MathDB

3

Part of 2011 Postal Coaching

Problems(6)

Find f(2) and f(1)

Source:

12/31/2011
Suppose f:R⟶Rf : \mathbb{R} \longrightarrow \mathbb{R} be a function such that 2f(f(x))=(x2−x)f(x)+4−2x2f (f (x)) = (x^2 - x)f (x) + 4 - 2x for all real xx. Find f(2)f (2) and all possible values of f(1)f (1). For each value of f(1)f (1), construct a function achieving it and satisfying the given equation.
functionalgebrafunctional equationalgebra unsolved
Prove that grasshopper can jump from any vertex to another

Source:

12/31/2011
Let CC be a circle, A1,A2,…,AnA_1 , A_2,\ldots ,A_n be distinct points inside CC and B1,B2,…,BnB_1 , B_2 ,\ldots ,B_n be distinct points on CC such that no two of the segments A1B1,A2B2,…,AnBnA_1B_1 , A_2 B_2 ,\ldots ,A_n B_n intersect. A grasshopper can jump from ArA_r to AsA_s if the line segment ArAsA_r A_s does not intersect any line segment AtBt(t≠r,s)A_t B_t (t \neq r, s). Prove that after a certain number of jumps, the grasshopper can jump from any AuA_u to any AvA_v .
combinatorics unsolvedcombinatorics
Polynomial with integer coefficients

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12/31/2011
Let P(x)P (x) be a polynomial with integer coefficients. Given that for some integer aa and some positive integer nn, where P(P(…P⏟n times(a)…))=a,\underbrace{P(P(\ldots P}_{\text{n times}}(a)\ldots)) = a, is it true that P(P(a))=aP (P (a)) = a?
algebrapolynomialalgebra unsolved
Construct a triangle given three special points

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12/31/2011
Construct a triangle, by straight edge and compass, if the three points where the extensions of the medians intersect the circumcircle of the triangle are given.
geometrycircumcircleconicsgeometry unsolved
Concurrency with tangents to nine point circle

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12/31/2011
Let ABCABC be a scalene triangle. Let lAl_A be the tangent to the nine-point circle at the foot of the perpendicular from AA to BCBC, and let lA′l_A' be the tangent to the nine-point circle from the mid-point of BCBC. The lines lAl_A and lA′l_A' intersect at A′A' . Define B′B' and C′C' similarly. Show that the lines AA′,BB′AA' , BB' and CC′CC' are concurrent.
geometry unsolvedgeometry
Find maximum and minimum value

Source:

12/31/2011
Let f:N⟶Nf : \mathbb{N} \longrightarrow \mathbb{N} be a function such that (x+y)f(x)≤x2+f(xy)+110(x + y)f (x) \le x^2 + f (xy) + 110, for all x,yx, y in N\mathbb{N}. Determine the minimum and maximum values of f(23)+f(2011)f (23) + f (2011).
functionalgebra unsolvedalgebra