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Difference between revisions of "G285 2021 MC10A"

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==Problem 12==
 
==Problem 12==
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Find the number of 3-digit positive integers <math>abc</math> such that <math>\phi(a)+\phi(b) \equiv 1 \pmod{3}</math> and <math>\phi(b)+\phi(c) \equiv 0 \pmod{4}</math>
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Let <math>\phi(n)</math> denote the number of integers less than <math>n</math> such that each is relatively prime to <math>n</math>. Find the number of 3-digit positive integers <math>abc</math> such that <math>\phi(a)+\phi(b) \equiv 1 \pmod{3}</math> and <math>\phi(b)+\phi(c) \equiv 0 \pmod{4}</math>
  
 
[[G285 MC10A Problems/Problem 12|Solution]]
 
[[G285 MC10A Problems/Problem 12|Solution]]
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[[G285 MC10A Problems/Problem 13|Solution]]
 
[[G285 MC10A Problems/Problem 13|Solution]]
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==Problem 14==
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Let an ellipsoid centered at the origin have radii <math>\{x,y,z \} = \{3,18,24 \}</math>. If a cross-section of the figure is taken at an angle of <math>30^o</math> to the horizontal base that lies along the <math>x</math> and <math>y</math> axes, find the area of the cross-section.
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[[G285 MC10A Problems/Problem 14|Solution]]

Revision as of 10:12, 20 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 $a+b+c+d$.

Solution

Problem 12

Let $\phi(n)$ denote the number of integers less than $n$ such that each is relatively prime to $n$. Find the number of 3-digit positive integers $abc$ such that $\phi(a)+\phi(b) \equiv 1 \pmod{3}$ and $\phi(b)+\phi(c) \equiv 0 \pmod{4}$

Solution

Problem 13

Let a recursive sequence $a_1=1$ and $a_2=13$ be defined as: \[a_n = \frac{2a_{n-1}+3a_{n-2}}{6n}\] for $n \ge 2$. Let $Q(x)$ be a monic polynomial with real roots $\{r_1,r_2,r_3,r_4 \}$. If each root is the reciprocal of the $4$ smallest $a_n$ such that $Q(a_n)>0$, find the reciprocal of the smallest possible value of $\left \lceil Q(-1) \right \rceil$

Solution

Problem 14

Let an ellipsoid centered at the origin have radii $\{x,y,z \} = \{3,18,24 \}$. If a cross-section of the figure is taken at an angle of $30^o$ to the horizontal base that lies along the $x$ and $y$ axes, find the area of the cross-section.

Solution

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