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# INMO 2012

1. Let ABCD be a quadrilateral inscribed in a circle. Suppose AB =$$\sqrt {2 + \sqrt {2} }$$ and AB subtends 1350 at the center of the circle. Find the maximum possible area of ABCD.
2. Let $$p_1<p_2< p_3< p_4$$ and $$q_1<q_2<q_3<q_4$$ be two sets of prime numbers such that $$p_4 – p_1 = 8$$  and $$q_4 – q_1 = 8$$ . Suppose $$p_1>5$$ and $$q_1>5$$ . Prove that 30 divides $$p_1 – q_1$$.
3. Define a sequence $$<fn(x)>$$ n∈N of functions as $$f_0(x )=1, f_1(x )=x$$, $$(f_n(x))^2 – 1 = f_{n-1} (x) f_{n+1} (x)$$, for $$n \ge 1$$ . Prove that each $$f_n(x )$$ is a polynomial with integer coefficients.
4. Let ABC be a triangle. An interior point P of ABC is said to be good if we can find exactly 27 rays emanating from P intersecting the sides of the triangle ABC such that the triangle is divided by these rays into 27 smaller triangles of equal area. Determine the number of good points for a given triangle ABC.
5. Let ABC be an acute angled triangle. Let D, E, F be points on BC, CA, AB such that AD is the median, BE is the internal bisector and CF is the altitude. Suppose that angle FDE = angle C and angle DEF = angle A and angle EFD = angle B. Show that ABC is equilateral.
6. Let $$f :Z \mapsto Z$$be a function satisfying $$f(0) \neq 0$$ , $$f(1)=0$$ and
1. $$f(xy) + f(x)f(y) = f(x) + f(y)$$,
2. $$(f(x-y) – f(0) ) f(x) f(y) = 0$$ for all x , $$y in Z$$ simultaneously.
1. Find the set of all possible values of the function f.
2. If $$f(10) \neq 0$$ and $$f(2) = 0$$, find the set of all integers n such that $$f(n) \neq 0$$ .
September 24, 2013