The number of ways in which coins of denominations of rupees 2, 5 and 10 can be combined to make a value of rupees 50 is

Positive reals \(x, y\) satisfy \(x \neq y\) and \(\frac{x^2 + y^2}{xy} = k\). If replacing \(x\) by \(x+y\) and \(y\) by \(|x-y|\) leaves the value of \(k\) unchanged, then \(k\) equals

If \(a\), \(b\), and \(c\) are three prime numbers such that \(abc = 23(a+b+c)\), then the maximum possible value of \(a+b-c\) is

The possible values of \(x\) in the set \(\{1, 5, 13\}\) for which the mean of eight observations \(5, 8, 3x+2, 15, 27, 29, 36, 5x-2\) equals their median are

The number of integer solutions \((x, y)\) of the inequality \(x^2 + y^2 \leq 10\) is

A circle of non-zero radius has origin as its centre. If it passes through the point of intersection of two curves \(y^2 = 4ax\) and \(x^2 = 4ay\), then its equation is

If \(\log_{18} 24 = p\), then \(\log_{96} 108\) equals

A person walks one lap along a circle at a speed \(v\). Thereafter, he runs one lap along the boundary of the largest square that can be inscribed in the circle at a speed \(3v\). The ratio of the time he walks to the time he runs is

Let \(S = \{1, 2, \ldots, 180\}\). Define \(A\) as the set of all multiples of 4 in \(S\), \(B\) as the set of all multiples of 6 in \(S\), and \(C\) as the set of all multiples of 9 in \(S\). The number of elements in \(S\) that belong to exactly one of \(A, B, C\) is