2021 HMC 10

Problem 1

Let $A=\frac{11}{5}$ , $B=\frac{9}{10}$ and $C=\frac{12}{25}$ . The value of $A+B+C$ can be expressed as $\frac{m}{n}$ in the simplest form. Find $m$ .

$\textbf{(A)} ~177 \qquad\textbf{(B)} ~178 \qquad\textbf{(C)} ~179 \qquad\textbf{(D)} ~180 \qquad\textbf{(E)} ~181$

Problem 2

James is standing in the origin of a plane. He walks $3$ units to the south, and turns $90^{\circ}$ to the left, walked $4$ units. Find the distance between his position now and the origin.

$\textbf{(A)} ~\sqrt{7} \qquad\textbf{(B)} ~\sqrt{10} \qquad\textbf{(C)} ~4 \qquad\textbf{(D)} ~2\sqrt{5} \qquad\textbf{(E)} ~5$

Problem 3

In Martian civilization, when they are doing addition, in their perspective, $a+b=(\sqrt{a}+\sqrt{b})^2$ . Which of the following ordered pair of $(a,b)$ would result in this rule to be true?

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

Problem 4

There exists a rectangular prism such that the sum of all the side lengths is equal to $36$ units and the total surface area is $84$ units. Find the length of a space diagonal of this rectangular prism.

$\textbf{(A)} ~3\sqrt{5} \qquad\textbf{(B)} ~2\sqrt{13} \qquad\textbf{(C)} ~2\sqrt{15} \qquad\textbf{(D)} ~8 \qquad\textbf{(E)} ~4\sqrt{7}$

Problem 5

Let $ABC$ be a triangle with side length $AB=7, BC=24, AC=25$ . The angle bisector of $\angle{ABC}$ intersects $AC$ at $D$ . Find $AD$ .

$\textbf{(A)} ~\frac{170}{31} \qquad\textbf{(B)} ~\frac{175}{31} \qquad\textbf{(C)} ~\frac{180}{31} \qquad\textbf{(D)} ~6 \qquad\textbf{(E)} ~\frac{200}{31}$

Problem 6

Let $a\%_1b=a-(b-a)$ and let $a\%_2b=(a-b)-a$ . Find the value of $12\%_111-12\%_211$ .

$\textbf{(A)} ~-12 \qquad\textbf{(B)} ~-11 \qquad\textbf{(C)} ~-1 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~24$

Problem 7

Dr.Jeck is arranging books on a bookshelf. How many ways are there to arrange two identical math books, three identical science books and four identical English books such that the first and last books must be English books?

$\textbf{(A)} ~210 \qquad\textbf{(B)} ~420 \qquad\textbf{(C)} ~630 \qquad\textbf{(D)} ~840 \qquad\textbf{(E)} ~1260$

Problem 8

Let $k_!=k^2-\triangle_{k-1}$ and let $\triangle_n$ be the $n$ th triangular number. Find $({\triangle_5})_!+({\triangle_4})_!$

$\textbf{(A)} ~170 \qquad\textbf{(B)} ~175 \qquad\textbf{(C)} ~176 \qquad\textbf{(D)} ~200 \qquad\textbf{(E)} ~225$

Problem 9

Kashuv won a game, and he wanted to inform his friends about it. On the first day, he told his three best friends about the game. For each consecutive days, those who were informed about the victory of the game would introduce to four more best friends. Find the total number of people informed about the game on the $4$ th day assume that no people can be informed twice.

$\textbf{(A)} ~255 \qquad\textbf{(B)} ~256 \qquad\textbf{(C)} ~257 \qquad\textbf{(D)} ~258 \qquad\textbf{(E)} ~259$

Problem 10

Let $f(x)=x^2+ax+16$ and $g(x)=x^2+bx+12$ . If $a$ and $b$ are both nonzero integers, how many possible ordered pairs $(a,b)$ would result in $f(x)$ and $g(x)$ has exactly one common root?

$\textbf{(A)} ~3 \qquad\textbf{(B)} ~6 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~18 \qquad\textbf{(E)} ~24$

Problem 11

Let $a$ be a sequence of real numbers such that $a_n=\sqrt{a_{n-1}^2+n}$ and $a_1=1$ . How many terms from $a_1$ to $a_{50}$ , inclusive, are positive integers?

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

Problem 12

Joana is walking on a straight road that goes from south to north. Assume that all of his step is straight in the direction of north. Joana started from a point, walked $100m$ to north, and then he decided to walk backward $50m$ . After, he decided to walk $150m$ , and walks backward $75m$ . Then, he walked forward $200m$ and walked backward $100m$ . Continuing on this sequence that he walks $50(n+1)$ meter forward $n$ th time and then walks half the distance he walked forward immediately after that. Find the distance between his position after the $2$ nd time he walked backward and $4th$ time he walked forward.

$\textbf{(A)} ~250 \qquad\textbf{(B)} ~275 \qquad\textbf{(C)} ~350 \qquad\textbf{(D)} ~375 \qquad\textbf{(E)} ~450$

Problem 13

$\left(\sqrt{8+\sqrt{60}}\right)^3+\left(\sqrt{8-\sqrt{60}}\right)^3$ can be expressed as $a\sqrt{b}$ . Find $a+b$

$\textbf{(A)} ~17 \qquad\textbf{(B)} ~19 \qquad\textbf{(C)} ~21 \qquad\textbf{(D)} ~33 \qquad\textbf{(E)} ~39$

Problem 14

Let $f(x)$ be a polynomial with degree of $5$ such that $f(-2)=f(-1)=f(1)=f(2)=f(3)=13$ and the coefficient of the $x^5$ term is equal to the reciprocal of the constant term. What is the smallest possible value of $f(5)$ ?

$\textbf{(A)} ~7 \qquad\textbf{(B)} ~19 \qquad\textbf{(C)} ~84 \qquad\textbf{(D)} ~97 \qquad\textbf{(E)} ~1021$

Problem 15

Let $k=3^3+4^3+5^3+...+20^3$ . Find the number of positive integer factors of $k$ .

$\textbf{(A)} ~8 \qquad\textbf{(B)} ~12 \qquad\textbf{(C)} ~16 \qquad\textbf{(D)} ~24 \qquad\textbf{(E)} ~32$

Problem 16

In the large square with side length $4$ , a quarter circle arc of radius $\sqrt{10}$ intersects two sides that divided the side length into ratio $3:1$ , as shown in the diagram. Find the area of the shaded region.

$\textbf{(A)} ~\frac{5\pi}{2}-1 \qquad\textbf{(B)} ~\frac{5\pi}{2}+1 \qquad\textbf{(C)} ~4\pi \qquad\textbf{(D)} ~3\pi+3 \qquad\textbf{(E)} ~5\pi-2\sqrt{2}+3$

Problem 17

The largest root of the polynomial $x^4-4x^3+6x-4x=24$ can be expressed as $a+\sqrt{b}$ . Find $a+b$ .

$\textbf{(A)} ~7 \qquad\textbf{(B)} ~8 \qquad\textbf{(C)} ~10 \qquad\textbf{(D)} ~12 \qquad\textbf{(E)} ~16$

Problem 18

Jonas has a collection of toys, and he wanted to organize them into boxes. The collection contains $21$ toys, and he wants to put them into $21$ different boxes, each numbered from $1$ to $21$ , respectively. Also, each boxes can only contain one toy. However, due to his believe of equity, he decided to do a paper lottery by putting twenty one sheets of paper that represents each positive integer from $1$ to $21$ , inclusive. After he randomly choose any sheet, he would take out and not replace back into the cup. The number on the sheet he chose for a specific toy would determine the box that the toy would stay in. Jonas first rearranged all the toys and started this procedure. What is the probability that toy $1$ would be in box $\#1$ ?

$\textbf{(A)} ~\frac{1}{21} \qquad\textbf{(B)} ~\frac{3}{7} \qquad\textbf{(C)} ~\frac{1}{2} \qquad\textbf{(D)} ~\frac{10}{21} \qquad\textbf{(E)} ~\frac{11}{21}$

Problem 19

Find the largest positive integer less than or equal to $100$ that divides $2^{324}+3^{324}$ .

$\textbf{(A)} ~89 \qquad\textbf{(B)} ~91 \qquad\textbf{(C)} ~93 \qquad\textbf{(D)} ~97 \qquad\textbf{(E)} ~99$

Problem 20

Let $A$ be the area enclosed by the graph of equation $||2x|+|4\sqrt{2}y||=8$ . Find $A^2$

$\textbf{(A)} ~16 \qquad\textbf{(B)} ~32 \qquad\textbf{(C)} ~64 \qquad\textbf{(D)} ~128 \qquad\textbf{(E)} ~256$

Problem 21

Joshua wrote all the divisors of $5000$ , all the divisors of $2750$ , and all the divisors of $1600$ on different sheets of paper, such that only $1$ number is on each sheets. Then, he putted all the sheets into a jar and randomly choose $2$ sheets from the jar. Find the probability that the two sheets contains the same number.

Problem 22

Let $f(x)=x^5-3x^4+2$ . Let $r_1, r_2, r_3, r_4, r_5$ be the roots of $f(x)$ . Furthermore, let $g(x)=x^4$ . Find the value of $g(r_1+r_2+r_3+r_4)g(r_2+r_3+r_4+r_5)g(r_3+r_4+r_5+r_1)g(r_4+r_5+r_1+r_2)g(r_5+r_1+r_2+r_3)$

$\textbf{(A)} ~16 \qquad\textbf{(B)} ~36 \qquad\textbf{(C)} ~128 \qquad\textbf{(D)} ~192 \qquad\textbf{(E)} ~1024$

Problem 23

Let $f(x)$ be a monic polynomial with degree $8$ such that $\sqrt{2}+\sqrt[4]{5}$ is a root of $f(x)$ and all the coefficients of $f(x)$ are rational numbers. Find $f(1)$

$\textbf{(A)} ~-288 \qquad\textbf{(B)} ~-144 \qquad\textbf{(C)} ~-36 \qquad\textbf{(D)} ~72 \qquad\textbf{(E)} ~144$

Problem 24

At Janus high school, in a math class, students are playing an icebreaker game. They each wrote a name tag that has their name, and each names are different. Then, they gave their name tag to the teacher, and the teacher rearranged them and randomly give each student a name tag after rearrangement. Given that there are $32$ students in that class, find the expected value of the number of students who get their own name tag back.

$\textbf{(A)} ~\frac{1}{2} \qquad\textbf{(B)} ~\frac{31}{32} \qquad\textbf{(C)} ~\frac{1}{2} \qquad\textbf{(D)} ~\frac{32}{31} \qquad\textbf{(E)} ~16$

Problem 25

In the following diagram, the three concentric squares have areas $64$ , $36$ , $16$ and $2$ units, respectively. The smallest square was formed by rotating one of the larger squares by $45$ degrees. John has a uniform circular coin with radius $\frac{1}{4}$ , and he wants to place it in the sheet of three concentric squares such that a part of the coin lands in the sheet(Not necessarily the entire coin). To do that, he would first randomly choose and label a point for the coin's center(Possibly outside this diagram), and then he would place the coin. The probability that the coin would intersects this diagram at two points can be expressed as $\frac{1}{1152+\pi}(a+b\sqrt{2}+c\pi)$ Find $a+b+c$ .