Question bank IMC 2023

What is the simplified value of $1+(1+2+1)+(1+2+3+2+1)+\cdots+(1+2+3+\cdots+7+6+5+\cdots+2+1)$?

How many non-zero three-digit numbers have the sum of their digits equal to 6?

Clarification: each of the three digits is nonzero, as stated in the Vietnamese source.

A bookstore is planning to sell 600 notebooks, which were packed 40 notebooks in each box. After selling some, there were 4 packages remaining. Find the number of notebooks that were already sold.

Calculate $52+52\times0-0\times27-27$.

The figure is made up of 5 small identical rectangles. If the perimeter of the big rectangle is 48 cm, find the area of the big rectangle.

John bought a story book and count the digits in all the pages in the book (the book start from page 2), which have a total of 485. How many pages does this book have?

Grading convention: count the numbered pages actually listed, starting at 2; do not include an unnumbered page 1.

What is the value of $97\times97-98\times96$?

In a chess tournament, each player has to play one game with every other player. Five players, Albert, Ben, Charles, Dennis and Ethan participated in the tournament. So far during the tournament, Albert has played 4 games, Ben has played 3 games, Charles has played 2 games and Dennis has played 1 game. How many games has Ethan played?

Calculate $(1+2+3+\cdots+20)\times(1+2+3+\cdots+23):20:23:2023\times(2023-337\times6-1)$.

What is the simplified value of $76\times35+76\times47+82\times19+17\times95$?

A particular month has 5 Thursdays. The first and the last day of the month are not Thursday. What day is the first day of the month?

Knowing that $3\mathbin{\Phi}1=4$, $9\mathbin{\Phi}2=13$, $1\mathbin{\Phi}3=10$, calculate $4\mathbin{\Phi}5$.

Editorial grading convention: interpret $a\mathbin{\Phi}b$ as $a+b^2$, the simple rule matching the examples.

Arrange 256 students to form a square array. How many students are on the outer most side of the square?

Find the difference between $A$ and $B$ in the sequence $1,1,2,3,4,4,8,7,A,11,32,B,64,29,\ldots$.

Grading convention: take the nonnegative difference between the two missing values, using the indicated alternating number patterns.

In the equation below, each letter stands for a different digit. What is the highest possible value for H?

How many distinct 4-digit numbers are there such that sum of the first three digits is 20 and sum of the last three digits is 17?

Source-language note: the English and Vietnamese texts admit different readings of “distinct.” Both readings are analysed in the solution; the corresponding number is accepted.

Two boys, Robert and Dave play a game by taking turns in picking cards from a pokemon card box including 200 cards. Each turn, the boy have to “pick” by getting at least 1 but not more than 10 cards in whichever ways they want to pick. The one who gets the last card wins the game. If Robert goes first, how many cards should he pick on his first turn to ensure he will win the game?

Using each digits from 0 to 9 only once to make two different 5-digit number. What is the smallest possible difference between the two numbers?

The figure below is made of a number of blocks, some of them are invisible. Can you count how many invisible blocks are there?

Modelling convention: count the blocks needed to support the shown staircase in solid vertical columns; do not add extra blocks behind the depicted structure.

Which of the following is the sum of three consecutive natural numbers?