Difference between revisions of "2023 IOQM Problems"

Revision as of 16:38, 4 January 2024

Problem 1

Let $n$ be a positive integer such that $1 \leq n \leq 1000$ . Let $M_n$ be the number of integers in the set

$X_n = \left\{\sqrt{4n + 1}, \sqrt{4n + 2}, \ldots, \sqrt{4n + 1000}\right\}$ . Let $a = \max\{M_n : 1 \leq n \leq 1000\}$ , and $b = \min\{M_n : 1 \leq n \leq 1000\}$ .

Find $a - b$ .

Problem 2

Find the number of elements in the set

$\lbrace(a.b)\in N: 2 \leq a,b \leq2023,\:\: \log_{a}{b}+6\log_{b}{a}=5\rbrace$

Problem 3

Let α and β be positive integers such that $\frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16}$ Find the smallest possible value of β .

Problem 4

Let $x, y$ be positive integers such that $x^{4}=(x-1)(y^{3}-23)-1$ Find the maximum possible value of $x + y$ .

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Find the number of triples $(a, b, c)$ of positive integers such that (a) $ab$ is a prime;

(b) $bc$ is a product of two primes;

(c) $abc$ is not divisible by square of any prime and

(d) $abc\leq30$

Problem 10

The sequence $\{a_n\}_{n\geq0}$ is defined by $a_0 = 1$ , $a_1 = -4$ , and $a_{n+2} = -4a_{n+1} - 7a_n, \text{for } n \geq 0.$ Find the number of positive integer divisors of $a_{50}^2 - a_{49}a_{51}$ .

Problem 11

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 $x^3 − px^2 + qx − r = 0$ (Error compiling LaTeX. Unknown error_msg) , where $p, q, r$ are integers, find the integer nearest to $\sqrt{p+q+r}$

Problem 14

Problem 15

Problem 16

Problem 17

Consider the set $S = \{ (a, b, c, d, e) : 0 < a < b < c < d < e < 100 \}$ where $a, b, c, d, e$ are integers. If $D$ 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 $P$ be a convex polygon with $50$ vertices. A set $F$ of diagonals of $P$ is said to be minimally friendly if any diagonal $d$ ∈ $F$ intersects at most one other diagonal in $F$ at a point interior to $P$ . Find the largest possible number of elements in a minimally friendly set $F$ .

Problem 19

For $n$ ∈ $N$ , let $P(n)$ denote the product of the digits in $n$ and $S(n)$ denote the sum of the digits in $n$ . Consider the set $A = {n$ (Error compiling LaTeX. Unknown error_msg)∈$N : P(n) \text{is non-zero, square free and }S(n) \text{ is a proper divisor of } P(n) }$ (Error compiling LaTeX. Unknown error_msg) . Find the maximum possible number of digits of the numbers in $A$ .

Problem 20

For any finite non empty set X of integers, let max(X) denote the largest element of X and |X| denote the number of elements in X . If N is the number of ordered pairs (A, B) of finite non-empty sets of positive integers, such that max(A) × |B| = 12; and |A| × max(B) = 11 and N can be written as 100a + b where a, b are positive integers less than 100, find a + b .

Problem 21

For n ∈ N , consider non-negative integer-valued functions f on {1, 2, . . . , n} satisfying f(i) ≥ f(j) for i > j and Pn i=1 (i + f(i)) = 2023 . Choose n such that Pn i=1 f(i) 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, 1 ≤ n ≤ 100 , consider n − 1 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 $n > 1$ is called $beautiful$ if $n$ can be written in one and only one way as $n = a_1 + a_2 +... + a_k = a_1 a_2 ... a_k$ for some positive integers $a_1, a_2, . . . , a_k$ , where $k > 1$ and $a_1 \geq a_2 \geq ... \geq a_k$ . (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.

@@ Line 1: / Line 1: @@
-===Problem 1===
+==Problem 1==
 Let <math>n</math> be a positive integer such that <math>1 \leq n \leq 1000</math>. Let <math>M_n</math> be the number of integers in the set
@@ Line 7: / Line 7: @@
 Find <math>a - b</math>.
-===Problem 2===
+==Problem 2==
 Find the number of elements in the set
 <cmath>\lbrace(a.b)\in N: 2 \leq a,b \leq2023,\:\: \log_{a}{b}+6\log_{b}{a}=5\rbrace</cmath>
-===Problem 3===
+==Problem 3==
 Let α and β be positive integers such that<cmath>\frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16}</cmath>Find the smallest possible value of β .
-===Problem 4===
+==Problem 4==
 Let <math>x, y</math> be positive integers such that<cmath>x^{4}=(x-1)(y^{3}-23)-1</cmath>
 Find the maximum possible value of <math>x + y</math>.
-===Problem 5===
+==Problem 5==
-===Problem 6===
+==Problem 6==
-===Problem 7===
+==Problem 7==
-===Problem 8===
+==Problem 8==
-===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;
@@ Line 29: / Line 29: @@
 (d) <math>abc\leq30</math>
-===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>.
-===Problem 11===
+==Problem 11==
-===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>
-===Problem 14===
+==Problem 14==
-===Problem 15===
+==Problem 15==
-===Problem 16===
+==Problem 16==
-===Problem 17===
+==Problem 17==
 Consider the set<cmath>S = \{ (a, b, c, d, e) : 0 < a < b < c < d < e < 100 \}</cmath>where <math>a, b, c, d, e</math> are integers. If <math>D</math> 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===
+==Problem 18==
 Let <math>P</math> be a convex polygon with <math>50</math> vertices. A set <math>F</math> of diagonals of <math>P</math> is said to be minimally friendly if any diagonal <math>d</math> ∈ <math>F</math> intersects at most one other diagonal in <math>F</math> at a point interior to <math>P</math> . Find the largest possible number of elements in a minimally friendly set <math>F</math>.
-===Problem 19===
+==Problem 19==
 For <math>n</math> ∈ <math>N</math> , let <math>P(n)</math> denote the product of the digits in <math>n</math> and <math>S(n)</math> denote the sum of the digits in <math>n</math> . Consider the set <math>A = {n </math>∈<math> N : P(n) \text{is non-zero, square free and }S(n) \text{ is a proper divisor of } P(n) }</math> . Find the maximum possible number of digits of the numbers in <math>A</math> .
-===Problem 20===
+==Problem 20==
 For any finite non empty set X of integers, let max(X) denote the largest element of X and |X| denote the number of elements in X . If N is the number of ordered pairs (A, B) of finite non-empty sets of positive integers, such that max(A) × |B| = 12; and |A| × max(B) = 11 and N can be written as 100a + b where a, b are positive integers less than 100, find a + b .
-===Problem 21===
+==Problem 21==
 For n ∈ N , consider non-negative integer-valued functions f on {1, 2, . . . , n} satisfying f(i) ≥ f(j) for i > j and Pn i=1 (i + f(i)) = 2023 . Choose n such that Pn i=1 f(i) is the least. How many such functions exist in that case?
-===Problem 22===
+==Problem 22==
-===Problem 23===
+==Problem 23==
-===Problem 24===
+==Problem 24==
-===Problem 25===
+==Problem 25==
-===Problem 26===
+==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 27==
-===Problem 28===
+==Problem 28==
 On each side of an equilateral triangle with side length n units, where n is an integer, 1 ≤ n ≤ 100 , consider n − 1 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===
+==Problem 29==
 A positive integer <math>n > 1</math> is called <math>beautiful</math> if <math>n</math> can be written in one and only one way as <math>n = a_1 + a_2 +... + a_k = a_1 a_2 ... a_k</math> for some positive integers <math>a_1, a_2, . . . , a_k</math> , where <math>k > 1</math> and <math>a_1 \geq a_2 \geq ... \geq a_k</math> . (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===
+==Problem 30==

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