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Problem If $\sum_{n=1}^{\infty} \frac{1}{n^2} =\frac{{\pi}^2}{6}$ then $\sum_{n=1}^{\infty} \frac{1}{(2n-1)^2}$ is equal to (A) $\frac{{\pi}^2}{24}$ (B) $\frac{{\pi}^2}{8}$ (C) $\frac{{\pi}^2}{6}$ (D) $\frac{{\pi}^2}{3}$ Hint Try to write the summation as sum of square of reciprocal of odd numbers and even numbers and take the advantage of the infinite sum Solution $\sum_{n=1}^{\infty} \frac{1}{n^2} =\frac{{\pi}^2}{6}$ $\Rightarrow \sum_{n=1}^{\infty} \frac{1}{(2n)^2} + \sum_{n=1}^{\infty} \frac{1}{(2n-1)^2}= […]

Probability theory is nothing but common sense reduced to calculation. Pierre-Simon Laplace Today we will be discussing a problem from the second chapter of A First Course in Probability(Eighth Edition) by Sheldon Ross. Let's see what the problem says: Describing the Problem The problem(prob-48) says: Given 20 people, what is the probability that among the […]

PROBLEM Let $p$ be an odd prime.Then the number of positive integers less than $2p$ and relatively prime to $2p$ is: (A)$p-2$ (B) $\frac{p+1}{2} $(C) $p-1$(D)$p+1$ SOLUTION This is a number theoretic problem .We can solve this problem in 2 different methods. Let us see them both one by one Method -1 Let us look […]

There should be no such thing as boring mathematics. Edsger W. Dijkstra In one of our previous post, we have discussed on Mandelbrot Set. That set is one of the most beautiful piece of art and mystery. At the end of that post, I have said that we can calculate the value of $\pi $ […]

Author: Kazi Abu Rousan The pure mathematician, like the musician, is a free creator of his world of ordered beauty. Bertrand Russell Today we will be discussing one of the most fascinating idea of number theory, which is very simple to understand but very complex to get into. Today we will see how to find […]

Problem Let $f$:$\mathbb{R} \rightarrow \mathbb{R}$ be a continous function such that for all$x \in \mathbb{R}$ and all $t\geq 0$ f(x)=f(ktx) where $k>1$ is a fixed constant Hint Case-1 choose any 2 arbitary nos $x,y$ using the functional relationship prove that $f(x)=f(y)$ Case-2 when $x,y$ are of opposite signs then show that $$f(x)=f(\frac{x}{2})=f(\frac{x}{4})\dots$$ use continuity to […]

PROBLEM Let $f (0,\infty)\rightarrow \mathbb{R}$ be a continous function such that for all $x \in (0,\infty)$ $f(x)=f(3x)$ Define $g(x)= \int_{x}^{3x} \frac{f(t)}{t}dt$ for $x \in (0,\infty)$ is a constant function HINT Use leibniz rule for differentiation under integral sign SOLUTION using leibniz rule for differentiation under integral sign we get $g'(x)=f(3x)-f(x)$ $\Rightarrow g'(x)=0$ [ Because f(3x)=f(x)] […]

PROBLEM Show that there are exactly $2$ numbers $a$ in the set $\{1,2,3\dots9400\}$ such that $a^2-a$ is divisible by $10000$ HINT Use Modular arithmetic and concepts of coprime numbers SOLUTION we know $10000=2^4*5^4$ In order for $10000$ to divide $a^2-a$ both $2^4$ and $5^4$ must divide $ a^2-a $ Write $a^2-a=a(a-1)$ Note that $a$ and […]

Author: Kazi Abu Rousan $\pi$ is not just a collection of random digits. $\pi$ is a journey; an experience; unless you try to see the natural poetry that exists in $\pi$, you will find it very difficult to learn. Today we will see a python code to find the value of $\pi $ up to […]

Author: Kazi Abu Rousan Pi is not merely the ubiquitous factor in high school geometry problems; it is stitched across the whole tapestry of mathematics, not just geometry’s little corner of it. $\pi$ is truly one of the most fascinating things exist in mathematics. It's not just there in geometry, but it's also there in pendulum, […]

Access Australian Mathematics Competition past year paper of 2014 year 5 - 6 Upper Primary to sharpen your problem-solving skills.

Access Australian Mathematics Competition past year paper of 2013 year 5 - 6 Upper Primary to sharpen your problem-solving skills.

Access Australian Mathematics Competition past year paper of 2012 year 5 - 6 Senior to sharpen your problem-solving skills.

Access Australian Mathematics Competition past year paper of 2011 year 5 - 6 Upper Primary to sharpen your problem-solving skills.

Access Australian Mathematics Competition past year paper of 2010 year 5 - 6 Upper Primary to sharpen your problem-solving skills.

Access Australian Mathematics Competition past year paper of 2009 year 5 - 6 Upper Primary to sharpen your problem-solving skills.

Access Australian Mathematics Competition past year paper of 2008 year 5 - 6 Upper Primary to sharpen your problem-solving skills.

Dive into the discussion of the solution to Problem 17 from the 2023 AMC (Australian Mathematics Competition), Intermediate Year category.

Dive into the discussion of the solution to Problem 27 from the 2023 AMC (Australian Mathematics Competition), Junior Year category.

American Math Competition 8 (AMC 8) 2024 Problems, Solutions, Concepts and discussions.