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6

Part of 1996 Putnam

Problems(2)

Putnam 1996 A6

Source:

6/4/2014
Let c≥0c\ge 0 be a real number. Give a complete description with proof of the set of all continuous functions f:R→Rf: \mathbb{R}\to \mathbb{R} such that f(x)=f(x2+c)f(x)=f(x^2+c) for all x∈Rx\in \mathbb{R}.
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Putnam 1996 B6

Source:

6/6/2014
Let (a1,b1),(a2,b2),…,(an,bn)(a_1,b_1),(a_2,b_2),\ldots ,(a_n,b_n) be the vertices of a convex polygon containing the origin in its interior. Prove that there are positive real numbers x,yx,y such that : (a1,b1)xa1yb1+(a2,b2)xa2yb2+…+(an,bn)xanybn=(0,0) (a_1,b_1)x^{a_1}y^{b_1}+(a_2,b_2)x^{a_2}y^{b_2}+\ldots +(a_n,b_n)x^{a_n}y^{b_n}=(0,0)
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