AMO 2021 - Secondary 2 (Grade 8)

Question 1

Sam has six different fruits. He chooses five different fruits for each cup of juice. How many different flavours can he make?

1. Five - 5
2. Six - 6
3. Twelve - 12
4. Fifteen - 15
5. Thirty - 30

Question 2

The average of ten numbers is 12. Replacing one number with 10 changes the average to 9. What was the original value of the replaced number?

1. Not enough information
2. 10
3. 20
4. 30
5. 40

Question 3

After a 20% price cut, a sport shirt sells for 36 dollars before taxes. What percentage increase restores its original price?

1. 20%
2. 25%
3. 40%
4. 15%
5. 36%

Question 4

$ABCDEFGH$ is a cube with edge $AB=1$ inch. Triangle $ABG$ is constructed inside it as shown. What is its perimeter?

1. $3.5$ inches
2. $\sqrt5$ inches
3. $\sqrt6$ inches
4. $(\sqrt3+\sqrt2+1)$ inches
5. $\sqrt7$ inches

Question 5

Two circles have center $A$, with $AB=2$ inches and $AC=4$ inches. What is the ratio of the shaded ring’s area to the small circle’s area?

1. 6 : 1
2. 4 : 2
3. 3 : 2
4. 3 : 1
5. 2 : 1

Question 6

Points $A=(1,2)$, $B=(4,4)$ and $C=(4,1)$ are joined. What is the area of triangle $ABC$ in square units?

1. 3.5
2. 4
3. 4.5
4. 5.6
5. 6

Question 7

Mrs. Smith’s students measured diameters and circumferences of circular objects and fitted the linear model $y=2.6x-0.5$ shown below. Here $x$ is diameter and $y$ is circumference, both in centimeters. Based on this model, what circumference is predicted for a diameter of 50 cm?

1. 129.5
2. 157.5
3. 314.5
4. 78.5
5. 628.5

Question 8

In a dark room, box A contains four green and six red balls; box B contains eight green and two red balls. Amy randomly picks a box and then a ball. The ball is green. What is the probability it came from box A? Clarification: the two boxes are equally likely to be selected, and each ball in the chosen box is equally likely.

1. $\frac12$
2. $\frac13$
3. $\frac14$
4. $\frac15$
5. $\frac16$

Question 9

A circular coffee filter has radius 3 inches. Linda cuts out one-third of the disk, as shown, and rolls that sector into a cone without overlap. What is the radius of the cone’s base?

1. $\frac12$ inch
2. $\frac{3\pi}{2}$ inches
3. $\pi$ inches
4. 1 inch
5. $\frac\pi2$ inches

Question 10

Katrina has a rectangular sheet 10 inches by 12 inches. She cuts a square of side 2 inches from each corner and folds the remaining sheet into an open-top box. What is its volume in cubic inches?

1. 240
2. 64
3. 84
4. 1205
5. 96

Question 11

Lucy places a marble at $P$ on a rectangular notebook cover $ABCD$, labeled as shown. Distances $PA$, $PB$ and $PC$ are 2, 4 and 10 inches respectively. What is $PD$?

1. Not enough information
2. $2\sqrt{30}$ inches
3. $2\sqrt{22}$ inches
4. $2\sqrt{23}$ inches
5. 8 inches

Question 12

Parallelogram $ABCD$ has area 24 square inches. Point $P$ lies on $AD$. What is the area of triangle $BCP$?

1. Not enough information
2. 6 square inches
3. 18 square inches
4. 12 square inches
5. $2\sqrt6$ square inches

Question 13

Star $S$ is dilated from center $A$ to star $S^{\prime}$ as shown, with $AF=FF^{\prime}$ and $AC=CC^{\prime}$. Star $S$ has area 3.8 square inches. What is the area of $S^{\prime}$?

1. Not enough information
2. 7.6 square inches
3. 11.4 square inches
4. 15.2 square inches
5. 19.0 square inches

Question 14

In the figure, $ABC$ is equilateral with $AB=1$ foot. Points $D,E$ are the midpoints of $AC,BC$. Segment $FG$ is half as long as $AB$, and $EF=\frac{\sqrt[4]3}{2}$ feet. Find $DH$. As marked, $F,G$ lie on $AB$ and $DH\perp EF$ at $H$.

1. Not enough information
2. $\frac{\sqrt[3]2}{4}$ feet
3. $\frac{\sqrt[3]3}{2}$ feet
4. $\frac{\sqrt3}{3}$ feet
5. $\frac{\sqrt[4]3}{4}$ feet

Question 15

A right triangle has legs 2 inches and 1 inch. Its hypotenuse, in inches, lies between which pair of fractions?

1. $\frac94$ and $\frac{10}4$
2. $\frac{17}8$ and $\frac{18}8$
3. $\frac14$ and $\frac13$
4. $\frac{14}4$ and $\frac{15}4$
5. $\frac34$ and $\frac54$

Question 16

Three lights, Blue, Green, and Red, are mounted in a row and each has its own switch. Each light is either on or off. How many different color patterns can be created?

Question 17

For what value of $x$ is $2^x=8^{3x-8}$ true?

Question 18

The source describes decimal notation using “ten digits $\{0,1,2,3,4,5,7,8,9\}$” and gives $825=8\times10^2+2\times10+5$. In base three the digits are $\{0,1,2\}$, and $1212_3=1\times3^3+2\times3^2+1\times3+2$. What is the base-three representation of decimal 35? Clarification: the decimal digit list accidentally omits 6. Enter the base-three digit string.

Question 19

Find the value of $2022^2-2021^2$.

Question 20

$A$ is the center of the circle through $B,C,D$. Points $C,A,E$ are collinear, and $B,D,E$ are collinear, in the order shown. Given $\angle CAB=48^\circ$, find $\angle AED$ in degrees. The original problem is underdetermined. For this adapted exercise, add $DE=DA$; this is an extra assumption, not a stated source fact.