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 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 […]
Question 1 What is the difference between the sum of the first 2003 even counting numbers and the sum of the first 2003 odd counting numbers? (a) 0 (b) 1 (c) 2 (d) 2003 (e) 4006 Question 2 Members of the Rockham Soccer League buy socks and T-shirts. Socks cost \($ 4\) per pair and […]
Question 1 You and five friends need to raise \($ 1500\) in donations for a charity, dividing the fundraising equally. How many dollars will each of you need to raise? (a) 250 (b) 300 (c) 1500 (d) 7500 (e) 9000 Question 2 For any three real numbers \(a, b\), and \(c\), with \(b \neq c\), […]
Question 1 While eating out, Mike and Joe each tipped their server 2 dollars. Mike tipped \(10 %\) of his bill and Joe tipped \(20 %\) of his bill. What was the difference, in dollars between their bills? (a) 2 (b) 4 (c) 5 (d) 10 (e) 20 Question 2 For each pair of real […]
Question 1 Sandwiches at Joe's Fast Food cost \($3\) each and sodas cost \($ 2\) each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas? (a) 31 (b) 32 (c) 33 (d) 34 (e) 35 Question 2 Define \(x \otimes y=x^{3}-y\). What is \(h \otimes(h \otimes h)\) ? (a) \(-h\) (b) […]
Question 1 What is $10 \cdot\left(\frac{1}{2}+\frac{1}{5}+\frac{1}{10}\right)^{-1}$ ? (a) 3 (b) 8 (c) $\frac{25}{2}$ (d) $\frac{170}{3}$ (e) 170 Question 2 Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing 6 cans […]
Question 1 What is the value of $9901 \cdot 101-99 \cdot 10101$ ? (a) 1 (b) 20 (c) 200 (d) 202 (e) 2020 Question 2 A 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 $a$ […]
Question 1 What is the value of $$ 3+\frac{1}{3+\frac{1}{3+\frac{1}{3}} ?} $$ (a) $\frac{31}{10}$ (b) $\frac{49}{15}$ (c) $\frac{33}{10}$ (d) $\frac{109}{33}$ (e) $\frac{15}{4}$ Question 2 Mike cycled 15 laps in 57 minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first 27 minutes? (a) 5 (b) 7 (c) […]
Question 1 What is the value of $$ \left(2^{2}-2\right)-\left(3^{2}-3\right)+\left(4^{2}-4\right) ? $$ (a) 1 (b) 2 (c) 5 (d) 8 (e) 12 Question 2 Portia's high school has 3 times as many students as Lara's high school. The two high schools have a total of 2600 students. How many students does Portia's high school have? (a) […]