Last year, \(\frac{1}{8}\) of the employees of Raman Furniture House were originally from outside Jaipur. During the past year, for every \(3\) employees hired from outside Jaipur, \(1\) Jaipur-native employee left the company. Now, \(\frac{1}{3}\) of all employees of Raman Furniture House are from outside Jaipur. If the \(75\) employees hired during the last year were from outside Jaipur, what is the smallest possible number of employees Raman Furniture House could now have?

The product of \(6\) distinct positive integers written on \(6\) cards is \(6^6\). What is the minimum possible value of the average of these \(6\) integers?

A retailer purchased a mixer grinder for Rs. \(1000\) and marked its price as Rs. \(2160\). He then offered three successive discounts of \(a%\), \(b%\), and \(c%\), where \(a + b + c = 50\). If his final profit is \(x%\), then what is the difference between the maximum possible value of \(x\) and the minimum possible value of \(x\)?

For two real numbers \(m\) and \(n\), it is known that \(m^2 < 225\) and \(n - m = -10\). What is the positive difference between the smallest possible integer value of \(3m + 2n\) and the greatest possible integer value of \(3m + 2n\)?

There are \(25\) prime numbers less than \(100\). In a mock test at an MBA coaching class, \(130\) students each chose a two-digit prime number. What is the greatest integer \(x\) such that there must be at least \(x\) students who chose the same prime number?

A pharmacist needs to purchase a set of \(n\) metal weights, each having an integer weight in grams. Using a combination of one or more of these weights, every integer weight from \(1\) gram to \(300\) grams, inclusive, must be measurable. What is the minimum number of metal weights the pharmacist must purchase?

An activity centre offers four short documentary shows that run continuously throughout the day, with each show starting again as soon as it ends. The first show runs every \(15\) minutes, the second every \(30\) minutes, the third every \(45\) minutes, and the fourth every \(40\) minutes. The first screening of each show starts at \(10\) am, and the last screening of each show ends at \(4\) pm. A group can watch the shows in any order, but needs at least \(10\) minutes between two shows to regroup. What is the least amount of time the group can take to watch all four shows?

To be allowed to participate in a national sports trial, an athlete must clear fitness tests conducted by three different panels. Out of \(50\) athletes, \(40\) cleared the test conducted by the first panel, \(35\) cleared the test conducted by the second panel, and \(30\) cleared the test conducted by the third panel. If every athlete cleared the test conducted by at least one panel, which of the following represents the lowest and highest possible numbers of athletes who could have cleared the tests conducted by all three panels?

A drawer in Rohan's hostel room contains \(10\) pairs of socks of each of \(3\) colours. Rohan takes out socks one at a time from the drawer at random without looking at their colours. What is the smallest number of socks he must take out so that the socks drawn definitely form at least \(10\) pairs of colour-matched socks?

If the straight line \(y = x + c\) is tangent to the circle \((x - 1)^2 + (y + 2)^2 = 4\), what is the maximum possible value of the constant \(c\)?

In an MBA coaching batch, the number of students enrolled in three modules is as follows: Quant: \(40\) Verbal: \(50\) DI-LR: \(35\)

If the number of students enrolled in exactly two modules is maximum and no student is enrolled in all three modules, then what is the minimum number of students enrolled in exactly one module?

For non-zero real numbers \(p, q, r,\) and \(s\), it is given that \(4 \le p \le 9\), \(-0.25 \le q \le 0.36\), \(-0.49 \le r \le -0.01\), and \(\frac{r}{s^2} = \frac{p^2}{q}\). What is the difference between the maximum and minimum possible values of \(s\)?

Let \(a_1, a_2, \ldots, a_{52}\) be positive integers such that \(a_1 < a_2 < \cdots < a_{52}\). Suppose the arithmetic mean of all \(52\) integers is \(1\) less than the arithmetic mean of \(a_2, a_3, \ldots, a_{52}\). If \(a_{52} = 100\), then what is the largest possible value of \(a_1\)?

In a school cricket league of \(8\) teams, each team played every other team \(10\) times. The number of wins of the \(8\) teams formed an arithmetic sequence. What is the least possible number of games won by the champion?

For any integer \(n\), \(f(n)\) is defined as the greatest multiple of \(10\) less than or equal to \(n + 5\). If \(a, b, c,\) and \(d\) are positive integers such that \(abcd = 360\), what is the greatest possible value of \(f(a) + f(b) + f(c) + f(d) - (a + b + c + d)\)?

A data set contains \(7\) integers. Its median is \(9\), its range is \(10\), and its average is \(9\). What is the lowest possible value of the smallest integer in the data set?

For a real value of \(a\), what is the minimum possible value of the sum of the squares of the roots of the equation \(x^2 + (a + 3)x - (a + 5) = 0\)?

A gym in Delhi plans to purchase weight plates in four sizes: \(1\) kg, \(5\) kg, \(10\) kg, and \(25\) kg. The plates cost Rs. \(1\), Rs. \(8\), Rs. \(24\), and Rs. \(100\) respectively. The gym wants to purchase plates so that every integer weight from \(1\) kg to \(100\) kg can be assembled using some subset of the purchased plates, with each purchased plate used at most once. If the total cost of the plates must be less than Rs. \(350\), what is the smallest number of plates the gym can purchase?

Nisha's cafe sells small, medium, and large coffees for Rs. \(2\), Rs. \(3\), and Rs. \(4\) respectively. On a particular day, she collected Rs. \(404\) from coffee sales. The number of medium coffees sold was greater than the number of small coffees and also greater than the number of large coffees. Also, at least one coffee of each size was sold. What is the difference between the greatest and smallest possible total numbers of coffees sold that day?

In a college fest group, \(95%\) of the members speak Hindi, \(85%\) speak English, \(80%\) speak Marathi, and \(90%\) speak Tamil. What is the minimum possible percentage of members in the group who speak all four languages?

At a mobile parts factory, less than \(0.3%\) of the chips produced last week were defective, and exactly \(5%\) of the defective chips were installed in budget phones. If more than \(60%\) of the non-defective chips produced last week were installed in flagship phones, what is the smallest possible number of chips produced last week that were installed in flagship phones?

Two \(3\)-digit registration numbers, \(N\) and \(M\), are formed using the digits \(1, 2, 3, 6, 7,\) and \(8\) exactly once. What is the smallest possible positive difference between \(N\) and \(M\)?

If \(|x + 3| = |y - 4|\), where \(x\) and \(y\) are non-zero integers. If \(|x| < 5\) and \(|y| < 5\), what is the maximum possible value of \(-|xy|\)?

If \(T = \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \frac{1}{4 \times 5} + \cdots + \frac{1}{n(n+1)}\), for some positive integer \(n\), what is the smallest value of \(n\) such that \(T > 0.97\)?

For positive integers \(k\) and \(n\), the \(k\)-power remainder of \(n\) is defined as \(r\) in the equation \(n = k^w + r\), where \(w\) is the largest integer such that \(r\) is non-negative. For example, the \(3\)-power remainder of \(13\) is \(4\), since \(13 = 3^2 + 4\). In terms of \(k\) and \(w\), what is the largest possible value of \(r\)?