IMC 2023 Grade 6 — ROUND 1

Three fourths of the people at a party are sitting in two thirds of the chairs. There are still 9 empty chairs. How many people are there at the party?

Four friends have a total of 6 identical pencils, and each one has at least one pencil. In how many ways can this happen?

Mr. Zack is think of a 3-digit number. If he subtract a 2-digit number from it, and then add another 3-digit number to the result, what can be the greatest value of the final result he can get?

Editorial grading convention: the two three-digit numbers are distinct; interpret “another” in this sense.

Katherine had 80 potatoes to peel. She started peeling at the rate of 12 potatoes in 2 minutes. 5 minutes later, her sister joined and peeled at the rate of 4 potatoes per minute. How many more potatoes does Katherine peel than her sister?

Joe gets 6 on a 10-problem test, 7 on a 30-problem test and 8 on a 60-problem test. If the three tests are combined into one 100-problem test, what is his score?

Editorial grading convention: 6, 7 and 8 are scores out of 10, with equally weighted problems.

What is the 2nd number in the arithmetic sequence: $10,\_\_\_,9,10,12,15,19,24,\ldots$?

Editorial grading convention: consecutive differences increase by 1 throughout the displayed list.

Mr. Zack has plenty of other numbers but only 13 2's. How far can he number the pages of his comic book?

Original choices: A. 91. B. 26. C. 62. D. 41.

Editorial response convention: none of the printed options matches the stated data; enter the numerical result.

A long rope of 72 cm is cut into 3 pieces in the ratio of 3:4:5. What is the length of the longest piece?

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 quadrilateral $ABCD$ is q trapezoid. $AE=\frac12 AB$, $BF=\frac23 BC$, and the area of $ADE$ is 2021, and the area of $DFC$ is 2022, then find the area of $DEF$.

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.