Answer Keys (Unofficial)

5) 10

14) 40

20) 43

23) 36

26) 19

27) 91

28) 67

29) 95

30) 18

Problem Set

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 $X_n={\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots, \sqrt{4 n+1000}}$. Let $$a=\max\{M_n: 1 \leq n \leq 1000\} \text {, and } b=\min \{M_n: 1 \leq n \leq 1000\} \text {. }$$
   Find $a-b$.
2. Find the number of elements in the set
   $$ \{(a, b) \in \mathbb{N}: 2 \leq a, b \leq 2023, \log _a(b)+6 \log _b(a)=5\}$$
3. Let $\alpha$ and $\beta$ be positive integers such that
   $$
   \frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16}
   $$
   Find the smallest possible value of $\beta$.
4. Let $x, y$ be positive integers such that
   $$
   x^4=(x-1)\left(y^3-23\right)-1 .
   $$
   Find the maximum possible value of $x+y$.
5. In a triangle $A B C$, let $E$ be the midpoint of $A C$ and $F$ be the midpoint of $A B$. The medians $B E$ and $C F$ intersect at $G$. Let $Y$ and $Z$ be the midpoints of $B E$ and $C F$ respectively. If the area of triangle $A B C$ is 480 , find the area of triangle $G Y Z$.
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 $X$.
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.
8. Given a $2 \times 2$ tile and seven dominoes ( $2 \times 1$ tile), find the number of ways of tiling (that is, cover without leaving gaps and without overlapping of any two tiles) a $2 \times 7$ rectangle using some of these tiles.
9. Find the number of triples $(a, b, c)$ of positive integers such that
   (a) $a b$ is a prime; 23
   (b) $b c$ is a product of two primes; $3 \times 5=15$
   (c) $a b c$ is not divisible by square of any prime and
   (d) $a b c \leq 30$.
10. The sequence $\left\langle a_n\right\rangle_{n \geq 0}$ is defined by $a_0=1, a_1=-4$ and $a_{n+2}=-4 a_{n+1}-7 a_n$, for $n \geq 0$. Find the number of positive integer divisors of $a_{50}^2-a_{49} a_{51}$.
11. A positive integer $m$ has the property that $m^2$ is expressible in the form $4 n^2-5 n+16$ where $n$ is an integer (of any sign). Find the maximum possible value of $|m-n|$.
12. Let $P(x)=x^3+a x^2+b x+c$ be a polynomial where $a, b, c$ are integers and $c$ is odd. Let $p_i$ be the value of $P(x)$ at $x=i$. Given that $p_1^3+p_2^3+p_3^3=3 p_1 p_2 p_3$, find the value of $p_2+2 p_1-3 p_0$.
13. The ex-radii of a triangle are $10 \frac{1}{2}, 12$ and 14. If the sides of the triangle are the roots of the cubic $x^3-p x^2+q x-r=0$, where $p, q, r$ are integers, find the integer nearest to $\sqrt{p+q+r}$.
14. Let $A B C$ be a triangle in the $x y$ plane, where $B$ is at the origin $(0,0)$. Let $B C$ be produced to $D$ such that $B C: C D=1: 1, C A$ be produced to $E$ such that $C A: A E=1: 2$ and $A B$ be produced to $F$ such that $A B: B F=1: 3$. Let $G(32,24)$ be the centroid of the triangle $A B C$ and $K$ be the centroid of the triangle $D E F$. Find the length $G K$.
15. Let $A B C D$ be a unit square. Suppose $M$ and $N$ are points on $B C$ and $C D$ respectively such that the perimeter of triangle $M C N$ is 2. Let $O$ be the circumcentre of triangle $M A N$, and $P$ be the circumcentre of triangle $M O N$. If $\left(\frac{O P}{O A}\right)^2=\frac{m}{n}$ for some relatively prime positive integers $m$ and $n$, find the value of $m+n$.
16. The six sides of a convex hexagon $A_1 A_2 A_3 A_4 A_5 A_6$ are colored red. Each of the diagonals of the hexagon is colored either red or blue. If $N$ is the number of colorings such that every triangle $A_i A_j A_k$, where $1 \leq i<j<k \leq 6$, has at least one red side, find the sum of the squares of the digits of $N$.
17. Consider the set
    $$
    \mathcal{S}={(a, b, c, d, e): 0<a<b<c<d<e<100}
    $$
    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 $\mathcal{S}$, find the largest integer less than or equal to $D$.
18. Let $\mathcal{P}$ be a convex polygon with 50 vertices. A set $\mathcal{F}$ of diagonals of $\mathcal{P}$ is said to be minimally friendly if any diagonal $d \in \mathcal{F}$ intersects at most one other diagonal in $\mathcal{F}$ at a point interior to $\mathcal{P}$. Find the largest possible number of elements in a minimally friendly set $\mathcal{F}$.
19. For $n \in \mathbb{N}$, 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
    $A={n \in \mathbb{N}: P(n) \text{ is non-zero, square free and } S(n) \text{ is a proper divisor of }P(n)}$.
    Find the maximum possible number of digits of the numbers in $A$.
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
    $$
    \begin{aligned}
    & \max (A) \times|B|=12 ; \text { and } \
    & |A| \times \max (B)=11
    \end{aligned}
    $$
    and $N$ can be written as $100 a+b$ where $a, b$ are positive integers less than 100 , find $a+b$.
21. For $n \in \mathbb{N}$, consider non-negative integer-valued functions $f$ on ${1,2, \ldots, n}$ satisfying $f(i) \geq f(j)$ 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?
22. In an equilateral triangle of side length 6 , pegs are placed at the vertices and also evenly along each side at a distance of 1 from each other. Four distinct pegs are chosen from the 15 interior pegs on the sides (that is, the chosen ones are not vertices of the triangle) and each peg is joined to the respective opposite vertex by a line segment. If $N$ denotes the number of ways we can choose the pegs such that the drawn line segments divide the interior of the triangle into exactly nine regions, find the sum of the squares of the digits of $N$.
23. In the coordinate plane, a point is called a lattice point if both of its coordinates are integers. Let $A$ be the point $(12,84)$. Find the number of right angled triangles $A B C$ in the coordinate plane where $B$ and $C$ are lattice points, having a right angle at the vertex $A$ and whose incenter is at the origin $(0,0)$.
24. A trapezium in the plane is a quadrilateral in which a pair of opposite sides are parallel. A trapezium is said to be non-degenerate if it has positive area. Find the number of mutually non-congruent, non-degenerate trapeziums whose sides are four distinct integers from the set ${5,6,7,8,9,10}$.
25. Find the least positive integer $n$ such that there are at least 1000 unordered pairs of diagonals in a regular polygon with $n$ vertices that intersect at a right angle in the interior of the polygon.
26. In the land of Binary, the unit of currency is called Ben and currency notes are available in denominations $1,2,2^2, 2^3, \ldots$ 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 Govermment.
27. A quadruple $(a, b, c, d)$ of distinct integers is said to be balanced if $a+c=b+d$. Let $\mathcal{S}$ be any set of quadruples $(a, b, c, d)$ where $1 \leq a<b<d<c \leq 20$ and where the cardinality of $S$ is 4411. Find the least number of balanced quadruples in $\mathcal{S}$.
28. On each side of an equilateral triangle with side length $n$ units, where $n$ is an integer, $1 \leq n \leq 100$, 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.
29. A positive integer $n>1$ is called beautiful if $n$ can be written in one and only one way as $n=a_1+a_2+\cdots+a_k=a_1 \cdot a_2 \cdots a_k$ for some positive integers $a_1, a_2, \ldots, a_k$, where $k>1$ and $a_1 \geq a_2 \geq \cdots \geq a_k$. (For example 6 is beautiful since $6=3 \cdot 2 \cdot 1=3+2+1$, and this is unique. But 8 is not beautiful since $8=4+2+1+1=4 \cdot 2 \cdot 1 \cdot 1$ as well as $8=2+2+2+1+1=2 \cdot 2 \cdot 2 \cdot 1 \cdot 1$, so uniqueness is lost.) Find the largest beautiful number less than 100 .
30. Let $d(m)$ denote the number of positive integer divisors of a positive integer $m$. If $r$ is the number of integers $n \leq 2023$ for which $\sum_{i=1}^n d(i)$ is odd, find the sum of the digits of $r$.