Maths Olympiad Exam

Let f : Z ! Z be a function satisfying f(0) = 0, f(1) = 0 and
(i) f(xy) + f(x)f(y) = f(x) + f(y);
(ii)f(x y) âˆ’ f(0)

f(x)f(y) = 0,
for all x, y 2 Z, simultaneously.
(a) Find the set of all possible values of the function f.
(b) If f(10) 6= 0 and f(2) = 0, find the set of all integers n such that
f(n) 6= 0

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Maths Olympiad Other Question

Define a sequence hf0(x), f1(x), f2(x), . . .i of functions by
f0(x) = 1, f1(x) = x,
􀀀
fn(x)
2
− 1 = fn+1(x)fn−1(x), for n 1.
Prove that each fn(x) is a polynomial with integer coefficients Let ABC be an acute-angled triangle, and let D, E, F be points on BC, CA, AB respectively such that AD is the median, BE is the internal angle bisector and CF is the altitude. Suppose FDE = C, DEF = A and EFD = B. Prove that ABC is equilateral.