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 ?

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2024 IOQM Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30