
Advent Calendar Archive
Penta-Week 1
Question 1
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\ $$\ \ \ \forall \ i,j \in S$
Find the sum of the cardinalities of all possible $S$.
Answer: 1
Question 2
Pranjal has a challenge for Achintya. Let $\varepsilon$ denote the non-terminating decimal expansion of a fraction (Take $\frac{1}{4}$ as an example, its $\varepsilon$ is $0.24\overline{9}$). Pranjal asks him to write the $\varepsilon$ of $\frac{1}{k}$, for any natural number $k>1$.
Achintya is then required to find the number of digits in the non-repeating part of $\varepsilon$. (It is $2$ ($0.\textbf{24}\overline{9}$) in the case of $\frac{1}{4}$). Can you help him?
What would be the number of digits in the non-repeating part of $\varepsilon$ for k = $2^{1234}\times3^{2345}\times4^{3456}\times5^{4567}$
Answer: 8146
Question 3
Let $f:\mathbb{N}\rightarrow\mathbb{R}$ which satisfies the following property:
$f(1) + 2^2 f(2) + $$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 $\log_{10} \left(\frac{f(100)}{f(1)}\right)$.
Answer: -196
Question 4
Answer: 1728
Question 5
Answer: 2
Penta-Week 2
Question 1
Evaluate the sum, $f(x) = $$\sum_{k=0}^{\infty}\left\lfloor \frac{2^k+x}{2^{k+1}} \right\rfloor$.
where $\lfloor x \rfloor$ denotes the Greatest integer Function.
Let $f(50.2)=n$. When $100$ coins are tossed, what is the probability that exactly $n$ are heads? Round your answer to two decimal places.
Answer: 0.08
Question 2
Let $f(n) = $$\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.
Answer: 1.33
Question 3
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,
$$\prod_{i=0}^{\infty} \left(1-\frac{1}{s_i}\right)$$
Answer: 0.43
Question 4
References:
Answer: -1
Question 5
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 $$\left(\sum_{k=2}^{2024}\psi(k)\right) – 20$$
Answer: 2024
Penta-Week 3
Question 1
Answer: 8
Question 2
Let x,y,z,w be positive real numbers satisfying $$(x+z)(y+w)=xz+yw$$
Find the smallest possible value of $$P=\frac{x}{y}+\frac{y}{z}+\frac{z}{w}+\frac{w}{x}$$
Answer: 14
Question 3
Answer: $\frac{\pi}{4}$ or 0.79
Question 4
Answer: 0.49
Question 5
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 $A$.
Answer: 0.12 or $\frac{7}{60}$
Penta-Week 4
Question 1
Answer: 1
Question 2
Answer: 183
Question 3
Answer: 14
Question 4
Answer: 2
Question 5
Answer: 100 or 0
Penta-Week 5
Question 1
(Note: Some internet use (or a good calculator) might be required for the exact value.)
Answer: 198th Fibonacci number
Question 2
(Note: Some internet use (or a good calculator) might be required for the exact value.)
Answer: -15.125
Question 3
Answer: $2+\sqrt{2}$
Question 4
Answer: 3000
Question 5
Answer: 35