2024 IOQM Problems
Contents
[hide]- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
Problem 1
The smallest positive integer that does not divide is:
Problem 2
The number of four-digit odd numbers having digits , each occuring exactly once, is:
Problem 3
The number obtained by taking the last two digits of in the same order is:
Problem 4
Problem 5
Problem 6
Find the number of triples of real numbers such that
.
Problem 7
Determine the sum of all possible surface areas of a cube two of whose vertices are and
.
Problem 8
Let be the smallest integer such that the sum of digits of
is divisible by
as well as the sum of
digits of
is divisible by
. What are the first two digits of
in the same order?
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Three positive integers with
satisfy the folowing equations:
.
Find the value of
.
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
On a natural number , you are allowed two operations: (1) multiply by 2 or (2) subtract 3 from
. For example starting with
you can reach
as follows:
.You need two steps and you cannot do in less than two steps. Starting from
, what is the least number of steps required to reach 121 ?