Difference between revisions of "2023 IOQM Problems"

Revision as of 07:13, 6 September 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 $a - b$ .

Solution

Video link containing all the solutions for this paper https://youtu.be/NXzyDJKbM1k?si=uUByYIteDqHY9-_L

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

In a triangle $ABC$ , let $E$ be the midpoint of $AC$ and $F$ be the midpoint of $AB$ . The medians $BE$ and $CF$ intersect at $G$ . Let $Y$ and $Z$ be the midpoints of $BE$ and $CF$ respectively. If the area of triangle $ABC$ is $480$ , find the area of triangle $GYZ$ .

Problem 6

Let $X$ be the set of all even positive integers $n$ such that the measure of the angle of some regular polygon is $n$ degrees. Find the number of elements in $X$ .

Problem 7

Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ 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 $(a, b, c)$ 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

A positive integer $m$ haas the property that $m^2$ is expressible in the form $4n^2-5n+16$ , where n is an integer. Find the maximum value of $|m-n|$ .

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 $x^3-px^2+qx-r=0$ , where $p, q, r$ are integers, find the integer nearest to $\sqrt{p+q+r}$

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 $A=\{n \in N : P(n) \text{is non-zero, square free and } S(n) \text{ is a proper divisor of } P(n) \}$ . Find the maximum possible number of digits of the numbers in $A$ .

Problem 20

For any finite non empty set $X$ of integers, let $\text{max} (X)$ 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 $\text{max} (A)$ × $|B|=12$ ; and $|A|$ × $\text{max} (B)=11$ 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 function f on {1,2,...,n} satisfying $f(i)\ge f(j)$ for $i>j$ and $\sum_{i=1}^{n} i+f(i) = 2023$ . Choose n such that $\sum_{i=1}^{n}f(i)$ 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, $1 \le n \le 100$ , 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.

Problem 30

Let $d(m)$ denote the number of positive integer divisors of a positive integer $m$ . If $r$ is the no of integers $n \leq 2023$ for which $\sum_{i = 1}^{n} d(i)$ is odd, find the sum of the digits of $r$ .

@@ Line 6: / Line 6: @@
 Find <math>a - b</math>.
-==Solution==
-All of the sets from <math>1 \leq n \leq 1000</math> will have 1000 elements the most the number of square numbers will be in <math>X_1</math> and least number of perfect squares in <math>X_{1000}</math>. So <math>a=M_1</math> and <math>b=M_{1000}</math>. This is because the gaps between square number is less when n is greater then when n is less. By checking for <math>X_1</math> there would be squares from {<math>3^2, 4^2,..., 31^2</math>}, a total of 29 numbers while in <math>X_{1000}</math> there would be squares from {<math>64^2, 65^2,..., 70^2</math>} a total of 7 numbers, so <math>a=29</math> and <math>b=7</math>, giving us <math>\boxed{a-b=\textbf{22}}</math>
+== Solution ==
+Video link containing all the solutions for this paper https://youtu.be/NXzyDJKbM1k?si=uUByYIteDqHY9-_L
-~ Lakshya Pamecha
 ==Problem 2==
@@ Line 22: / Line 21: @@
 Find the maximum possible value of <math>x + y</math>.
 ==Problem 5==
+In a triangle <math>ABC</math>, let <math>E</math> be the midpoint of <math>AC</math> and <math>F</math> be the midpoint of <math>AB</math>.
+The medians <math>BE</math> and <math>CF</math> intersect at <math>G</math>. Let <math>Y</math> and <math>Z</math> be the midpoints of <math>BE</math>
+and <math>CF</math> respectively. If the area of triangle <math>ABC</math> is <math>480</math>, find the area of triangle <math>GYZ</math>.
 ==Problem 6==
+Let <math>X</math> be the set of all even positive integers <math>n</math> such that the measure of the angle of some regular polygon is <math>n</math> degrees. Find the number of elements in <math>X</math>.
 ==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.
-==Solution==
-We can fix 1 and 2 anywhere now we can permute the remaining faces like in a necklace (not a circle because anticlockwise and clockwise would be same), so the number of ways to permute them around the remaining 4 faces would be <math>\frac{3!}{2}</math> and number of ways to colour them <math>2\cdot2\cdot2</math> ways. So the answer would be <math>\boxed{\frac{3!}{2} \cdot8=\textbf{24}}</math>
-~Lakshya Pamecha
 ==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.
-==Solution==
-Let <math>a_n</math> be the total number of ways of tiling a 2 x n rectangle using only dominoes. So <math>a_n=a_{n-1}+a_{n-2}</math>. Obviously <math>a_1=1</math> and <math>a_2=2</math> (remember right now we are not considering the 2 x 2 tile). This will give us <math>a_7=21</math>. Now we introduce the 2 x 2 tile. Number of ways of tiling the 2 x 7 board will totally depend on the position of the 2 x 2 tile.
-*Case I: If the 2 x 2 tile is on the 1st column of the 2 x 7 board then the numbers of ways for tiling the rest of the board would be <math>a_5=8</math>.
-*Case II: If the 2 x 2 tile is on the 2nd column of the 2 x 7 board then numbers of ways of tiling the rest of the board will be <math>a_1a_4=5</math>.
-*Case III: If the 2 x 2 tile is on the 3rd column of the board then the number of ways of tiling the rest of the board will be <math>a_2a_3=6</math>.
-Now we will add all the cases and multiply by 2 (Why? Because the cases will repeat again). <math>2(8+5+6)=38</math>, remember this is when we consider a 2 x 2 tile, without the 2 x 2 tile there are <math>a_7=21</math> ways. So the total number of ways of tiling a 2 x 7 board using a 2 x 2 tile and 2 x 1 dominos is <math>\boxed{38+21=\textbf{59}}</math>
-~Lakshya Pamecha
 ==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;
@@ Line 53: / Line 44: @@
 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 12==
+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 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 10
 ==Problem 15==
@@ Line 71: / Line 61: @@
 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==
-For <math>n</math> ∈ <math>N</math> , consider non-negative integer-valued functions <math>f</math> on <math>{1, 2, . . . , n}</math> satisfying <math>f(i) \le f(j)</math> for <math>i > j</math> and Pn i=1 (i + f(i))=2023 . Choose <math>n</math> such that Pn i=1 f(i) 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?
 ==Problem 22==
 ==Problem 23==
@@ Line 86: / Line 75: @@
 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==
+Let <math>d(m)</math> denote the number of positive integer divisors of a positive integer <math>m</math>. If <math>r</math> is the no of integers <math>n \leq 2023</math> for which <math>\sum_{i = 1}^{n} d(i)</math> is odd, find the sum of the digits of <math>r</math>.

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