AMO 2021 — Secondary 1 (Grade 7)

Question 1

Find the value of $\sqrt{\sqrt{0.0081}+\sqrt{0.0001}+\sqrt{0.81}}$.

1. $1$
2. $-1$
3. Both 1 and −1
4. Decimal numbers do not have square roots
5. $\sqrt{0.019}$

Question 2

If 5 is added to both the numerator and denominator of $\frac6{11}$, how does the fraction change?

1. No change
2. Increase
3. Decrease
4. Increases by $\frac{11}{16}$
5. Becomes 1

Question 3

A mother makes $x$ cakes in 2 hours and $y$ pizzas in 3 hours. How many cakes and pizzas altogether does she make in 6 hours? Assume she makes both together at the same rates as when making them separately.

1. $6(\frac x2+\frac y3)$
2. $6(\frac y2+\frac x3)$
3. $6(2x+3y)$
4. $6(3x+2y)$
5. $30$

Question 4

Sasha’s image is $1\frac12$ inches wide and $2\frac14$ inches tall. She enlarges it proportionally to fit exactly into a screen box 4 inches wide. How tall must the box be, in inches?

1. $\frac9{32}$
2. $\frac{27}{32}$
3. $6$
4. $4\frac34$
5. $2\frac12$

Question 5

Four numbers $P,Q,R,S$ are in ascending order. The average of $P,Q,R$ is 22, and the average of $Q,R,S$ is 24. The largest number is 27. Which statement describes the smallest number?

1. It is prime
2. It is a multiple of 9
3. It is a multiple of 11
4. It is a multiple of 7
5. It cannot be determined

Question 6

Form the largest and smallest numbers using each of the digits 3, 4, 5, 6, 7 exactly once. What is the difference between the place values of digit 7 in those two numbers?

1. $69993$
2. $41976$
3. $76536$
4. $9999$
5. $70$

Question 7

All sections within each shape have equal area. Count only the flower, circle, or polygon itself, excluding its surroundings. Which fraction below is in simplest form and represents the shaded part of at least one shape?

1. $\frac6{13}$
2. $\frac4{10}$
3. $\frac14$
4. $\frac23$
5. $\frac12$

Question 8

A tessellation covers a plane without gaps or overlaps; a triangle example is shown. Consider three identical regular pentagons. In how many ways can they be arranged to form a tessellation? Interpret the question as asking whether identical regular pentagons can tile the plane, with three meeting at a vertex.

1. Zero
2. One
3. Two
4. Four
5. Infinitely many

Question 9

Which of the listed sections of the distance–time graph has the highest average rate of change?

1. $A$ to $B$
2. $B$ to $C$
3. $A$ to $C$
4. $D$ to $E$
5. $E$ to $F$

Question 10

The triangular paper is folded and punched as shown. Which pattern of holes appears when it is unfolded?

1. 
2. 
3. 
4. 
5. None of these

Question 11

Which Venn diagram represents soccer players, high-school mathematics enthusiasts, and students? Use the usual relationships: all high-school mathematics enthusiasts are students; soccer players can belong to either group or lie outside the student group.

1. Diagram A
2. Diagram B
3. Diagram C
4. Diagram D
5. None of these

Question 12

A movie theatre has 40 people. Of them, 26 like popcorn but not chips, and 32 like popcorn. Everyone likes at least one of the two snacks. How many like both? Clarification: the source switches from “people” to “students”; all counts are interpreted as referring to the same 40 people.

1. $6$
2. $8$
3. $32$
4. $40$
5. Indeterminate

Question 13

The surface areas of two spheres are in the ratio $4:9$. What is the ratio of their volumes, in the same order?

1. $2:3$
2. $16:31$
3. $3:2$
4. $4:9$
5. $8:27$

Question 14

How many three-digit natural numbers are divisible by 7?

1. $141$
2. $142$
3. $143$
4. $128$
5. $129$

Question 15

A lattice point has integer coordinates $x$ and $y$. How many lattice points lie on the line through $(2,4)$ and $(3,6)$? The entire line, not just the segment, is intended.

1. $12$
2. $6$
3. $1$
4. $2$
5. Infinitely many

Question 16

A positive integer has 2021 digits. Let $P$ be its digit sum, $Q$ the digit sum of $P$, and $R$ the digit sum of $Q$. Each is one ordinary sum, not a repeated digital-root operation. Find the largest possible $Q$.

Question 17

Given $\overline{XYZ}+\overline{ZYX}=\overline{ABAA}$, where $A,B,X,Y,Z$ are pairwise distinct positive digits, how many possible combinations of $X$ and $Z$ satisfy the equation? Count ordered pairs: swapping $X$ and $Z$ is a different assignment.

Question 18

A steel cuboid measuring 270 cm by 100 cm by 64 cm is reshaped into a cube without losing material. Find the sum of the digits of the cube’s surface area measured in square centimetres.

Question 19

The ratio $(3x-4):(y+15)$ is fixed. When $x=\frac73$, $y=12$. Find $x+y$ when $x=2$.

Question 20

What is the sum of the prime numbers between 2000 and 2021?