Recent questions tagged isi2023-pcb

10. Given a permutation \( \sigma\) of \( \{1, 2, . . . ,n \} \) , we define the symmetric binary matrix \(A^{\sigma} \) as follows: \[A^{\sigma}_{ij} = \begin{equation...

N9. Consider the set of points \( \{(x, y, z) \ | \ x, y, z \in \{0, 1, . . . ,n\}\} \).Determine how many of these points lie on the Euclidean plane that contains the ...

N8. Consider the sequence \(a_n\), defined as\[ a_{n+1} = \frac{1}{2} \left( a_n + \frac{4}{a_n} \right) \]Prove that $\lim_{n\to\infty} a_n$ an exists. Compute the valu...

N7. An oil-exploration company is considering 10 possible drilling sites \(S_1, S_2, . . . ,S_{10}\). The drilling costs associated with these sites are \(C_1, C_2, . . ....

N6. Find all values of \( a \in \mathbb{C} \) for which the equation\[x^4 + ax^2 + a^2x −1 = 0\]has all roots of the same absolute value.

N5. A person \(A\) rolls a fair die until the outcome is 6. After that, \(B\) rolls the die as many times as \(A\) had rolled it. Compute the probability that \(B\) will ...

C7. For a graph, a \( proper \ vertex \ colouring \) assigns a colour to each vertex such that no two adjacent vertices have the same colour.A vertex colouring is \(opt...

C6. Let \(A\) be an \(n \times n\) matrix with entries \(a_{ij} = \pm 1\) for all \(1 \leq i, j \leq n\) . Prove that \(2^{n−1}\) divides the determinant of \(A\) .