2021 JMC 10

Problem 1

What is the value of $\dfrac{20}{2\cdot1} - \dfrac{2+0}{2/1}?$ $\textbf{(A) } 3 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 9 \qquad\textbf{(E) } 13$

Solution

Problem 2

There exist irrational numbers $e\approx 2.72$ and $\pi \approx 3.14.$ How can $|\pi - |e - | e - \pi|||$ be expressed in terms of $\pi$ and $e?$

$\textbf{(A) }\pi-e \qquad\textbf{(B) }2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2\pi \qquad\textbf{(E) }2\pi +e$

Solution

Problem 3

A group of $8$ people are either honest or liars, where honest people always tell the truth and liars always lie. People $P_1,P_2, \cdots, P_8$ stand in a line, and person $P_i$ calls $P_{i+1}$ a liar where $P_1 = P_9.$ Out of these eight people, how many liars are there?

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

Solution

Problem 4

A day in the format $mm/dd$ is called binary if all of the digits are either $0$ s or $1$ s with leading zeros allowed. How many days in a year are binary?

$\textbf{(A) } 5 \qquad\textbf{(B) } 9 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 15 \qquad\textbf{(E) } 16$

Solution

Problem 5

A mixture has $96$ grams of aluminum and $4$ grams of barium. Nir the chemist uses magic to remove some aluminum. Now, exactly $90\%$ of the mixture consists of aluminum. How many grams of the mixture now remain?

$\textbf{(A) } 25 \qquad\textbf{(B) } 40 \qquad\textbf{(C) } 50 \qquad\textbf{(D) } 70 \qquad\textbf{(E) } 72$

Solution

Problem 6

The sum of the ages of a family equals $78.$ Fifteen years later, the sum of their ages is equal to $153.$ How many people are in this family?

$\textbf{(A) } 4 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 7 \qquad\textbf{(D) } 9 \qquad\textbf{(E) } 10$

Solution

Problem 7

For some real $x,$ the area of a square equals $3x+1$ and the product of the lengths of its diagonals equals $5x+7.$ What is the perimeter of this square?

$\textbf{(A) } 16 \qquad\textbf{(B) } 17 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 19 \qquad\textbf{(E) } 20$

Solution

Problem 8

A positive integer is pretentious if it has both even and odd digits. For example, $74$ and $83$ are pretentious. How many pretentious three-digit numbers are odd?

$\textbf{(A) } 325 \qquad\textbf{(B) } 375 \qquad\textbf{(C) } 370 \qquad\textbf{(D) } 450 \qquad\textbf{(E) } 775$

Solution

Problem 9

In Malachar, the number system is identical to ours, but all real numbers are written with digits in reverse order. A citizen in Malachar writes $15\cdot 73.$ What does this Malacharian write as the answer?

$\textbf{(A) } 1095 \qquad\textbf{(B) } 1887 \qquad\textbf{(C) } 3723 \qquad\textbf{(D) } 5901 \qquad\textbf{(E) } 7881$

Solution

Problem 10

Let $ABCD$ be a square with sides of length $4.$ Point $X$ is on side ${CD}$ and point $Y$ is on side ${BC}$ such that $AX = 5$ and angle ${AYX}$ is right. What is $AY \cdot XY?$

$\textbf{(A) } 10 \qquad\textbf{(B) } 9\sqrt{2} \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 6\sqrt{5} \qquad\textbf{(E) } 7\sqrt{5}$

Solution

Problem 11

There exist positive integers $k$ that satisfy $k = 3\gcd(20,k).$ What is the sum of all possible values of $k?$

$\textbf{(A) } 84 \qquad\textbf{(B) } 108 \qquad\textbf{(C) } 120 \qquad\textbf{(D) } 126 \qquad\textbf{(E) } 132$

Solution

Problem 12

Mihir draws line $y=2x$ and Nathan draws line $x+y = n$ for an integer $n.$ The two lines divide the region $y \ge x^2$ into four regions, with regions possibly having infinite area. What is the sum of all possible values of $n?$

$\textbf{(A) } 12 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 23 \qquad\textbf{(D) } 29 \qquad\textbf{(E) } 32$

Solution

Problem 13

An angle chosen from $1^{\circ},2^{\circ}, \dots,90^{\circ}$ and an angle chosen from $1^{\circ},2^{\circ},\dots,89^{\circ}$ determine two angles of a triangle. What is the probability this triangle is obtuse?

$\textbf{(A) } \dfrac{11}{45} \qquad\textbf{(B) } \dfrac{22}{45} \qquad\textbf{(C) } \dfrac{1}{2} \qquad\textbf{(D) } \dfrac{5}{9} \qquad\textbf{(E) } \dfrac{11}{15}$

Solution

Problem 14

For a certain $b,$ the base $b$ numbers $24_b,n,57_b,72_b, \ldots$ form an increasing arithmetic sequence in that specific order. Then, what is the value of $n,$ expressed in base $10?$

$\textbf{(A) } 47 \qquad \textbf{(B) } 51 \qquad \textbf{(C) } 63 \qquad \textbf{(D) } 64 \qquad \textbf{(E) } 75$

Solution

Problem 15

Let $a_0,a_1,a_2,\dots,a_{2021}$ be a sequence such that $a_0=1$ and $a_n = 10^n\cdot a_{n-1}+1$ for positive integers $n \ge1.$ How many terms of this sequence are divisible by $99?$

$\textbf{(A) } 0 \qquad\textbf{(B) } 20 \qquad\textbf{(C) } 21 \qquad\textbf{(D) } 91 \qquad\textbf{(E) } 112$

Solution

Problem 16

If $a$ and $b$ are randomly chosen numbers between $-5$ and $5$ , what is the probability that $\lfloor |a| \rfloor - | \lfloor b \rfloor |=0 ?$ (Recall that $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r.$ )

$\textbf{(A) } \dfrac{1}{20} \qquad\textbf{(B) } \dfrac{2}{25} \qquad\textbf{(C) } \dfrac{1}{10} \qquad\textbf{(D) } \dfrac{9}{50} \qquad\textbf{(E) } \dfrac{1}{5}$

Solution

Problem 17

One lit lightbulb is $2$ units above the top of spherical ball with a radius of $8.$ The spherical ball, lying atop a flat floor, casts a shadow. What is the area of this shadow?

$\textbf{(A) } 100\pi \qquad\textbf{(B) } 144\pi \qquad\textbf{(C) } 288\pi \qquad\textbf{(D) } 576\pi \qquad\textbf{(E) } 1024\pi$

Solution

Problem 18

If $x,y,$ and $z$ are positive real numbers that satisfy the equation $xy+yz+xz=96(y+z)-x^2 = 24(x+z) -y^2 = 54(x+y) -z^2,$ what is the value of $xyz$ ?

$\textbf{(A) } 1720 \qquad\textbf{(B) } 2720 \qquad\textbf{(C) } 7560 \qquad\textbf{(D) } 9600 \qquad\textbf{(E) } 15120$

Solution

Problem 19

Two distinct divisors of $6^4=1296$ are mutual if their difference divides their product. For instance, $(4,2)$ is mutual as $(4-2)\mid 4\cdot2.$ Suppose a mutual pair $(d_1,d_2)$ exists where $d_1 = kd_2$ for a positive integer $k.$ What is the sum of all possible $k?$

$\textbf{(A) } 14 \qquad\textbf{(B) } 18 \qquad\textbf{(C) } 19 \qquad\textbf{(D) } 20 \qquad\textbf{(E) } 23$

Solution

Problem 20

A particle is in a $5 \times 5$ grid. Each second, it moves to an adjacent cell and when traveling from a cell to another cell, it takes one of the paths with shortest time. The particle starts at cell $A$ and travels to cell $B$ in $3$ seconds, to cell $C$ in $4$ seconds, and finally back to cell $A$ in $5$ seconds. How many possible triples $\{A,B,C\}$ exist?

$\textbf{(A) } 56 \qquad \textbf{(B) }96 \qquad \textbf{(C) } 104 \qquad \textbf{(D) } 136 \qquad \textbf{(E) } 168$

Solution

Problem 21

Two identical circles $\omega_{a}$ and $\omega_{b}$ with radius $1$ have centers that are $\tfrac{4}{3}$ units apart. Two externally tangent circles $\omega_1$ and $\omega_2$ of radius $r_1$ and $r_2$ respectively are each internally tangent to both $\omega_a$ and $\omega_b$ . If $r_1 + r_2 = \tfrac{1}{2}$ , what is $r_1r_2$ ?

$\textbf{(A) }\dfrac{1}{21}\qquad\textbf{(B) }\dfrac{1}{14}\qquad\textbf{(C) }\dfrac{5}{63}\qquad\textbf{(D) }\dfrac{2}{21}\qquad\textbf{(E) }\dfrac{1}{7}$

Solution

Problem 22

Let $r_1,r_2,r_3,r_4$ be the roots of $P(x)= x^4+4x^3-3x^2+2x-1.$ Suppose $Q(x)$ is the monic polynomial with all six roots in the form $r_{i}+r_{j}$ for integers $1\le i < j \le 4.$ What is the coefficient of the $x^4$ term in the polynomial $Q(x)?$

$\textbf{(A) } 32 \qquad \textbf{(B) } 36 \qquad \textbf{(C) } 42 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 56$

Solution

Problem 23

An invisible ant and an anteater, at the same constant speed of $1$ edge length per second, start at (not necessarily distinct) randomly chosen vertices of a cube. Each second, the ant first pings its location to the anteater, then randomly chooses one of the $3$ edges emerging from its vertex to traverse immediately. The anteater traverses the edge on the closest path to the ping at the same time the ant travels. If multiple optimal paths exist, one is randomly chosen. The anteater eats the ant if at some time they are both at the same point, not necessarily a vertex. What is the ant's expected lifespan in seconds?

$\textbf{(A) }\dfrac{41}{16}\qquad\textbf{(B) }\dfrac{11}{4}\qquad\textbf{(C) }\dfrac{49}{16}\qquad\textbf{(D) }\dfrac{27}{8}\qquad\textbf{(E) }\dfrac{55}{16}$

Solution

Problem 24

In cyclic convex hexagon $AZBXCY,$ diagonals $\overline{AX}$ , $\overline{BY}$ , and $\overline{CZ}$ concur at the circumcenter of the hexagon, and quadrilateral $BCYZ$ has area $1.$ If the sum of the areas of $\triangle{ABC}$ and the original hexagon is equal to $3,$ what is the sum of the areas of quadrilaterals $XZAC$ and $YXBA?$

$\textbf{(A) }\dfrac{3}{2}\qquad\textbf{(B) }2\qquad\textbf{(C) }\dfrac{8}{3}\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

Solution

Problem 25

How many ordered pairs of positive integers $(a,b)$ with $a \le 100$ and $b \le 10$ exist such that neither the numerator nor denominator of the below fraction, when completely simplified (i.e. numerator and denominator are relatively prime), are divisible by five? $\frac{4^b +121^b}{2^a -3^b}$