Solutions of INMO 2023 Problem 1 Let $S$ be a finite set of positive integers. Assume that there are precisely $2023$ ordered pairs $(x, y)$ in $S \times S$ so that the product $xy$ is a perfect square. Prove that one can find at least four distinct elements in $S$ so that none of their […]
Understand how Cheenta Math Circles are critical for training students for math olympiads and other advanced mathematical competitions.
Problems, solutions and discussion on IOQM 2022 (India).
Being in grade 4, Mrinalini and Sharanyaa, have developed keen interest in Mathematics, such that they work with the students of Grade 4 and 5 from SSN, Sundarbans.
How to Prepare for Chennai Mathematical Institute, CMI? Learn from Avishek Hazra who got into the CMI Entrance 2022.
How to Prepare for Chennai Mathematical Institute, CMI? Learn from John Tom who got into the CMI Entrance 2022.
How to Prepare for Chennai Mathematical Institute, CMI? Learn from Dwitimaya Sahoo who got into the CMI Entrance 2022.
How to Prepare for Chennai Mathematical Institute, CMI? Learn from Mayur N Sastry who got into the CMI Entrance 2022.
How to Prepare for Chennai Mathematical Institute, CMI? Learn from Vemparala Bhuvan who got into the CMI Entrance 2022.
Question 1 How many integer values of $x$ satisfy $|x|<3 \pi$ ? (a) 9 (b) 10 (c) 18 (d) 19 (e) 20 Question 2 What is the value of $\sqrt{(3-2 \sqrt{3})^{2}}+\sqrt{(3+2 \sqrt{3})^{2}}$ ? (a) 0 (b) $4 \sqrt{3}-6$ (c) 6 (d) $4 \sqrt{3}$ (e) $4 \sqrt{3}+6$ Question 3 In an after-school program for juniors and […]
Question 1 What is the value of $1234+2341+3412+4123$ ? (a) 10,000 (b) 10,010 (c) 10,110 (d) 11,000 (e) 11,110 Question 2 What is the area of the shaded figure shown below? (a) 4 (b) 6 (c) 8 (d) 10 (e) 12 Question 3 The expression $\frac{2021}{2020}-\frac{2020}{2021}$ is equal to the fraction $\frac{p}{q}$ in which $p$ […]
Question 1 Define $x \diamond y$ to be $|x-y|$ for all real numbers $x$ and $y$. What is the value of $$ (1 \diamond(2 \diamond 3))-((1 \diamond 2) \diamond 3) ? $$ (a) -2 (b) -1 (c) 0 (d) 1 (e) 2 Question 2 In rhombus $A B C D$, point $P$ lies on segment […]
Question 1 Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely full but runs out of juice when the fourth glass is only $\frac{1}{3}$ full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the […]
Question 1 In a long line of people arranged left to right, the $1013^{\text {th }}$ person from the left is also the $1010^{\text {th }}$ person from the right. How many people are in the line? (a) 2021 (b) 2022 (c) 2023 (d) 2024 (e) 2025 Question 2 What is $10!-7!\cdot 6!$ ? (a) […]
Question 1 The instructions on a $350-$ gram bag of coffee beans say that proper brewing of a large mug of pour-over coffee requires 20 grams of coffee beans. What is the greatest number of properly brewed large mugs of coffee that can be made from the coffee beans in that bag? (a) 16 (b) […]
Question 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 […]
Question 1 In the year 2001, the United States will host the International Mathematical Olympiad. Let $I, M$, and $O$ be distinct positive integers such that the product $I \cdot M \cdot O=2001$. What's the largest possible value of the sum $I+M+O$ ? (a) 23 (b) 55 (c) 99 (d) 111 (e) 671 Question 2 […]
Question 1 The median of the list \[ n, n+3, n+4, n+5, n+6, n+8, n+10, n+12, n+15 \] is 10 . What is the mean? (a) 4 (b) 6 (c) 7 (d) 10 (e) 11 Question 2 A number \(x\) is 2 more than the product of its reciprocal and its additive inverse. In which […]
Question 1 The ratio \(\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}\) is closest to which of the following numbers? (a) 0.1 (b) 0.2 (c) 1 (d) 5 (e) 10 Question 2 For the nonzero numbers \(a, b, c\), define \((a, b, c)=\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\). Find \((2,12,9)\). (a) 4 (b) 5 (c) 6 (d) 7 (e) 8 Question 3 According to the standard convention […]
What happens when elementary school mathematics leaves the textbook and enters the garden, the night sky, and embroidered cloth? We share three open-ended problems for grades 1 to 6, each built on two levers, discovery and creation, and each ending with an invitation to invent a definition.
During Week 12 of the Teacher Training Program, the trainees actively participated in a comprehensive assessment covering three important areas: Literacy, Numeracy, and Digital Literacy. The assessment was designed to evaluate their knowledge, understanding, and practical application of the concepts learned during the training. All participants completed the assessment with dedication and enthusiasm, demonstrating their […]
Question 1 Two wave pulses I and II have the same wavelength. They are travelling in the directions as shown by the single headed arrows. The resultant sketch of the two wave pulses at some instant of time when $P$ coincides with $R$ is ____ . Question 2 The equivalent resistance of two resistances in […]
Semiconductors sit at the heart of nearly every modern technology, and this minicourse traces the full arc from materials to finished chips. It opens with inorganic semiconductor devices, laying out the physics of diodes, transistors, and junctions that have powered electronics for decades. Organic semiconductor devices follow, exploring how carbon-based materials enable flexible, low-cost alternatives […]
IX-XAnalytical Questions Q1 500 kg ভরের একটি কামান থেক 5 kg ভরের একটি গোলা ছোঁড়ায় কামানটি 1ms-1 বেগে পিছনে ছিটকে আসে । যদি গোলাটি 1000 kg ভরের একটি স্থির ঘোড়ার গাড়ির উপর আঘাত হেনে এতে গতির সঞ্চার করে তাহলে গাড়ির ও ঘোড়ার সম্মিলিত বেগ কত ? (অভিকর্ষ, বাতাসের বাধা, ঘর্ষণ ইত্যাদির প্রভাব উপেক্ষণীয়) A cannon […]
Question 1 A particle moves along a straight line. Its displacement S varies with time t according to the law $s^{2}=a t^{2}+2 b t+c$ ( $\mathrm{a}, \mathrm{b}$ and c are constants). The acceleration of this particle varies as (a) $S^{0}$ (b) $s^{-1}$ (c) $S^{-2}$ (d) $s^{-3}$ Question 2 A ball A (mass $m_{1}$ ) moving […]
Question 1 A point mass m moves in a straight line under a retardation $k v^{2}$ [where $k$ is a positive constant and $v$ is the instantaneous velocity]. The initial velocity of the point mass is $u$. The displacement of the point mass at time t is (a) $\frac{1}{k} \ln (1+k u t)$ (b) $\frac{1}{k} […]
During Week 11 of the Teacher Training Program, the trainees participated in an assessment covering three essential areas: Literacy, Numeracy, and Digital Literacy. The purpose of the assessment was to evaluate their knowledge, practical understanding, and ability to apply the skills acquired during the training. The activities included reading and writing tasks, numeracy exercises, and […]
Question 1 Let $\alpha$ and $\beta$ be the roots of $x^{2}-5 x+3=0$ with $\alpha>\beta$. If $a_{n}=\alpha^{n}-\beta^{n}$ for $n \geq 1$ then the value of $\frac{3 a_{6}+a_{8}}{a_{7}}$ is (a) 2 (b) 3 (c) 4 (d) 5 Question 2 The number of triples $(x, y, z)$ such that any one of these numbers is added to the […]
Question 1 A tiny ball of mass $m$ is initially at rest at height $H$ above a cake of uniform thickness $h$. At some moment the particle falls freely, touches the cake surface and then penetrates in it at such a constant rate that its speed becomes zero on just reaching the ground (bottom of […]