AMO 2027 Grades 6-7 SET 12 - Self-study

What is the remainder of $1234^{2017}:7$?


A. 1

B. 2

C. 3

D. 4

E. None of the above

Find the missing number.

a) Four 2 × 2 grids, rows top to bottom: [3, 9; 11, 6], [7, 21; 14, 9], [5, 15; 16, 11], [4, A; 20, C].

b) Three central squares with surrounding numbers (top, left, right, bottom; centre): (12, 18, 3, 12; 45), (19, ?, 13, 12; 55), (4, 8, 9, 3; 24).





**Editorial input note:** Enter A, C, and the missing number in (b), in that order, separated by semicolons with no spaces.

There are a total of 1290 children in 2 childcare centers, A and B. 20% of the children in Center A and 40% of the children in Center B are girls. If both centers have 336 girls, how many children does Center B have?

In the diagram ST is parallel to UV.
What is the value of x?

Let n be a whole number greater than 1. It leaves a remainder of 1 when divided by any single digit whole number greater than 1. Find the smallest possible value of n.

In the figure below, the shaded part is the overlapping part of the two identical squares. The perimeter of the shaded part is 26cm. What is the perimeter of the rectangle?

Four children met together: Adam, Bobby, Charley and Daniel. Adam shook hands with 3 of these children, Bobby shook hands with 2, and Charley shook hands with 1. How many of the children’s hands did David shake?

**Editorial note for this self-study exercise:** Interpret “David” as “Daniel”, the fourth child listed above.

Joel bought 4 cakes and 3 packs of candy for $\$93$. If each cake cost $\$4$ more than each pack of candy, find the cost of 1 pack of candy.

Evaluate the following expression.

$44\times22-22\times66+66\times44$.


A. 0

B. 2420

C. 3388

D. 5324

E. None of the above

Here, we want to evaluate greatest common divisor of 4200, 3780, and 3528. First, we try to factorize these three numbers.

Three students A, B and C are asked to write down the height of a child, the circumference of a circle, the volume of a cup and the weight of a ball. Their responses are tabulated below:

Student | Height of the child (cm) | Circumference of the circle (cm) | Volume of the cup (cm³) | Weight of the ball (g)

A | 90 | 22 | 250 | 510

B | 70 | 21 | 245 | 510

C | 80 | 22 | 250 | 520

If each student has only two correct responses, and the height of the child is xcm, find the value of x.

On a particular afternoon, Eugene and Benjamin left their school at the same time and headed toward the bus stop. After running at an average speed of 8m/s for 30 second, Eugene ran past their classmate Dexter. Dexter was walking in the opposite direction the bus stop back to school, as he had left his pencil case in school. 20 seconds later, Benjamin walked past Dexter. Given that Benjamin was walking at an average speed of 1m/s, find Dexter’s average speed in m/s.

There are 61 mangoes in carton X and 59 mangoes in carton Y. How many mangoes must be transferred from carton X to carton Y so that the number of mangoes in carton Y is 3 times the number of mangoes in carton X?

The length of a glass tank is four times its width. Its height is twice its length. Find its volume if it has a height of 88cm. Give your answer in liters.

Daniel had some money. He spent $\$695$ on an iPhone. He then spent $\frac25$ of the remaining money on a pair of shoes. The amount of money spent was $\$65$ less than the amount of money he had left. How much did Daniel have at first?