Difference between revisions of "2023 IOQM Problems"
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==Problem 21== | ==Problem 21== | ||
For n ∈ N, consider non-negative integer valued function f on {1,2,...,n} satisfying <math>f(i)\ge f(j)</math> for <math>i>j</math> and <math>\sum_{i=1}^{n} i+f(i) = 2023</math>. Choose n such that <math>\sum_{i=1}^{n}f(i)</math> is the least. How many such functions exist in that case? | For n ∈ N, consider non-negative integer valued function f on {1,2,...,n} satisfying <math>f(i)\ge f(j)</math> for <math>i>j</math> and <math>\sum_{i=1}^{n} i+f(i) = 2023</math>. Choose n such that <math>\sum_{i=1}^{n}f(i)</math> is the least. How many such functions exist in that case? | ||
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==Problem 22== | ==Problem 22== | ||
==Problem 23== | ==Problem 23== |
Revision as of 13:45, 3 September 2024
Contents
- 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 13
- 13 Problem 15
- 14 Problem 16
- 15 Problem 17
- 16 Problem 18
- 17 Problem 19
- 18 Problem 20
- 19 Problem 21
- 20 Problem 22
- 21 Problem 23
- 22 Problem 24
- 23 Problem 25
- 24 Problem 26
- 25 Problem 27
- 26 Problem 28
- 27 Problem 29
- 28 Problem 30
Problem 1
Let be a positive integer such that . Let be the number of integers in the set
. Let , and .
Find .
Problem 2
Find the number of elements in the set
Problem 3
Let α and β be positive integers such thatFind the smallest possible value of β .
Problem 4
Let be positive integers such that Find the maximum possible value of .
Problem 5
In a triangle , let be the midpoint of and be the midpoint of . The medians and intersect at . Let and be the midpoints of and respectively. If the area of triangle is , find the area of triangle .
Problem 6
Let be the set of all even positive integers such that the measure of the angle of some regular polygon is degrees. Find the number of elements in .
Problem 7
Unconventional dice are to be designed such that the six faces are marked with numbers from to with and appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.
Problem 8
Given a 2 x 2 tile and seven dominoes (2 x 1 tile), find the number of ways of tilling a 2 x 7 rectangle using some of these tiles.
Problem 9
Find the number of triples of positive integers such that (a) is a prime;
(b) is a product of two primes;
(c) is not divisible by square of any prime and
(d)
Problem 10
The sequence is defined by , , and Find the number of positive integer divisors of .
Problem 11
A positive integer haas the property that is expressible in the form , where n is an integer. Find the maximum value of .
Problem 13
The ex-radii of a triangle are 10 1/ 2, 12 and 14. If the sides of the triangle are the roots of the cubic , where are integers, find the integer nearest to
Problem 15
Problem 16
Problem 17
Consider the setwhere are integers. If is the average value of the fourth element of such a tuple in the set, taken over all the elements of S, find the largest integer less than or equal to D.
Problem 18
Let be a convex polygon with vertices. A set of diagonals of is said to be minimally friendly if any diagonal ∈ intersects at most one other diagonal in at a point interior to . Find the largest possible number of elements in a minimally friendly set .
Problem 19
For ∈ , let denote the product of the digits in and denote the sum of the digits in . Consider the set . Find the maximum possible number of digits of the numbers in .
Problem 20
For any finite non empty set of integers, let denote the largest element of and denote the number of elements in . If is the number of ordered pairs of finite non-empty sets of positive integers, such that × ; and × and can be written as where are positive integers less than , find .
Problem 21
For n ∈ N, consider non-negative integer valued function f on {1,2,...,n} satisfying for and . Choose n such that is the least. How many such functions exist in that case?
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
In the land of Binary, the unit of currency is called Ben and currency notes are available in denominations 1, 2, 2 2 , 2 3 , . . . Bens. The rules of the Government of Binary stipulate that one can not use more than two notes of any one denomination in any transaction. For example, one can give a change for 2 Bens in two ways: 2 one Ben notes or 1 two Ben note. For 5 Ben one can give 1 one Ben note and 1 four Ben note or 1 one Ben note and 2 two Ben notes. Using 5 one Ben notes or 3 one Ben notes and 1 two Ben notes for a 5 Ben transaction is prohibited. Find the number of ways in which one can give change for 100 Bens, following the rules of the Government.
Problem 27
Problem 28
On each side of an equilateral triangle with side length n units, where n is an integer, , consider points that divide the side into n equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a net of equilateral triangles of side length one unit. On each of the vertices of these small triangles, place a coin head up. Two coins are said to be adjacent if the distance between them is 1 unit. A move consists of flipping over any three mutually adjacent coins. Find the number of values of n for which it is possible to turn all coins tail up after a finite number of moves.
Problem 29
A positive integer is called if can be written in one and only one way as for some positive integers , where and . (For example 6 is beautiful since 6 = 3 · 2 · 1 = 3 + 2 + 1 , and this is unique. But 8 is not beautiful since 8 = 4 + 2 + 1 + 1 = 4 · 2 · 1 · 1 as well as 8 = 2 + 2 + 2 + 1 + 1 = 2 · 2 · 2 · 1 · 1 , so uniqueness is lost.) Find the largest beautiful number less than 100.
Problem 30
Let denote the number of positive integer divisors of a positive integer . If is the no of integers for which is odd, find the sum of the digits of .