Difference between revisions of "2023 IOQM Problems"
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Find <math>a - b</math>. | Find <math>a - b</math>. | ||
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==Problem 2== | ==Problem 2== | ||
Find the number of elements in the set | Find the number of elements in the set | ||
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==Problem 6== | ==Problem 6== | ||
==Problem 7== | ==Problem 7== | ||
+ | Unconventional dice are to be designed such that the six faces are marked with numbers from <math>1</math> to <math>6</math> with <math>1</math> and <math>2</math> 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== | ==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. | + | 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== | ==Problem 9== | ||
− | Find the number of triples <math>(a, b, c)</math> of positive integers such that (a) <math>ab</math> is a prime; | + | Find the number of triples <math>(a, b, c)</math> of positive integers such that |
+ | (a) <math>ab</math> is a prime; | ||
(b) <math>bc</math> is a product of two primes; | (b) <math>bc</math> is a product of two primes; | ||
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==Problem 10== | ==Problem 10== | ||
The sequence <math>\{a_n\}_{n\geq0}</math> is defined by <math>a_0 = 1</math>, <math>a_1 = -4</math>, and <math>a_{n+2} = -4a_{n+1} - 7a_n, \text{for } n \geq 0.</math> Find the number of positive integer divisors of <math>a_{50}^2 - a_{49}a_{51}</math>. | The sequence <math>\{a_n\}_{n\geq0}</math> is defined by <math>a_0 = 1</math>, <math>a_1 = -4</math>, and <math>a_{n+2} = -4a_{n+1} - 7a_n, \text{for } n \geq 0.</math> Find the number of positive integer divisors of <math>a_{50}^2 - a_{49}a_{51}</math>. | ||
− | == | + | ==Problem 11== |
− | + | A positive integer <math>m</math> haas the property that <math>m^2</math> is expressible in the form <math>4n^2-5n+16</math>, where n is an integer. Find the maximum value of <math>|m-n|</math>. | |
− | |||
==Problem 12== | ==Problem 12== | ||
==Problem 13== | ==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 <math>x^3-px^2+qx-r=0</math> , where <math>p, q, r</math> are integers, find the integer nearest to <math>\sqrt{p+q+r}</math> | 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 <math>x^3-px^2+qx-r=0</math> , where <math>p, q, r</math> are integers, find the integer nearest to <math>\sqrt{p+q+r}</math> | ||
− | |||
− | |||
==Problem 15== | ==Problem 15== | ||
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For any finite non empty set <math>X</math> of integers, let <math>\text{max} (X)</math> denote the largest element of <math>X</math> and <math>|X|</math> denote the number of elements in <math>X</math> . If <math>N</math> is the number of ordered pairs <math>(A, B)</math> of finite non-empty sets of positive integers, such that <math>\text{max} (A)</math> × <math>|B|=12</math>; and <math>|A|</math> × <math>\text{max} (B)=11</math> and <math>N</math> can be written as <math>100a + b</math> where <math>a, b</math> are positive integers less than <math>100</math>, find <math>a + b</math> . | For any finite non empty set <math>X</math> of integers, let <math>\text{max} (X)</math> denote the largest element of <math>X</math> and <math>|X|</math> denote the number of elements in <math>X</math> . If <math>N</math> is the number of ordered pairs <math>(A, B)</math> of finite non-empty sets of positive integers, such that <math>\text{max} (A)</math> × <math>|B|=12</math>; and <math>|A|</math> × <math>\text{max} (B)=11</math> and <math>N</math> can be written as <math>100a + b</math> where <math>a, b</math> are positive integers less than <math>100</math>, find <math>a + b</math> . | ||
==Problem 21== | ==Problem 21== | ||
− | For | + | 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? |
+ | ==Solution== | ||
+ | We have <math>\sum_{i=1}^{n} i+f(i) = 2023</math>, so <math>\sum_{i=1}^{n} f(i) = 2023-\frac{n(n+1)}{2}</math>, for this to be least <math>n=63</math>. Where <math>\sum_{i=1}^{n} f(i)=7</math>. Total number of such functions in natural numbers is number of ways of adding up to <math>p(7)=\boxed{15}</math>. | ||
+ | |||
+ | ~Lakshya Pamecha | ||
==Problem 22== | ==Problem 22== |
Revision as of 03:05, 4 May 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 12
- 13 Problem 13
- 14 Problem 15
- 15 Problem 16
- 16 Problem 17
- 17 Problem 18
- 18 Problem 19
- 19 Problem 20
- 20 Problem 21
- 21 Solution
- 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
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
Problem 6
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 12
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?
Solution
We have , so
, for this to be least
. Where
. Total number of such functions in natural numbers is number of ways of adding up to
.
~Lakshya Pamecha
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.