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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, […]

Problem 1 (A) 0(B) 1(C) 2(D) 3(E) 4 Answer: (C) 2 Problem 2What is the hundreds digit of $(20!-15!)$ ?(A) 0(B) 1(C) 2(D) 4(E) 5 Answer: (A) 0 Problem 3Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was 5 times as old as Bonita. This […]

Problem 1What is the value of $9901 \cdot 101-99 \cdot 10101$ ?(A) 2(B) 20(C) 200(D) 202(E) 2020 Answer: (A) 2 Problem 2A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T=a L+b G$, where $a$ and $b$ are constants, […]

Problem 1 What is the value of $\frac{(2112-2021)^{2}}{169}$ ?(A) 7(B) 21(C) 49(D) 64(E) 91 Answer: (C) 49 Problem 2 Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by 1 inch, the card would have area 18 square inches. What would the area of the […]

Problem 1 Andy and Betsy both live in Mathville. Andy leaves Mathville on his bicycle at $1: 30$, traveling due north at a steady 8 miles per hour. Betsy leaves on her bicycle from the same point at $2: 30$, traveling due east at a steady 12 miles per hour. At what time will they […]

The AMC 8 (2005) is a 40-minute, 25-question multiple-choice contest for middle-school students (Grade 8 and below).
It tests problem-solving in arithmetic, algebra, geometry, counting, and probability (not complex calculus).

Problem 1 Andy and Betsy both live in Mathville. Andy leaves Mathville on his bicycle at $1: 30$, traveling due north at a steady 8 miles per hour. Betsy leaves on her bicycle from the same point at $2: 30$, traveling due east at a steady 12 miles per hour. At what time will they […]

Problem 1 What is the ones digit of $$222,222-22,222-2,222-222-22-2 ?$$ (A) 0(B) 2(C) 4(D) 6(E) 8 Answer: (B) 2 Problem 2 What is the value of this expression in decimal form? $$\frac{44}{11}+\frac{110}{44}+\frac{44}{1100}$$ (A) 6.4(B) 6.504(C) 6.54(D) 6.9(E) 6.94 Answer: (C) 6.54 Problem 3 Four squares of side length $4,7,9$, and 10 units are arranged in […]

AMC 8 2004 is a classic middle-school math contest featuring 25 engaging problems in algebra, geometry, counting, probability, and logical reasoning. It tests speed, accuracy, and smart problem-solving strategies in a fun competitive format.

Problem 1Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of 10 hours per week helping around the house for 6 weeks. For the first 5 weeks, she helps around the house for $8,11,7,12$ and 10 hours. How many hours must she work during the final […]