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Revision as of 19:46, 31 January 2021

Here is where I'll fully "publish" mock tests I've written. So far there's only one, but there will definitely be more to come.

I'll also add a user count here. It won't change whether the action occurs, but instead how I feel about the action.

(am clout chaser)

User Count

Increase the user count below if this is your first time discovering this page.

1
RedFireTruck was here</color>


User Count

Increase the user count below if this is your first time discovering this page and you have used these problems as practice.

0

Apocalyptic AMC 8 (2020)

This contest took place between September 8th and November 6th. All problems were written by me (bissue) and testsolved by nikenissan, bobthegod78, knightime1010, truffle, cw357, and ApraTrip.

The mock was separated into two sections: The AMC 8 and The Tiebreakers.

Standard AMC 8 rules were used. All participants had 40 minutes to complete as many of the 25 problems as they could. Correct answers were worth 1 point each, while incorrect or blank answers were worth 0 points each.

The Tiebreakers were used to break ties between participants with the same score on the AMC 8. The rules were the same as those used for the ARML tiebreaker. For more information, see the original post in the AoPS Mock Contests Forum here:

https://artofproblemsolving.com/community/c594864h2255517

Full statistics and discussion threads can be found using the link above as well.

Problem 1

To walk up a single floor in her eighteen floor apartment building, Sarah needs to take nine steps up a flight of stairs. If Sarah starts on Floor $3$ and walks up $100$ steps, she would end up on the flight of stairs connecting which two floors?

$\textbf{(A)} ~ \mbox{11 and 12} \qquad \textbf{(B)} ~ \mbox{12 and 13} \qquad \textbf{(C)} ~ \mbox{13 and 14} \qquad \textbf{(D)} ~ \mbox{14 and 15} \qquad \textbf{(E)} ~ \mbox{15 and 16}$

Problem 2

Abby, Barb, and Carlos each have $35$, $42$, and $31$ trading cards respectively. If they share their trading cards equally between each other, how many more trading cards would Carlos have than before?

$\textbf{(A)} ~ 4 \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 6 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 11$

Problem 3

In triangle $ABC$ the measure of angle $\angle A$ is the average of the measures of angles $\angle B$ and $\angle C$. What is the measure of angle $\angle A$?

$\textbf{(A)} ~ 45^{\circ} \qquad \textbf{(B)} ~ 60^{\circ} \qquad \textbf{(C)} ~ 75^{\circ} \qquad \textbf{(D)} ~ 90^{\circ} \qquad \textbf{(E)} ~ 120^{\circ}$

Problem 4

A spruce tree grows by $25$ feet, increasing its height by $25 \%$. If the tree grows for a second time by $25$ feet, by what percent would its height increase?

$\textbf{(A)} ~ 5 \% \qquad \textbf{(B)} ~ 15 \% \qquad \textbf{(C)} ~ 20 \% \qquad \textbf{(D)} ~ 25 \% \qquad \textbf{(E)} ~ 30 \%$

Problem 5

Find the sum of the digits of $\dfrac{5 \times 10^{2020}}{2}$.

$\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 5 \qquad \textbf{(D)} ~ 7 \qquad \textbf{(E)} ~ 8$

Problem 6

Square $B$ with side length three is attached to a side of square $A$ with side length four, as shown in the figure below. Find the area of the shaded region. [asy] size(150); draw((0, 0)--(4, 0)--(4, 4)--(0, 4)--cycle); draw((4, 1)--(7, 1)--(7, 4)--(4, 4)--cycle); filldraw((0, 0)--(4, 0)--(4, 2.285714)--cycle, grey); filldraw((4, 1)--(7, 1)--(7, 4)--(4, 2.285714)--cycle, grey); label("A", (2, 2)); label("B", (5.5, 2.5)); [/asy] $\textbf{(A)} ~ 10 \qquad \textbf{(B)} ~ 10 \frac{1}{2} \qquad \textbf{(C)} ~ 11 \qquad \textbf{(D)} ~ 14 \qquad \textbf{(E)} ~ 14 \frac{1}{2}$

Problem 7

When expressed as a decimal rounded to the nearest ten-thousandth, what is the value of $\dfrac{125+3}{125 \times 3}$?

$\textbf{(A)} ~ 0.3412 \qquad \textbf{(B)} ~ 0.3413 \qquad \textbf{(C)} ~ 0.3414 \qquad \textbf{(D)} ~ 0.3415 \qquad \textbf{(E)} ~ 0.3416$

Problem 8

What is the value of \[(1+2+3)-(2+3+4)+(3+4+5)-\cdots -(98+99+100)?\] $\textbf{(A)} ~ -150 \qquad \textbf{(B)} ~ -147 \qquad \textbf{(C)} ~ -144 \qquad \textbf{(D)} ~ 147 \qquad \textbf{(E)} ~ 150$

Problem 9

Kayla writes down the first $N$ positive integers. What is the sum of all possible values of $N$ if Kayla writes five multiples of $13$ and six multiples of $12$?

$\textbf{(A)} ~ 447 \qquad \textbf{(B)} ~ 453 \qquad \textbf{(C)} ~ 518 \qquad \textbf{(D)} ~ 525 \qquad \textbf{(E)} ~ 548$

Problem 10

In Murphy's seventh grade homeroom, $\frac{7}{12}$ of the students like tennis, $\frac{2}{3}$ of the students like badminton, and $\frac{1}{12}$ of the students like neither. What is the minimum possible number of students who like both tennis and badminton?

$\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 3 \qquad \textbf{(D)} ~ 4 \qquad \textbf{(E)} ~ 6$

Problem 11

For how many values of $N$ does there exist a regular $N$ sided polygon whose vertices all lie on the vertices of a regular $24$ sided polygon?

$\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 10$

Problem 12

Quadrilateral $WXYZ$ has its vertices on the sides of rectangle $ABCD$ with $AB=7$ and $BC=5$, as shown below. What is the area of quadrilateral $WXYZ$? [asy] size(150); draw((0, 0)--(7, 0)--(7, 5)--(0, 5)--cycle); label("A", (0, 0), SW); label("B", (7, 0), SE); label("C", (7, 5), NE); label("D", (0, 5), NW); filldraw((0, 1)--(4, 0)--(7, 3)--(4, 5)--cycle, grey); label("W", (0, 1), W); label("X", (4, 0), S); label("Y", (7, 3), E); label("Z", (4, 5), N); label("4", (2, -0.5)); label("3", (5.5, -0.5)); label("4", (2, 5.5)); label("3", (5.5, 5.5)); [/asy] $\textbf{(A)} ~ 15 \dfrac{1}{2} \qquad \textbf{(B)} ~ 16 \qquad \textbf{(C)} ~ 16 \dfrac{1}{2} \qquad \textbf{(D)} ~ 17 \qquad \textbf{(E)} ~ 17 \dfrac{1}{2}$

Problem 13

To drive to the supermarket, Mable drives for $m$ miles, then drives $12$ miles per hour faster for the remaining $\frac{4}{3}m$ miles. The amount of time Mable spent driving at each of the two speeds was equal. What was Mable's average speed during her drive to the supermarket, in miles per hour?

$\textbf{(A)} ~ \dfrac{81}{2} \qquad \textbf{(B)} ~ \dfrac{288}{7} \qquad \textbf{(C)} ~ 42 \qquad \textbf{(D)} ~ \dfrac{300}{7} \qquad \textbf{(E)} ~ 50$

Problem 14

Six circles of radius one are cut out of the rectangle below. What is the area of the shaded region? [asy] size(150); filldraw((0, 0)--(6, 0)--(6, 4)--(0, 4)--cycle, grey); filldraw(circle((1, 1), 1), white); filldraw(circle((3, 1), 1), white); filldraw(circle((5, 1), 1), white); filldraw(circle((1, 3), 1), white); filldraw(circle((3, 3), 1), white); filldraw(circle((5, 3), 1), white); [/asy] $\textbf{(A)} ~ 20-6\pi \qquad \textbf{(B)} ~ 24-6\pi \qquad \textbf{(C)} ~ 28-6\pi \qquad \textbf{(D)} ~ 30-6\pi \qquad \textbf{(E)} ~ 32-6\pi$

Problem 15

One metronome beeps at a steady rate of $72$ beeps per minute, while another metronome beeps at a steady rate of $96$ beeps per minute. If both metronomes beep at the same time once, how long will it take, in seconds, until they first beep at the same time again?

$\textbf{(A)} ~ 2 \dfrac{1}{2} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 10 \qquad \textbf{(D)} ~ 18 \qquad \textbf{(E)} ~ 24$

Problem 16

A square with side length two is placed on a table, forming a $30$ degree angle with the table's surface. How much higher is the top vertex of the square than the table? [asy] size(150); draw((0, 0)--(0.882, 0.4714)--(0.4106, 1.3534)--(-0.4714, 0.882)--cycle); draw((-0.5, 0)--(1, 0), linewidth(3)); draw((-0.75, 1.3534)--(-0.65, 1.3534)); draw((-0.7, 1.3534)--(-0.7, 0)); draw((-0.75, 0)--(-0.65, 0)); [/asy] $\textbf{(A)} ~ \dfrac{5}{2} \qquad \textbf{(B)} ~ \sqrt{3}+1 \qquad \textbf{(C)} ~ \dfrac{4\sqrt{3}}{3} \qquad \textbf{(D)} ~ 3 \qquad \textbf{(E)} ~ \dfrac{3\sqrt{3}}{2}+1$

Problem 17

Kurtis' school schedule is made up of four classes, followed by lunch, followed by three more classes. In how many ways can Kurtis arrange his schedule if two of his classes (Reading and Writing) must occur one immediately after the other?

$\textbf{(A)} ~ 600 \qquad \textbf{(B)} ~ 840 \qquad \textbf{(C)} ~ 1200 \qquad \textbf{(D)} ~ 1440 \qquad \textbf{(E)} ~ 1680$

Problem 18

When the number $25$ is added to a list of numbers with total sum $S$, the average of all the numbers increases by one. What is the sum of the digits of the greatest possible value of $S$?

$\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 12$

Problem 19

A magician randomly picks a three digit positive integer to put into her hat and pulls out the same number with its digits in reverse order. (For example $496$ would become $694$ and $720$ would become $27$.) What is the probability the magician pulls out a multiple of $22$?

$\textbf{(A)} ~ \dfrac{1}{15} \qquad \textbf{(B)} ~ \dfrac{1}{18} \qquad \textbf{(C)} ~ \dfrac{1}{20} \qquad \textbf{(D)} ~ \dfrac{1}{25} \qquad \textbf{(E)} ~ \dfrac{1}{30}$

Problem 20

Tyrone has three books to read in six days. He reads one-half of a single book every day. In how many ways can he finish all the books if he may not read the same book two days in a row?

$\textbf{(A)} ~ 12 \qquad \textbf{(B)} ~ 18 \qquad \textbf{(C)} ~ 24 \qquad \textbf{(D)} ~ 30 \qquad \textbf{(E)} ~ 36$

Problem 21

There exists a circle that is tangent to $\overline{AB}$ and $\overline{BC}$ at $A$ and $C$, respectively. If $AB=BC=13$ and $AC=10$, what is the radius of the circle? [asy] size(150); draw((-5, 0)--(5, 0)--(0, -12)--cycle); draw(circle((0, 2.08333), 5.41666)); label("A", (-5, 0), W); label("C", (5, 0), E); label("B", (0, -12), S); label("13", (-2.7, -6), W); label("13", (2.7, -6), E); label("10", (0, 0.2), N); [/asy] $\textbf{(A)} ~ \dfrac{60}{13} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ \dfrac{26}{5} \qquad \textbf{(D)} ~ \dfrac{65}{12} \qquad \textbf{(E)} ~ \dfrac{156}{25}$

Problem 22

For each of the distinct sets of numbers containing only positive integers between $1$ and $9$ inclusive, Jordan writes the sum of the numbers in that set. What is the sum of the numbers Jordan writes?

$\textbf{(A)} ~ 11520 \qquad \textbf{(B)} ~ 11565 \qquad \textbf{(C)} ~ 11610 \qquad \textbf{(D)} ~ 11655 \qquad \textbf{(E)} ~ 11700$

Problem 23

In rectangle $ABCD$, the perpendicular from $B$ to diagonal $\overline{AC}$ bisects segment $\overline{CD}$. Which of the following is closest to $\frac{AB}{BC}$?

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

Problem 24

How many ordered triples of positive integers $(a, b, c)$ satisfy $\text{gcd}(a, b, c)=20$ and $\text{lcm}(a, b, c)=240$?

$\textbf{(A)} ~ 18 \qquad \textbf{(B)} ~ 24 \qquad \textbf{(C)} ~ 36 \qquad \textbf{(D)} ~ 54 \qquad \textbf{(E)} ~ 72$

Problem 25

Cheyanne rolls two standard six sided dice, then repeatedly rerolls all dice which show an odd number and stops as soon as all dice show an even number. What is the probability Cheyanne stops after exactly four rounds of rerolling?

$\textbf{(A)} ~ \dfrac{61}{1024} \qquad \textbf{(B)} ~ \dfrac{1}{16} \qquad \textbf{(C)} ~ \dfrac{67}{1024} \qquad \textbf{(D)} ~ \dfrac{9}{128} \qquad \textbf{(E)} ~ \dfrac{29}{256}$

Tiebreaker Problem 1

A whiteboard has positive real numbers $1$ and $m$ written on it. Every second, if the numbers $x$ and $y$ are on the whiteboard, a ghost will replace those numbers with $|x^2-y^2|$ and $2xy$. The ghost stops once one number on the whiteboard is $m$ times the other. For how many positive real numbers $m$ does the ghost stop after exactly $16$ seconds?

Tiebreaker Problem 2

The perpendicular bisectors of triangle $ABC$ can be described in the coordinate plane as lines $y=0$, $y=x$, and $y=\sqrt{3}x$. Given that triangle $ABC$ has circumradius $1$, find its area.

Tiebreaker Problem 3

The diagram below is constructed by attaching an equilateral triangle, a square, a regular pentagon, and a regular hexagon together. Compute the measure of the obtuse angle formed by the three red vertices. [asy] import graph; size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -29.36, xmax = -9.8, ymin = 4.78, ymax = 17.66; /* image dimensions */ draw((-22,12)--(-19,12)--(-20.5,14.598076211353318)--cycle, linewidth(2)); draw((-20.5,14.598076211353318)--(-19,12)--(-16.401923788646684,13.5)--(-17.90192378864668,16.098076211353316)--cycle, linewidth(2)); draw((-16.401923788646684,13.5)--(-19,12)--(-18.376264927546725,9.065557197798585)--(-15.392699241441907,8.751971807995622)--(-14.172489312214504,11.492608180923423)--cycle, linewidth(2)); draw((-18.376264927546725,9.065557197798585)--(-19,12)--(-21.85316954888546,12.927050983124847)--(-24.082604025317647,10.919659164048271)--(-23.45886895286437,7.985216361846854)--(-20.60569940397891,7.058165378722009)--cycle, linewidth(2)); Label laxis; laxis.p = fontsize(10); string blank(real x) {return "";} xaxis(xmin, xmax, Ticks(laxis, blank, Step = 1, Size = 2, NoZero),EndArrow(6), above = true); yaxis(ymin, ymax, Ticks(laxis, blank, Step = 1, Size = 2, NoZero),EndArrow(6), above = true); /* draws axes; NoZero hides '0' label */ /* draw figures */ draw((-22,12)--(-19,12), linewidth(2)); draw((-19,12)--(-20.5,14.598076211353318), linewidth(2)); draw((-20.5,14.598076211353318)--(-22,12), linewidth(2)); draw((-20.5,14.598076211353318)--(-19,12), linewidth(2)); draw((-19,12)--(-16.401923788646684,13.5), linewidth(2)); draw((-16.401923788646684,13.5)--(-17.90192378864668,16.098076211353316), linewidth(2)); draw((-17.90192378864668,16.098076211353316)--(-20.5,14.598076211353318), linewidth(2)); draw((-16.401923788646684,13.5)--(-19,12), linewidth(2)); draw((-19,12)--(-18.376264927546725,9.065557197798585), linewidth(2)); draw((-18.376264927546725,9.065557197798585)--(-15.392699241441907,8.751971807995622), linewidth(2)); draw((-15.392699241441907,8.751971807995622)--(-14.172489312214504,11.492608180923423), linewidth(2)); draw((-14.172489312214504,11.492608180923423)--(-16.401923788646684,13.5), linewidth(2)); draw((-18.376264927546725,9.065557197798585)--(-19,12), linewidth(2)); draw((-19,12)--(-21.85316954888546,12.927050983124847), linewidth(2)); draw((-21.85316954888546,12.927050983124847)--(-24.082604025317647,10.919659164048271), linewidth(2)); draw((-24.082604025317647,10.919659164048271)--(-23.45886895286437,7.985216361846854), linewidth(2)); draw((-23.45886895286437,7.985216361846854)--(-20.60569940397891,7.058165378722009), linewidth(2)); draw((-20.60569940397891,7.058165378722009)--(-18.376264927546725,9.065557197798585), linewidth(2)); /* dots and labels */ dot((-22,12),red); dot((-19,12),red); dot((-21.526119073220972,12.820785841919117),linewidth(4pt) + dotstyle+red); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Answer Key

AMC 8: DBBCD / CBBAD / AECBA / BCDDD / DACEA

Tiebreakers: ($65280$, $\dfrac{3-\sqrt{3}}{4}$, $102$)

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