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Difference between revisions of "G285 2021 MC10A"

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==Problem 6==
 
==Problem 6==
−
Find <cmath>\sum_{j=1}^{50} s^3 \sum_{h=3}^{10} {4h+5}</cmath>
+
If <cmath>k=wv_1+xv_2</cmath><cmath>l=yv_1^2+zv_2^2</cmath><cmath>y=2w , z=2x</cmath> Find <math>v_2</math> in terms of <math>y,z,k,l</math>
−
 
 
−
<math>\textbf{(A)}\ 323400\qquad\textbf{(B)}\ 336600\qquad\textbf{(C)}\ 673200\qquad\textbf{(D)}\ 646800\qquad\textbf{(E)}\ 2124150</math>
 
  
 
[[G285 MC10A Problems/Problem 6|Solution]]
 
[[G285 MC10A Problems/Problem 6|Solution]]
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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>
 +
 +
[[G285 MC10A Problems/Problem 9|Solution]]
 +
 +
==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]]
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 +
==Problem 11==
 +
 +
 +
[[G285 MC10A Problems/Problem 11|Solution]]
 +
 +
==Problem 12==
 +
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]]
 +
 +
==Problem 13==
 +
Let a recursive sequence <math>a_1=1</math> and <math>a_2=13</math> be defined as: <cmath>a_n = \frac{2a_{n-1}+3a_{n-2}}{6n}</cmath> for <math>n \ge 2</math>. Let <math>Q(x)</math> be a monic polynomial with real roots <math>\{r_1,r_2,r_3,r_4 \}</math>. If each root is the reciprocal of the <math>4</math> smallest <math>a_n</math> such that <math>Q(a_n)>0</math>, find the reciprocal of the smallest possible value of <math>\left \lceil Q(-1) \right \rceil</math>
 +
 +
[[G285 MC10A Problems/Problem 13|Solution]]
 +
 +
==Problem 14==
 +
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.
 +
 +
[[G285 MC10A Problems/Problem 14|Solution]]
 +
 +
==Problem 15==
 +
Find <cmath>\sum_{j=1}^{50} s^3 \sum_{h=3}^{10} {4h+5}</cmath>
 +
 +
<math>\textbf{(A)}\ 323400\qquad\textbf{(B)}\ 336600\qquad\textbf{(C)}\ 673200\qquad\textbf{(D)}\ 646800\qquad\textbf{(E)}\ 2124150</math>
 +
 +
[[G285 MC10A Problems/Problem 15|Solution]]
 +
 +
==Problem 16==
 +
Suppose <math>a_n</math> is a recursive sequence with <math>a_1=29</math> and <math>a_{n+1}=n \cdot a_{n-1}</math> for <math>n>1</math>. If <math>\frac{a_{5}}{a_{k}}=\frac{5}{3}</math> for <math>2<k<7</math>, find <cmath>\sum a_2</cmath>
 +
 +
[[G285 MC10A Problems/Problem 16|Solution]]
 +
 +
==Problem 23==
 +
Let regular hexagon <math>ABCDEF</math> of side length <math>4</math> be centered at <math>(-2,2- 2\sqrt{3})</math> on the Cartesian Plane, where points <math>P</math>,<math>X</math>, and <math>Y</math> lie in its interior. Let the ratio of the area of <math>\triangle AEP</math> to <math>\triangle BDP</math> is <math>-2+ \sqrt{3}:6- \sqrt{3}</math>, the ratio of the area of <math>AXB</math> to <math>EXD</math> is <math>7:1</math>, <math>PY \parallel AE</math> and <math>PX=PY</math>. Now, suppose <math>\triangle PXY</math> can be rotated about point <math>B</math> <math>m</math> degrees counterclockwise to form a new triangle <math>P'X'Y'</math>, such that if the coordinates of <math>P' = (x_1,y_1)</math>, <math>X' = (x_2,y_2)</math> and <math>Y' = (x_3,y_3)</math>, <cmath>\sum_{i=0}^{3} x_i = - \frac{3 \sqrt{147}}{5} \sum_{j=0}^{3} y_j</cmath><cmath>x_3-x_2=x_2-x_1= -\frac{\sqrt{3}}{2} y_3-y_2 = -\frac{\sqrt{3}}{2} y_1-y_2</cmath> If <math>m+x_1+x_2+x_3+y_1+y_2+y_3</math> can be represented as <math>d-\frac{n \sqrt{a}}{p}</math>, find <math>d+n+a+p</math>
 +
 +
<math>\textbf{(A)}\ 64\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ 74\qquad\textbf{(D)}\ 81\qquad\textbf{(E)}\ 84</math>
 +
 +
[[G285 MC10A Problems/Problem 23|Solution]]
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 +
==Problem 24==
 +
Let <math>v_p (a)</math> denote the number of <math>p</math>'s in the prime factorization of <math>a</math>. If <math>k</math> and <math>n</math> are positive integers such that <math>\sqrt{n} < k < n</math>, find the largest sum <math>k+n</math> such that <cmath>\sum_{i=k}^{n} \binom{n-i}{i} v_p(i) > (1+k)^n</cmath>
 +
 +
[[G285 MC10A Problems/Problem 24|Solution]]
 +
 +
==Problem 25==

Latest revision as of 11:55, 23 June 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

If \[k=wv_1+xv_2\]\[l=yv_1^2+zv_2^2\]\[y=2w , z=2x\] Find $v_2$ in terms of $y,z,k,l$

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

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

Problem 15

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 16

Suppose $a_n$ is a recursive sequence with $a_1=29$ and $a_{n+1}=n \cdot a_{n-1}$ for $n>1$. If $\frac{a_{5}}{a_{k}}=\frac{5}{3}$ for $2<k<7$, find \[\sum a_2\]

Solution

Problem 23

Let regular hexagon $ABCDEF$ of side length $4$ be centered at $(-2,2- 2\sqrt{3})$ on the Cartesian Plane, where points $P$,$X$, and $Y$ lie in its interior. Let the ratio of the area of $\triangle AEP$ to $\triangle BDP$ is $-2+ \sqrt{3}:6- \sqrt{3}$, the ratio of the area of $AXB$ to $EXD$ is $7:1$, $PY \parallel AE$ and $PX=PY$. Now, suppose $\triangle PXY$ can be rotated about point $B$ $m$ degrees counterclockwise to form a new triangle $P'X'Y'$, such that if the coordinates of $P' = (x_1,y_1)$, $X' = (x_2,y_2)$ and $Y' = (x_3,y_3)$, \[\sum_{i=0}^{3} x_i = - \frac{3 \sqrt{147}}{5} \sum_{j=0}^{3} y_j\]\[x_3-x_2=x_2-x_1= -\frac{\sqrt{3}}{2} y_3-y_2 = -\frac{\sqrt{3}}{2} y_1-y_2\] If $m+x_1+x_2+x_3+y_1+y_2+y_3$ can be represented as $d-\frac{n \sqrt{a}}{p}$, find $d+n+a+p$

$\textbf{(A)}\ 64\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ 74\qquad\textbf{(D)}\ 81\qquad\textbf{(E)}\ 84$

Solution

Problem 24

Let $v_p (a)$ denote the number of $p$'s in the prime factorization of $a$. If $k$ and $n$ are positive integers such that $\sqrt{n} < k < n$, find the largest sum $k+n$ such that \[\sum_{i=k}^{n} \binom{n-i}{i} v_p(i) > (1+k)^n\]

Solution

Problem 25

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