G285 2021 MC10A

Problem 1

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

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


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 $n=4$?

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


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 $EF^2$?

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


Problem 4

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

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


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 $k_{2021} = k-2022p+1$?

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


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 $y,z,k,l$


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 $m+n+r+e$.

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


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 $y=\sin 20^o$.

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


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?

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


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 $EI$.

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


Problem 11


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 $\phi(b)+\phi(c) \equiv 0 \pmod{4}$


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 $\left \lceil Q(-1) \right \rceil$


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.


Problem 15

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

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


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 \[\sum a_2\]


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 $d+n+a+p$

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


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 \[\sum_{i=k}^{n} \binom{n-i}{i} v_p(i) > (1+k)^n\]


Problem 25

Invalid username
Login to AoPS