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AMO 2027 Grade 5 SET 1 - Self-study

Question 1

Brenda multiplies a number by 3, adds 3, divides by 3, adds 6, and subtracts 3 from the result to get 9. What is six times the original number?

Choose one option.
  1. 5
  2. 3
  3. 30
  4. 54
  5. 126

Question 2

Which statement is true? For option D, interpret the two heptagons as joined along one complete side, with no overlap, and count the outer boundary sides.

Choose one option.
  1. If a circle and a regular octagon have the same area, the radius of the circle equals the distance from the centre to a vertex of the octagon.
  2. The difference between the numbers of sides of an octagon and a trapezium is a prime number.
  3. All regular quadrilaterals have sides of equal length.
  4. Four more than the number of sides of a decagon equals the number of sides of the figure formed by joining two identical regular heptagons next to each other.
  5. Only right-angled triangles have area $(\text{base}\times\text{height})\div2$.

Question 3

Gerald likes to create patterns with dots. The first four figures are shown below. How many dots will the 16th figure contain?

Question illustration

Continue the same diamond-shaped outer-layer growth shown in the figures. Enter one whole number.

Question 4

Amy spent $\frac14$ of the cash in her wallet at the game store, $\frac23$ of the remaining cash at a clothing store, $\frac13$ of what was left for lunch, and $\frac34$ of what remained at the shoe store. After spending all that money, she had 10 dollars left. How much cash did she have at first?

Choose one letter A–E.
  1. 24 dollars
  2. 120 dollars
  3. 160 dollars
  4. 200 dollars
  5. 240 dollars

Question 5

Three pentagons are joined as shown. Every side is 3 cm. What is the difference between the sum of the three individual perimeters and the perimeter of the joined figure?

Question illustration

Choose one option.
  1. 12 cm
  2. 6 cm
  3. 45 cm
  4. 33 cm

Question 6

What is the greatest three-digit number divisible by both 32 and 36?

Choose one option.
  1. 988
  2. 888
  3. 864
  4. 998
  5. 846

Question 7

What is the positive difference between the sum of the multiples of 3 less than 100 and the sum of the multiples of 4 less than 100?

Use the nonnegative multiples below 100. Choose one option.
  1. 483
  2. 1200
  3. 1683
  4. 3366
  5. 2400

Question 8

Each of the whole numbers 1 through 9 is written on 9 cards, one number per card. Three cards are dealt to each of the three players. The sum of Player One's cards is 15. The sum of Player Two's cards is 23. What is the highest single card Player Three could have?

Enter only the whole-number answer, without units.

Question 9

Given $\blacktriangle+\blacksquare+\blacklozenge=14$, $\blacksquare+\blacklozenge+\bullet=19$, $\blacklozenge+\bullet+\blacktriangle=16$, and $\bullet+\blacktriangle+\blacksquare=11$, what is the value of $\blacktriangle$?

Question illustration

Choose one option.
  1. 1
  2. 4
  3. 6
  4. 9
  5. 20

Question 10

Tom’s PIN is a five-digit number with all digits different. The thousands digit is twice the ten-thousands digit; the hundreds digit is 50% more than the thousands digit; the tens digit is 1 more than the ten-thousands digit; and the ones digit is 2 more than the tens digit. What is Tom’s PIN divided by 65?

Choose one option.
  1. 379
  2. 24635
  3. 6565
  4. 3790
  5. 24700

Question 11

Jethro is now 3 times as old as his son Hezekiah. Six years ago, Jethro was 4 times as old as Hezekiah was then. Find the sum of their present ages.

Enter one whole number, without units.

Question 12

A $5\times5\times5$ cube is painted red on all its faces. Find the number of smaller $1\times1\times1$ cubes with at most two faces painted.

Enter only the whole-number answer, without units.

Question 13

Summer vacation lasts for 85 consecutive days. What is the greatest number of Fridays that could occur during the vacation?

Choose one option.
  1. 12
  2. 13
  3. 15
  4. 18
  5. 20

Question 14

The five-digit number 653A2 is a multiple of 3. How many possible digits A are there?

Enter one whole number, without units.

Question 15

Last holiday, Jonathan received 134 dollars more than three times the money Sally received. They received 710 dollars altogether. How much did Jonathan receive?

Choose one option.
  1. 144 dollars
  2. 432 dollars
  3. 566 dollars
  4. 576 dollars
  5. 288 dollars