2020 FMC 12A Problems

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Here are the problems that were on the 2020 FMC 12A.

Problem 1

The price of a bottle of soda in Fidgetville has increased from $1.5$ dollars to $d$ dollars from $2019$ to $2020$ due to inflation. A standard pack of soda is defined as a collection of $15$ identical bottles. Given that Amy spent $3$ more dollars to buy a standard pack of soda in $2020$ than in $2019,$ what is the cost per bottle, in dollars, of soda in $2020$?

$\textbf{(A)}\ 1.6 \qquad\textbf{(B)}\ 1.65 \qquad\textbf{(C)}\ 1.7 \qquad\textbf{(D)}\ 1.75 \qquad\textbf{(E)}\ 1.8$

Solution

Problem 2

How many distinct real solutions are there to the equation\[x^4 + 5x^2 + 6 = 0?\]$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4 \qquad$

Solution

Problem 3

Call a repeating decimal standard if the repetend is exactly one digit long. For example, the repeating decimals $0.6\overline{5}$ and $0.\overline{3}$ are standard, but the repeating decimal $0.4\overline{20}$ is not. Find the sum of all positive integers $n$ such that $\frac{1}{n!}$ can be expressed as a standard repeating decimal that can not be expressed as a terminating decimal.


$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 21$

Solution

Problem 4

Adam, Brooks, Caleb, and Daniel are playing a game of Zheng Shang You, and they play for a total of four rounds. Assuming that for each round, each person has equal probability of winning, the probability that each person wins one round can be expressed as $\frac{m}{n},$ for relatively prime positive integers $m$ and $n$. Find $m+n.$

$\textbf{(A)}\ 19 \qquad\textbf{(B)}\ 35\qquad\textbf{(C)}\ 67\qquad\textbf{(D)}\ 73\qquad\textbf{(E)}\ 337$

Solution

Problem 5

The numbers $108,3,2,72,8,27$ are placed in the following $6$ tokens that are arranged in a hexagonal order. The pairwise products of all opposite tokens are all equal. Some numbers are already placed on the tokens, what is the absolute difference between the possible numbers placed on the shaded token?

[asy]  size(3cm); draw(circle((4,0),2)); draw(circle((12,0),2)); draw(circle((0,4sqrt(3)),2)); draw(circle((16,4sqrt(3)),2)); draw(circle((4,8sqrt(3)),2)); draw(circle((12,8sqrt(3)),2)); filldraw(circle((4,8sqrt(3)),2), gray); label("2",(0,4sqrt(3))); label("3", (12,8sqrt(3)));  [/asy]

$\textbf{(A) } 19 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 35 \qquad\textbf{(D) } 64 \qquad\textbf{(E) } 106$


Solution

Problem 6

How many positive integers $1 \leq n \leq 768$ are there such that $\frac{n}{768}$ can be expressed as\[0.\underline{a} \text{ } \underline{b} \text{ } \underline{c} \text{ } \underline{d} \text{ } \underline{e},\]where $a, b, c, d, e$ are digits in base $10$ and $e$ is nonzero?

$\textbf{(A)}\ 8 \qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 48$

Solution

Problem 7

Let $a$ and $b$ be positive integers such that\[a^2 + b^2 + (a + b)^2 + 10ab = 274.\]Find the positive value of $a - b$.

$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9$

Solution

Problem 8

It is given that an arithmetic sequence $\{ a_n \}$ satisfies that the initial term $a_0 = 0,$ the final term $a_n = 400,$ and that $\sum_{i=0}^{n} a_i \geq 2020.$ Find the minimum possible value of $n.$

$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 11\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 17$

Solution

Problem 9

$u = a + bi$ is a complex number such that \[(2+i)^2 u^2 = 50i.\] What is the positive value of $a+b?$

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8$

Solution

Problem 10

Alpha and Beta each choose numbers $\alpha$ and $\beta$ respectively with $\alpha, \beta \in \{1,2,\dots,9,10\}$ without telling each other. Alpha wins if $|\alpha - \beta| \le 4.$ Beta chooses first. To minimize Alpha's chance of winning, what is the sum of the two numbers Beta could choose?

$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15$

Solution

Problem 11

Let $S$ denote the set of all positive integers $n$ that satisfy $0 \leq n \leq 100$ and $n^2$ in base $10$ is a four digit number that has an odd tens digit. Find the number of elements in $S$.

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 14$

Solution

Problem 12

Consider a series of $50$ consecutive days. Mitsuha and Taki will switch bodies each day with a $\frac{1}{2}$ probability, independently from other days. The expected number of instances when Mitsuha and Taki switch bodies on two consecutive days can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. (For instance, the number of such instances for the series $SSNNNSSSNS$ is $3,$ where $S$ denotes a day with switching bodies, and $N$ a day without switching bodies.)

$\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 27 \qquad\textbf{(D) } 53 \qquad\textbf{(E) } 55$

Solution

Problem 13

Jensen likes tossing his ball into baskets $b = 1,2,3,4,5,6, \dots, 2020$. When Jensen tosses his ball into basket $b \neq 2020$, he is forced to also toss his ball into basket $b+1$. In general the ball has $\tfrac{i-1}{i}$ chance of missing its target for basket $i$. Jensen chooses a random basket from $1$ to $2019$, and the chance he makes both shots is equal to $\tfrac{m}{n}$ for relatively prime positive integers $m$ and $n$. The value of $m+n$ is

$\textbf{(A)}\ 79\qquad\textbf{(B)}\ 2020\qquad\textbf{(C)}\ 2021\qquad\textbf{(D)}\ 2023 \qquad\textbf{(E)}\ 4039$

Solution

Problem 14

In a right triangle $ABC$, with $\angle BAC=90^\circ$, let $BE$ be one of its internal angle bisector, with $E$ on $CA$. Let $D$ be the foot of altitude from $A$ onto $BC$. Let $\odot(BCE)$ meet the line $AB$ at point $F$. Let $P$ be a point on $AB$, such that $BP=BD=\frac{BC}4$. Then, $\frac{AP}{AF}$ equals

$\textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac14 \qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ \text{None of these}$

Solution

Problem 15

How many $12$-digit multiples of $37$ can be written in the form\[\sum_{j=0}^{11} a_j \cdot 10^j\]where $a_j \in \{ 0, 1\}$ for $0 \leq j \leq 11$?

$\textbf{(A)}\ 108 \qquad\textbf{(B)}\ 164 \qquad\textbf{(C)}\ 173 \qquad\textbf{(D)}\ 251 \qquad\textbf{(E)}\ 346$

Solution

Problem 16

Let \[f(x) = (x^2-2\cos(1 ^ { \circ })x+1)\cdot (x^2-2\cos(2 ^ {\circ})x+1) \cdots (x^2-2\cos(179 ^ {\circ})x+1).\]Find $f(1)$.

$\textbf{(A)}\ -360\qquad\textbf{(B)}\ -180\qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 180 \qquad\textbf{(E)}\ 360$

Solution

Problem 17

In rectangle $ABCD$, diagonal $AC$ is drawn. Point $P$ is selected uniformly at random on diagonal $AC$. Suppose that the probability that the circumcenter of triangle $CDP$ lies inside the rectangle is $\frac{1}{3}$. What is the ratio of the longer side of the rectangle to the shorter one?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ \sqrt{2}\qquad\textbf{(C)}\ \sqrt{3} \qquad\textbf{(D)}\ 2\sqrt{2} \qquad\textbf{(E)}\ 2\sqrt{3}$

Solution

Problem 18

Starting from $n=2,$ define a conventional listing $a_1, a_2, ..., a_{n!}$ of permutations of the first $n$ positive integers to be an ordering of all aforementioned permutations such that the sets of conventional listings of permutations of the first $n$ positive integers (which are ordered accordingly) not including $1$ are ordered based on the placement of the $1$ in the permutation. For instance, the conventional listing of permutations of the first $3$ positive integers is\[a_1, a_2, ..., a_{3!} = \underline{1}23, \underline{1}32, 2\underline{1}3, 3\underline{1}2, 23\underline{1}, 32\underline{1}.\]Let $f_k(n)$ denote the least positive integer $i$ in a conventional listing of the first $k$ positive integers such that the permutation $a_i$ has leading number $n.$ Find the largest power of $2$ that divides $f_{2021} (2019) - f_{2020} (2018).$

$\textbf{(A) }2011\qquad\textbf{(B) }2012\qquad\textbf{(C) }2013\qquad\textbf{(D) }2014\qquad\textbf{(E) }2015$

Solution

Problem 19

Find the number of ordered quadruples of positive integers $(w, x, y, z)$ that satisfy\[4w + x + 2(y + z) < 32.\]$\textbf{(A)}\ 1365 \qquad\textbf{(B)}\ 2366 \qquad\textbf{(C)}\ 2380 \qquad\textbf{(D)}\ 3876 \qquad\textbf{(E)}\ 4200$

Solution

Problem 20

Henry's NT2 TA gave him this problem to work on: "Find the exponent of the largest power of $2$ that evenly divides $n!,$" where $n$ is a fixed positive integer. However, Henry is bad at NT, so he just puts $n$ as his answer for the aforementioned problem. Let $M$ be the number of integer values $1 \leq n \leq 2^{2020}$ such that the positive difference between Henry's answer and the actual answer of the problem is exactly $105.$ Find the largest power of $2$ that divides $M$.

$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$

Solution

Problem 21

Let the function $S(n, k)$ denote the least positive integer value of $a$ such that $n^a - 1$ is divisible by $k$. Find the remainder when\[S(1, 257) + S(2, 257) + S(3, 257) + ... + S(256, 257)\]is divided by $1000$.

$\textbf{(A)}\ 690\qquad\textbf{(B)}\ 691\qquad\textbf{(C)}\ 845\qquad\textbf{(D)}\ 846\qquad\textbf{(E)}\ 847$


Solution

Problem 22

Call a sequence of rolls $a_1, a_2, ..., a_n$ generated from a standard fair $6$-sided die fluctuating if the $n-1$ sets of two consecutive rolls, placed in order, alternately fluctuate between having a sum greater than or equal to $7$ and having a sum less than or equal to $7$. For example, the sequences $2, 5, 1, 6, 1$ and $3, 3, 4, 2, 6$ are fluctuating, but the sequence $2, 1, 3, 6, 4$ is not. The probability that Jamie will produce a fluctuating sequence by rolling the said die $20$ times can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find the number of positive factors that $n$ contains.

$\textbf{(A) } 361 \qquad\textbf{(B) } 380 \qquad\textbf{(C) } 399 \qquad\textbf{(D) } 400 \qquad\textbf{(E) } 420$


Solution

Problem 23

Let the $l$-set, for some fixed $l$, be the set {$\log_2 2l, \log_3 3l, \log_4 4l...$}. For some fixed $l$, let the $l$-product, be the product of the greatest $2020$ rational values from its $l$-set. For example the $3$-product would be the product of the greatest $2020$ rational values from the set ${\log_2 6, \log_3 9, \log_4 12, ...}$. Let $S$ the sum of all the $l$-products such that $l$ is divisible by exactly one prime factor $q$ such that $q<100$ (note that $l$ is not divisible by any other prime factors) and has at most $2020$ total factors. Find the value of $S$ modulo $1000.$

Solution

Problem 24

A company has a system of $2020$ levels of authority, and each level of authority has two people. One of the rules is that a person at the $n$th level of authority can fire another person at a level of authority that is no greater than $n-2$. Define a firing chain as a sequence of at least one firing event such that there exist a sequence of positive integers $a_1, a_2, a_3, ..., a_{k+1}$ such that member of the $a_i$th level of authority fires a member of the $a_{i+1}$th level of authority according to the firing rules for all $1 \leq i \leq k.$ Note that if a member of the company is already fired, he cannot make any more actions for the company (this includes firing). Let $N$ be the number of possible distinct firing chains starting from the $2020$th level of authority, or otherwise, the founder and the co-founder of the company. Find the remainder when $N$ is divided by $1000$.

$\textbf{(A) } 048 \qquad\textbf{(B) } 056 \qquad\textbf{(C) } 100 \qquad\textbf{(D) } 200 \qquad\textbf{(E) } 524$


Solution


Problem 25

Carlos is bored during quarantine, so he writes a repeating decimal $d$ given by \[d = \sum_{k=0}^{\infty}\frac{\tbinom{k}{6}}{10^{6(k+1)}}.\]Recall that $\binom{m}{n} = 0$ if $m < n$. If $P$ is the period of $d$, let $N$ be the number of positive factors of $P$. Find the remainder when $N$ is divided by $1000.$

$\textbf{(A) } 040 \qquad\textbf{(B) } 372 \qquad\textbf{(C) } 416 \qquad\textbf{(D) } 619 \qquad\textbf{(E) } 792$

Solution

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