Let a, b, c, d be positive integers such that a + b + c + d = 2023. If a : b = 2 : 5 and c : d = 5 : 2 then the maximum possible value of a + c is ______.

The remainder when \(1! + 2! + 3! + \cdots + 95!\) is divided by 15 is ______.

In the xy-plane let A = (-2,0), B = (2,0). Define the set S as the collection of all points C on the circle \(x^2 + y^2 = 4\) such that the area of the triangle ABC is an integer. The number of points in the set S is ______.

The polynomial \(4x^{10} - x^9 + 3x^8 - 5x^7 + cx^6 + 2x^5 - x^4 + x^3 - 4x^2 + 6x - 2\) when divided by x - 1 leaves a remainder 2. Then the value of c + 6 is ______.

Amisha can complete a particular task in twenty days. After working for four days she fell sick for four days and resumed the work on the ninth day but with half of her original work rate. She completed the task in another twelve days with the help of a co-worker who joined her f…

If \(f(n) = 1 + 2 + 3 + \cdots + (n+1)\) and \(g(n) = \sum_{k=1}^{n} \frac{1}{f(k)}\), then the least value of n for which g(n) exceeds the value \(\frac{99}{100}\) is ______.

In an election with only two contesting candidates, 15% of the voters did not turn up to vote, and 50 voters cast invalid votes. It is known that 44% of all the voters in the voting list voted for the winner. If the winner got 200 votes more than the other candidate, then the num…

If f(1) = 1 and f(n) = 3n - f(n - 1) for all integers n > 1, then the value of f(2023) is ______.

Assume it is the beginning of the year today. Ankita will earn INR 10,000 at the end of the year, which she plans to invest in a bank deposit immediately at a fixed simple interest of 0.5% per annum. Her yearly income will increase by INR 10,000 every year, and the fixed simple i…