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Found problems: 2

2007 ISI B.Math Entrance Exam, 6
Tags: isi entrance , combinatorics
In $ISI$ club each member is on two committees and any two committees have exactly one member in common . There are 5 committees . How many members does $ISI$ club have????

2021 ISI Entrance Examination, 2
Tags: functional equation , isi entrance
Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \neq 0 = f(1)$. Assume also that $f$ satisfies equations [b](A)[/b] and [b](B)[/b] below. \begin{eqnarray*}f(xy) = f(x) + f(y) -f(x) f(y)\qquad\mathbf{(A)}\\ f(x-y) f(x) f(y) = f(0) f(x) f(y)\qquad\mathbf{(B)} \end{eqnarray*} for all integers $x,y$. [b](i)[/b] Determine explicitly the set $\big\{f(a)~:~a\in\mathbb{Z}\big\}$. [b](ii)[/b] Assuming that there is a non-zero integer $a$ such that $f(a) \neq 0$, prove that the set $\big\{b~:~f(b) \neq 0\big\}$ is infinite.