Let the set \(P = \{2, 3, 4, \ldots, 25\}\). For each \(k \in P\), define \(Q(k)=\{x\in P \text{ such that } x>k \text{ and } k \text{ divides } x\}\). Then the number of elements in the set \(P-\bigcup_{k=2}^{25}Q(k)\) is

If a \(3\times 3\) matrix is filled with +1's and -1's such that the sum of each row and column of the matrix is 1, then the absolute value of its determinant is...

The number of whole metallic tiles that can be produced by melting and recasting a circular metallic plate, if each of the tiles has a shape of a right-angled isosceles triangle and the circular plate has a radius equal in length to the longest side of the tile (Assume that the t…

The circle \(x^2+y^2-6x-10y+k=0\) does not touch or intersect the coordinate axes. If the point (1, 4) does not lie outside the circle, and the range of k is [a, b] then a + b is____

If \(|x| < 100\) and \(|y| < 100\), then the number of integer solutions of (x, y) satisfying the equation \(4x + 7y = 3\) is

The number of pairs (x, y) satisfying the equation \(\sin x+\sin y=\sin(x+y)\) and \(|x|+|y|=1\) is

The average of five distinct integers is 110 and the smallest number among them is 100. The maximum possible value of the largest integer is

If x, y, z are positive real numbers such that \(x^{12} = y^{16} = z^{24}\), and the three quantities \(3\log_y x\), \(4\log_z y\), \(n\log_x z\) are in arithmetic progression, then the value of n is

Assume that all positive integers are written down consecutively from left to right as in 1234567891011...... The \(6389^{th}\) digit in this sequence is...