2020 FMC 10A Problems

Here are the problems that were on the 2020 FMC 10A.

Problem 1

Josh walks from his house to the store. Halfway there, he stops at the bank to get some money. Continuing his trip, he again stops halfway between the bank and the store to pick up some food from a local restaurant. Then, he decides he instead wants to go to the park and heads $\tfrac{3}{4}$ of the way back to his house. What fraction of the way (from his house) is he from the store now?

$\textbf{(A)}\ \dfrac{9}{16}\qquad\textbf{(B)}\ \dfrac{3}{16}\qquad\textbf{(C)}\ \dfrac{3}{8}\qquad\textbf{(D)}\ \dfrac{1}{3}\qquad\textbf{(E)}\ \dfrac{3}{4}$


Problem 2

Given that the sum of the lengths of the line segments containing all six possible pairs of vertices of square $P_1P_2P_3P_4$ is $2020,$ which of these choices is the closest to the square's side length?

$\textbf{(A)}\ 100\qquad\textbf{(B)}\ 200\qquad\textbf{(C)}\ 300\qquad\textbf{(D)}\ 400\qquad\textbf{(E)}\ 500$


Problem 3

Jimmy's balancing scale is off such that on the right side, it mistakenly adds or subtracts one pound off the actual weight. Jimmy places a $4$-pound object on the left side, and an $a \in \{1,2,3,4,\dots,9,10\}$ pound object on the right side. What is the probability Jimmy can still tell which object is heavier?

$\textbf{(A) } \tfrac{3}{10} \qquad\textbf{(B) } \tfrac{1}{2} \qquad\textbf{(C) } \tfrac{3}{5} \qquad\textbf{(D) } \tfrac{7}{10} \qquad\textbf{(E) } \tfrac{4}{5}$


Problem 4

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


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]

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


Problem 6

Real numbers $a$ and $b$ exist such that $|a-3| + a^2 - 6ab + 9b^2 = 0$. Find the value of $a+b$.

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


Problem 7

The niceness of a positive number $n$ with divisors $1=d_1<d_2<d_3<\dots<d_k=n$ is equal to the value of $d_{k-1} - d_2$. Find the sum of the first three positive numbers with zero niceness.

$\textbf{(A)}\ 26\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 34\qquad\textbf{(D)}\ 38\qquad\textbf{(E)}\ 47$


Problem 8

Let $ABCD$ be a parallelogram such that $AB || CD$, $BC || AD$. Furthermore, let $E$ and $F$ lie outside $ABCD$ such that $\triangle{AED}$ and $\triangle{DCF}$ are equilateral. Given that $BE = 6,$ $E$ lies on the line formed by extending segment $AB,$ and $F$ lies on the line formed by sxtending segment $BC,$ which of the following is closest to the area of $BEF$?

$\textbf{(A)}\ 14\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18$


Problem 9

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 $n.$

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


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?

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


Problem 11

Determine the number of integers $n=1,2,3,\dots,100$ such that $\tfrac{n! + (n-1)n!}{(n-2)!}$ is a perfect square.

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 10$


Problem 12

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 $S$.

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


Problem 13

Let $ABC$ be an isosceles triangle with $AB = AC = 10$ and $BC = 12$. Let the incenter of $ABC$ be $I$. The radius of the incircle of $BIC$ can be expressed as $a\sqrt{b} - c,$ where $a, b,$ and $c$ are positive integers with $b$ being square-free. Find $a+b+c.$

$\textbf{(A)}\ 21\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 23 \qquad\textbf{(D)}\ 24 \qquad\textbf{(E)}\ 25$


Problem 14

Toby the ant will start at $(0, 6)$ on the coordinate plane and each second, given that he is on $(x, y)$, he will randomly choose one point in the set $\{ (x-1, y-1), (x, y-1), (x+1, y-1) \}$ to travel to. The probability that Toby will eventually hit the origin can be expressed as $\frac{m}{n},$ where $m, n$ are relatively prime positive integers. Find $m+n$.

$\textbf{(A) } 290 \qquad\textbf{(B) } 292 \qquad\textbf{(C) } 868 \qquad\textbf{(D) } 872 \qquad\textbf{(E) } 874$


Problem 15

Given that the 2018 AMC 12A had an AIME cutoff of 93, let $N$ be the least AIME-qualifying score one can score on that test such that the said person's AIME score can always be uniquely determined from just looking at his/her USAMO index. Find the number of factors in $10N$. (Note that the AMC 12 is a 25-question test giving $6$ points for each correct answer, $1.5$ points for each blank answer, and $0$ points for each wrong answer. The AIME is a 15-question examination giving $10$ points each correct answer and $0$ points for each wrong or blank answer. An USAMO index is the sum of one's AMC 12 and AIME scores.)

$\textbf{(A) }8 \qquad \textbf{(B) }12 \qquad \textbf{(C) }16 \qquad \textbf{(D) } 24\qquad \textbf{(E) } 30\qquad$


Problem 16

Define the operator $\phi (n)$ as the number of positive integers less than $n$ that are relatively prime to $n$. What is the least positive integer $n$ such that $\frac{\phi(n!)}{n!} \leq \frac{1}{5}$?

$\textbf{(A)}\ 11 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 19 \qquad\textbf{(E)}\ 23$


Problem 17

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.)

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


Problem 18

Square $ABCD$ has side length $10$, and semicircle $P$, which is fully contained inside $ABCD$, has one vertex coinciding with $D$ and its $4$-unit diameter coinciding with $CD$. Circle $O$ is tangent to $BC$, $CD$, and the circumference of semicircle $P$. The radius of $O$ can be expressed as \[\ell - m\sqrt{n},\]where $\ell, m, n$ are positive integers and $n$ is square-free. Find $\ell+m+n$.

$\textbf{(A)}\ 21 \qquad\textbf{(B)}\ 22 \qquad\textbf{(C)}\ 23 \qquad\textbf{(D)}\ 24 \qquad\textbf{(E)}\ 25$


Problem 19

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

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


Problem 20

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 $0 \leq j \leq 11$?

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


Problem 21

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?

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


Problem 22

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


Problem 23

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 $1000$.

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


Problem 24

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.

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


Problem 25

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 $1000$.

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


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