Let $S$ be a non-empty finite set of positive integers satisfying:

$\displaystyle\left(\frac{1}{i}+\frac{1}{j}\right)\times \text{LCM}\left( i,j \right) \in S\ $$\ \ \ \forall \ i,j \in S$

Find the sum of the cardinalities of all possible $S$.

Pranjal has a challenge for Achintya. Let $\varepsilon$ denote the non-terminating decimal expansion of a fraction (Take $\frac{1}{4}$ as an example, its $\varepsilon$ is $0.24\overline{9}$). Pranjal asks him to write the $\varepsilon$ of $\frac{1}{k}$, for any natural number $k>1$.

Achintya is then required to find the number of digits in the non-repeating part of $\varepsilon$. (It is $2$ ($0.\textbf{24}\overline{9}$) in the case of $\frac{1}{4}$). Can you help him?

What would be the number of digits in the non-repeating part of $\varepsilon$ for k = $2^{1234}\times3^{2345}\times4^{3456}\times5^{4567}$

Let $f:\mathbb{N}\rightarrow\mathbb{R}$ which satisfies the following property:

$f(1) + 2^2 f(2) + $$3^3 f(3) + 4^4 f(4) $$…..n^n f(n)$$= \left( \frac{n^{n+1}}{3} + \frac{n^n}{2} + \frac{n^{n-1}}{6} \right) f(n).$

Find $\log_{10} \left(\frac{f(100)}{f(1)}\right)$.

Find number of all combinations $(a,b,c,p)$ of positive integers $a$,$b$,$c$ and prime number $p$ such that $$2^ap^b = (p+2)^c + 1$$ Assume $b$ $\leq$ 5 and $b$ $\leq$ $c$ . You might want to have a look at P-adic valuation and Lifting the exponent.

Evaluate the sum, $f(x) = $$\sum_{k=0}^{\infty}\left\lfloor \frac{2^k+x}{2^{k+1}} \right\rfloor$.

where $\lfloor x \rfloor$ denotes the Greatest integer Function.

Let $f(50.2)=n$. When $100$ coins are tossed, what is the probability that exactly $n$ are heads? Round your answer to two decimal places.

Let $f(n) = $$\sum_{m=0}^n\;\sum_{k=0}^m\;\binom{n}{k} \binom{n-k}{\lfloor\frac{m-k}{2}\rfloor} 2^k$ where $\lfloor x \rfloor$ denotes the Greatest integer Function and $\binom{n}{r}$ denotes the number of ways to choose r items from a set of n distinct items.

Find the value of \[\sum_{n=0}^{\infty}\;\frac{1}{f(n)}\]Round your answer to two decimal places.

There is a certain series whose 1st term, $s_0$, is $\frac{5}{2}$, and the i’th term $s_i$, is given by $s_i = s^2_{i-1}-2$ for $i\geq1$. Compute,

$$\prod_{i=0}^{\infty} \left(1-\frac{1}{s_i}\right)$$

Consider the Euclidean space $X = \mathbb{R}^n$ equipped with the metric (parameterized with the parameter $p \neq 0$) $$d_p(x, y) = \left(\frac{(x_1-y_1)^p+ (x_2-y_2)^p + ….. + (x_n-y_n)^p}{n}\right)^\frac{1}{p}$$ First, ensure that this is a valid metric for all $p \neq 0$. If it is not, answer $-1$. If it is, then assume $n=2$. If $$s = \sup_{y=0, x_1 = 1, x_2 > 0} \{p : d_p(x, y) \text{ is convex in variable $p$}\}$$ then find $-2s$. `Convex in $p$’ means fix $x=(1, x_2)$ and $y=(0, 0)$ and examine the behaviour of the metric as $p$ changes. \\(Hint: plug $p=1$, $p=-1$ and $p \rightarrow 0$ and see if the form of the metric seems familiar.)

References:

• Metric spaces
• Convexity
• Infimum and supremum

Let $\psi(k)$ denote the smallest positive integer such that for every $n \geq \psi(k)$ $(n, k \in \mathbb{Z})$ there always exists an integer of the form $p^4$ ($p \in \mathbb{Z}$) in the range $\left(n, k^2n\right]$. Find the value of $$\left(\sum_{k=2}^{2024}\psi(k)\right) – 20$$

Let $S$ be the set of ordered pairs $(x, y)$ such that $ 0 < x \leq 1, \quad 0 < y \leq 1, $ and $\left\lfloor \log_2 \left( \frac{1}{x} \right) \right\rfloor$, $\left\lfloor \log_5 \left( \frac{1}{y} \right) \right\rfloor$ are even. Given that the area of the graph of $S$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m + n$. The notation $\lfloor z \rfloor$ denotes the greatest integer that is less than or equal to $z$.

Let x,y,z,w be positive real numbers satisfying $$(x+z)(y+w)=xz+yw$$

Find the smallest possible value of $$P=\frac{x}{y}+\frac{y}{z}+\frac{z}{w}+\frac{w}{x}$$

For a positive integer $N$ , let $f_N(x)$ be the function defined by $$ f_N(x) = \sum_{n=0}^ {N} \frac{N +\frac{1}{2}-n}{(N+1)(2n+1)} \sin((2n+1)x) $$ Determine the smallest constant $M$ such that $f_N(x) \leq M$ for all $N$ and all real $x$. (Round your answer to two decimal places)

Let $d: \mathbb{N} \to \mathbb{N}$ be defined as follows: If $n = 2^{k} a_k + 2^{k-1} a_{k-1}$$ + \dots$$ +\ 2^{0} a_0$, where $a_i \in \{0, 1\}$, then, $d(n) = \sum a_i$. Let, \[ S = \sum_{k=1}^{2024} (-1)^{d(k)} k^3 \] Let, $P = S \mod 2024$, $P = uvwt$ (where $u, v, w, t$ are the digits of the number $P$). Two players each roll two standard dice, first player $A$, then player $B$. If player $A$ rolls a sum of $t$, they win. If player $B$ rolls a sum of $w$, they win. They take turns, back and forth, until someone wins. What is the probability that player $A$ wins? (Round your answer to two decimal places.)

Consider $\Delta ABC$ right-angled at $A$ with $BC=5$.
Let $BC$ be divided into $7$ segments of equal length.

Let the segment containing the midpoint of $BC$ be $l$.

Find $\frac{\tan \alpha }{h}$ where $\alpha$ is the angle subtended by $l$ at $A$ and $h$ is the altitude from $A$.

Let $ x_1, \dots, x_n $ and $ y_1, \dots, y_n $ be real numbers. Let $ A = (a_{ij})_{1 \leq i,j \leq n} $ be the matrix with entries defined as: $$ a_{ij} = \begin{cases} 1, & \text{if } x_i + y_j \geq 0, \newline 0, & \text{if } x_i + y_j < 0. \end{cases} $$ Suppose $ B $ is an $ n \times n $ matrix with entries $ 0 $ or $ 1 $ such that the sum of the elements in each row and each column of $ B $ is equal to the corresponding sum for the matrix $ A $. Determine the number of matrices $ B $ satisfying the above condition for any set of random numbers.

PV, KK, KV, and PJ play a game of tag. The game begins with PV being ‘it’ (PV is tagged initially), and each time some other player is tagged (Self-tags are not allowed). What are the number of distinct ways PV is ‘it’ again after 6 tags? (Note : PV can be ‘it’ in the middle also)

I have two strings x and y. String x is formed by using the letters of the word Atreya while y is formed by using the letters of the word Shivanshu. Let $n(i)$ denote the sum of the number of possibilities for x and y, when both are of length $i$. Find the sum of all possible values of gcd($n(j),n(k)$), where j and k are relatively prime.

Given $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n$, \[ \limsup_{n \to \infty} \left(\frac{\sum_{i=1}^{n}\sqrt{\frac{\sum_{k=i}^{n}x_{k}^{2}}{i}}}{\sqrt{n}\sum_{j=1}^{n}x_{j}}\right) = \, \, ? \]

Let Circles $S_1$ and $S_2$ touch internally and have centres on the x-axis. Let $C_0$ be the circle with centre on the x-axis touching both $S_1$ and $S_2$. Let $C_1$ be the circle touching $S_1$, $S_2$ and $C_0$. In general, let $C_n$ be the circle touching $S_1$, $S_2$ and $C_{n-1}$ . Suppose that $h_n$ is the distance of the centre of $C_n$ from the x-axis and $d_n$ is the diameter of $C_n$ . Compute- \[\frac{h_{100}}{d_{100}}\]

Let $a_1=1$, $a_n=\sum_{k=1}^{n-1}(n-k)a_k$, $\forall\ n>1$. Find $a_{100}$.
(Note: Some internet use (or a good calculator) might be required for the exact value.)

The roots of a $6^{th}$ degree polynomial (of the form $x+ iy$) lie on the ellipse $\frac{x^2}{2}$$+\frac{y^2}{3}=1$ such that they divide the ellipse into 6 regions of equal area. One of the roots is given to be $x=\sqrt{2}$. Find the product of roots.
(Note: Some internet use (or a good calculator) might be required for the exact value.)

Let $A$ and $E$ be opposite vertices of a regular octagon. A frog starts jumping from the vertex $A$. From any vertex except $E$ of the octagon, it can jump to either of the two adjacent vertices. When it reaches the vertex $E$, the frog stops and stays there. Let $e_n$ be the number of paths of exactly $n$ jumps that start at $A$ and end at $E$. Find \[ \lim_{n \to \infty} \frac{e_{2n+2}}{e_{2n}} \]

\[ f(n) = \begin{cases} n – 2, & \text{for } n > 3000; \newline f(f(n + 5)), & \text{for } n \leq 3000. \end{cases} \] Find the value of $f(2024)$

Let, $ab = \pi y$ Find the sum of first 6 significant figures of y. \[ a = \lim_{n \to \infty} \sum_{k=2}^n \ln\left(\frac{2^k}{2^k – 1}\right) \prod_{p \mid k} \frac{p – 1}{p}, \] where the product is taken over all prime divisors $p$ of $k$. \[ b = \lim_{n \to \infty} \sum_{k=1}^n \ln\left(\frac{2^k}{2^k – 1}\right) \sum_{d \mid k} \frac{f(d)}{d}, \] where the summation is taken over all divisors $d$ of k, \[ f(x) = \begin{cases} 1 & \text{if } x \text{ is a square-free positive integer with an even number of prime factors}, \newline -1 & \text{if } x \text{ is a square-free positive integer with an odd number of prime factors}, \newline 0 & \text{if } x \text{ has a squared prime factor}. \end{cases} \] A square-free positive integer is a positive integer that is not divisible by any perfect square other than 1