2024 IOQM Problems

Problem 1

The smallest positive integer that does not divide $1\times2\times3\times4\times5\times6\times7\times8\times9$ is:

Problem 2

The number of four-digit odd numbers having digits $1, 2, 3, 4$, each occuring exactly once, is:

Problem 3

The number obtained by taking the last two digits of $5^{2024}$ in the same order is:

Problem 4

Problem 5

Problem 6

Find the number of triples of real numbers $(a, b, c)$ such that $a^{20}+ b^{20} + c^{20} = a^{24} + b^{24} + c^{24} = 1$.

Problem 7

Determine the sum of all possible surface areas of a cube two of whose vertices are $(1, 2, 0)$ and $(3, 3, 2)$.

Problem 8

Let $n$ be the smallest integer such that the sum of digits of $n$ is divisible by $5$ as well as the sum of digits of $(n + 1)$ is divisible by $5$. What are the first two digits of $n$ in the same order?

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Three positive integers $a, b, c$ with $a>c$ satisfy the folowing equations: $ac + b + c = bc + a + 66, a + b + c = 32$. Find the value of $a$.

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

On a natural number $n$, you are allowed two operations: (1) multiply by 2 or (2) subtract 3 from $n$. For example starting with $8$ you can reach $13$ as follows: $8\rightarrow 16\rightarrow 13$.You need two steps and you cannot do in less than two steps. Starting from $11$, what is the least number of steps required to reach 121 ?

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30