Author: Kazi Abu Rousan Where are the zeros of zeta of s? G.F.B. Riemann has made a good guess; They're all on the critical line, saith he, And their density's one over 2 p log t. Source https://www.physicsforums.com/threads/a-poem-on-the-zeta-function.16280/ If you are a person who loves to read maths related stuff then sure you have came […]
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 […]
Question 1 One ticket to a show costs \($20\) at full price. Susan buys 4 tickets using a coupon that gives her a \(25%\) discount. Pam buys 5 tickets using a coupon that gives her a \(30%\) discount. How many more dollars does Pam pay than Susan? (a) 2 (b) 5 (c) 10 (d) 15 […]
Question 1 What is the value of \((2(2(2(2(2(2+1)+1)+1)+1)+1)+1)\) (a) 70 (b) 97 (c) 127 (d) 159 (e) 729 Question 2 Pablo buys popsicles for his friends. The store sells single popsicles for \($ 1\) each, 3popsicle boxes for \($ 2\) each, and 5-popsicle boxes for \($ 3\). What is the greatest number of popsicles that […]
Problem 1 What is the value of $$\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1 ?$$ (A) $\frac{5}{8}$(B) $\frac{11}{7}$(C) $\frac{8}{5}$(D) $\frac{18}{11}$(E) $\frac{15}{8}$ Answer: (B) $\frac{11}{7}$ Problem 2 Liliane has $50 \%$ more soda than Jacqueline, and Alice has $25 \%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alica have?(A) Liliane has $20 \%$ […]
Problem 1 What is the value of $(2(2(2(2(2(2+1)+1)+1)+1)+1)+1)$(A) 70(B) 97(C) 127(D) 159(E) 729 Answer: (C) 127 Problem 2 Pablo buys popsicles for his friends. The store sells single popsicles for $\$ 1$ each, 3popsicle boxes for $\$ 2$ each, and 5 -popsicle boxes for $\$ 3$. What is the greatest number of popsicles that Pablo […]
Problem 1. Let $x_1, x_2, x_3, \ldots$ be a sequence of positive integers defined as follows: $x_1=1$ and for each $n \geqslant 1$ we have $$x_{n+1}=x_n+\left\lfloor\sqrt{x_n}\right\rfloor$$ Determine all positive integers $m$ for which $x_n=m^2$ for some $n \geqslant 1$. (Here $\lfloor x\rfloor$ denotes the greatest integer less or equal to $x$ for every real number […]
1 What is the value of the following expression? 1+2-3+4+5-6+7+8-9+10+11-12 A. 18 B. 21 C. 24 D. 27 E. 30 Answer - A 2 In the array shown below, three 3 s are surrounded by 2 s, which are in turn surrounded by a border of 1 s . What is the sum of the […]
Problem 1What is the value of $\frac{(2112-2021)^{2}}{169}$ ?(A) 7(B) 21(C) 49(D) 64(E) 91 Answer: (C) 49 Problem 2Menkara 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 card be […]
Problem 1 What value of $\boldsymbol{x}$ satisfies $$x-\frac{3}{4}=\frac{5}{12}-\frac{1}{3} ?$$ (A) $-\frac{2}{3}$(B) $\frac{7}{36}$(C) $\frac{7}{12}$(D) $\frac{2}{3}$(E) $\frac{5}{6}$ Answer: (E) $\frac{5}{6}$ Problem 2 The numbers $3,5,7, a$ and $b$ have an average (arithmetic mean) of 15 . What is the average of $a$ and $b$ ?(A) 0(B) 15(C) 30(D) 45(E) 60 Answer: (C) 30 Problem 3 Assuming $a […]
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, […]
Question 1 Each morning of her five-day workweek, Jane bought either a 50 -cent muffin or a 75 -cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? (a) 1 (b) 2 (c) 3 (d) 4 (e) 5 Question 2 Which of the following is […]
Question 1 What is \(100(100-3)-(100 \cdot 100-3)\) ? (a) \(-20,000\) (b) \(-10,000\) (c) -297 (d) -6 (e) 0 Question 2 Makayla attended two meetings during her 9 -hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings? (a) […]
Question 1 What is \[ \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? \] (a) \(-1\) (b) \(\frac{5}{36}\) (c) \(\frac{7}{12}\) (d) \(\frac{147}{60}\) (e) \(\frac{43}{3}\) Question 2 Josanna's test scores to date are \(90,80,70,60,\) and 85. Her goal is to raise her test average at least 3 points with her next test. What is the minimum test score she would need to accomplish […]
Question 1 Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms? (a) 48 (b) 56 (c) 64 (d) 72 (e) 80 Question 2 A circle of radius 5 is inscribed in a rectangle as shown. […]
Question 1 What is \[ \frac{2+4+6}{1+3+5}-\frac{1+3+5}{2+4+6}? \] (a) \(-1\) (b) \(\frac{5}{36}\) (c) \(\frac{7}{12}\) (d) \(\frac{49}{20}\) (e) \(\frac{43}{3}\) Question 2 Mr.\ Green measures his rectangular garden by walking two of the sides and finds that it is 15 steps by 20 steps. Each of Mr.\ Green's steps is 2 feet long. Mr.\ Green expects half a […]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
Question 1 Leah has 13 coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth? (a) 33 (b) 35 (c) 37 (d) 39 (e) 41 Question 2 What […]
Week 28 of the Hundred Miles Project got underway on Sunday with a focused and purposeful energy. The session served primarily as a baseline assessment across all three learning modules, establishing the starting points against which each learner's growth will be measured in the weeks ahead through the Learning Velocity Indicator — a new assessment […]
Question 1 What is the value of $$ \frac{2 a^{-1}+\frac{a^{-1}}{2}}{a} $$ when $a=\frac{1}{2}$ ? (a) 1 (b) 2 (c) $\frac{5}{2}$ (d) 10 (e) 20 Question 2 If $n \circlearrowleft m=n^{3} m^{2}$, what is $\frac{294}{49^{2}}$ ? (a) $\frac{1}{4}$ (b) $\frac{1}{2}$ (c) 1 (d) 2 (e) 4 Question 3 Let $x=-2016$. What is the value of $|||x|-x|-|x||-x$ […]