IMC 2023 Grade 5 — ROUND 1

A meeting room can hold 120 people. How many people can 13 meeting rooms hold?

You can you each of the digits in the set {2, 3, 4, 5, 9, 6} once to make the greatest odd 6-digit number. What is the digit in the hundred place?

18 trees are equally spaced along one side of a straight road. The distance from the first tree to the thirteen tree is 60 m. What is the distance in meter between the first and last tree?

A is the greatest 2-digit number that is divisible by 2. B is the smallest 3-digit number that is divisible by 3. What is the product of A and B?

If $A*B$ means $\frac{A+2\times B}{2}$, then $(6*5)*7$ is

During recess, half of the students went outside, one fourth began to play cards in classroom. 5 students were left reading comic. How many students are there in this class?

$A$ and $B$ are whole numbers and $A\times B=72$. The largest possible value of $A+B$ is

What is the sum of all possible values of $Z$ so that the following expression is a positive integer? $\frac{64}{Z-6}$.

Editorial grading convention: take $Z$ to be an integer; the source does not specify its domain.

Three halves of the cost of a book is \$15 more than two thirds of it. How much does the book cost?

Mr. Zack is trying to cut a long log of wood into small pieces. After every cut, he takes a fifteen-second break. It takes him a total of 11 minutes to cut that log of wood into 10 pieces. How long (in minutes) would it take him to cut that log of wood into 18 pieces?

Editorial grading convention: cuts take equal time, and elapsed time ends at the final cut, before any subsequent rest.

Point A and point B are 9 cm apart. Point C and B are 5 cm apart. Point C and D are 2 cm apart. If point A and D are as close as possible, what would be the distance between them?

On Monday, a chicken walked from home to school at $\frac23$ m/s and he was 6 minutes late. On Tuesday, a dog chased after him, so he increased the speed to 1 m/s. As a result, he came to school 10 minutes in advance. How far is school from home?

Editorial modeling convention: both trips start at the same time, have the same required arrival time, and use the stated speed throughout.

A has 5 sisters and 6 brothers. One of A’s sister has s sisters and b brothers. What can be the greatest value of the product of s and b?

Mr. Zack walks to school every day. 10 trees are planted regularly all the way between his house and his school. He takes a half-a-minute break in the shade of every single tree. It takes him a total of 12 minutes to go from his house to the fifth tree. How long does it take him to get to school?

Editorial grading convention: the 10 trees are strictly between house and school; all 11 gaps, including the end gaps, are equal. Walking speed is constant. The 12 minutes ends on arrival at tree 5 before its rest, and the full trip ends on arrival at school.

If a worker makes 20 pencils in 4 minutes, he will go home 10 late. If he makes 30 pencils in 4 minutes, he will go home 8 minutes earlier. How many pencils does he have to make before he can go home?

A and B are siblings but of different genders. A has an even number of brothers and an even number of sisters. B has an odd number of brothers and an odd number of sisters. What is the minimum number of children are there in this family? (zero does not count).

“The stair way to heaven” is made up of 100 blocks of concrete of $1\times4$ cm in the way one block overlaps half way another. What is the perimeter of the whole staircase?

The number of pairs of red socks, the number of blue socks and the number of white socks Mr. Zack has are consecutive even numbers respectively. He always puts all of his socks in a box. It is known that when blinded, he has to take out 26 socks to make sure that he has exactly 1 pair of white socks. How many socks does he have in total? (The right sock is different from the left one and all the socks are paired).

Editorial grading convention: 26 is the minimum number of draws guaranteeing at least one white left-right pair. A white left sock can pair with any white right sock. Keep the printed distinction between red pairs and blue/white individual socks.