhappy</math> , <math>l^3+5l^2 \ge 2064l</math>. If a number is neither <math>happy</math> or <math>unhappy</math>, it will be <math>neutral</math>. What is the probability that <math>3</math> randomly selected numbers from the interval <math>[1,100]</math> are  <math>happy</math> , <math>unhappy</math>, and <math>neutral</math>, in any given order?
+
If a real number <math>k</math> is  <math>happy</math> , <math>k^3+5k-3 \ge (k-1)^4</math>. If a real number <math>l</math>  is  <math>unhappy</math> , <math>l^3+5l^2 \ge 2064l</math>. If a number is neither <math>happy</math> or <math>unhappy</math>, it will be <math>neutral</math>. What is the probability that <math>3</math> randomly selected numbers from the interval <math>[1,100]</math> are  <math>happy</math> , <math>unhappy</math>, and <math>neutral</math>, in any given order?
  
 
<math>\textbf{(A)}\ \frac{20,007}{5,000,000}\qquad\textbf{(B)}\ \frac{2}{9}\qquad\textbf{(C)}\ \frac{6,669}{1,000,000}\qquad\textbf{(D)}\ \frac{247}{35,937}\qquad\textbf{(E)}\ \frac{494}{11,979}</math>
 
<math>\textbf{(A)}\ \frac{20,007}{5,000,000}\qquad\textbf{(B)}\ \frac{2}{9}\qquad\textbf{(C)}\ \frac{6,669}{1,000,000}\qquad\textbf{(D)}\ \frac{247}{35,937}\qquad\textbf{(E)}\ \frac{494}{11,979}</math>
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==Problem 10==
 
==Problem 10==
 +
Suppose the area of <math>\triangle ABC</math> is equal to the sum of its side lengths. Let point <math>D</math> be on the circumcircle of <math>\triangle ABC</math> such that <math>AD</math> is a diameter. If <math>E</math> is the center of the circumcircle, and <math>I</math> is the center of the incircle of <math>\triangle ABC</math>, and <math>CI=4</math>, find <math>EI</math>.
 +
 +
<math>\textbf{(A)}\ 0\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{3}{4}\qquad\textbf{(D)}\ \frac{5}{2}\qquad\textbf{(E)}\ 2</math>
 +
 +
[[G285 MC10A Problems/Problem 10|Solution]]
 +
 +
==Problem 11==
 +
If <math>abcd_{11}</math> is a palindrome in base <math>7</math>, and <math>dcba</math> expressed in base <math>10</math> does not begin with a nonzero digit, find the  difference between the largest and smallest possible sum of <math>a+b+c+d</math>.
 +
 +
<math>\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12</math>
 +
 +
[[G285 MC10A Problems/Problem 11|Solution]]

Revision as of 23:22, 14 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 $m+n+r+e$.

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

Solution

Problem 8

If $(\cos 20^o + \sin 20^o)^2$ can be expressed as $\frac{\sqrt{t}+u}{vx}+w(y^2)$, where $t$ is square free and $gcd(u,v,w) = 1$, find $t+u+v+w$ if $x=\cos 20^o$ and $y=\sin 20^o$.

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12$

Solution

Problem 9

If a real number $k$ is $happy$ , $k^3+5k-3 \ge (k-1)^4$. If a real number $l$ is $unhappy$ , $l^3+5l^2 \ge 2064l$. If a number is neither $happy$ or $unhappy$, it will be $neutral$. What is the probability that $3$ randomly selected numbers from the interval $[1,100]$ are $happy$ , $unhappy$, and $neutral$, in any given order?

$\textbf{(A)}\ \frac{20,007}{5,000,000}\qquad\textbf{(B)}\ \frac{2}{9}\qquad\textbf{(C)}\ \frac{6,669}{1,000,000}\qquad\textbf{(D)}\ \frac{247}{35,937}\qquad\textbf{(E)}\ \frac{494}{11,979}$

Solution

Problem 10

Suppose the area of $\triangle ABC$ is equal to the sum of its side lengths. Let point $D$ be on the circumcircle of $\triangle ABC$ such that $AD$ is a diameter. If $E$ is the center of the circumcircle, and $I$ is the center of the incircle of $\triangle ABC$, and $CI=4$, find $EI$.

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{3}{4}\qquad\textbf{(D)}\ \frac{5}{2}\qquad\textbf{(E)}\ 2$

Solution

Problem 11

If $abcd_{11}$ is a palindrome in base $7$, and $dcba$ expressed in base $10$ does not begin with a nonzero digit, find the difference between the largest and smallest possible sum of $a+b+c+d$.

$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12$

Solution

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