MathDB

3

Part of 2010 Postal Coaching

Problems(5)

An interesting functional equation

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10/21/2010
Find all functions f:Z→Zf: \mathbb{Z} \rightarrow \mathbb{Z} such that 1 f(1)=1\boxed{1} \ f(1) = 1 2 f(m+n)(f(m)−f(n))=f(m−n)(f(m)+f(n)) ∀ m,n∈Z\boxed{2} \ f(m+n)(f(m)-f(n)) = f(m-n)(f(m)+f(n)) \ \forall \ m,n \in \mathbb{Z}
functionalgebra unsolvedalgebra
prime no. expression

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10/15/2010
Prove that a prime pp is expressible in the form x2+3y2;x,y∈Zx^2+3y^2;x,y\in Z if and only if it is expressible in the form m2+mn+n2;m,n∈Z m^2+mn+n^2;m,n \in Z.Can pp be replaced by a natural number nn?
number theory unsolvednumber theory
A tough find of angles in a quadrilateral

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10/22/2010
In a quadrilateral ABCDABCD, we have ∠DAB=110∘,∠ABC=50∘\angle DAB = 110^{\circ} , \angle ABC = 50^{\circ} and ∠BCD=70∘\angle BCD = 70^{\circ} . Let M,N M, N be the mid-points of ABAB and CDCD respectively. Suppose PP is a point on the segment MNM N such that AMCN=MPPN\frac{AM}{CN} = \frac{MP}{PN} and AP=CPAP = CP . Find ∠APC\angle AP C.
geometryangle bisectorgeometry unsolved
Acquaintances and Parties

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12/9/2010
In a group of kk people, some are acquainted with each other and some are not. Every evening, one person invites all his acquaintances to a party and introduces them to each other(if they have not already acquainted). Suppose that after each person has arranged at least one party, some two people do not know each other. Prove that they do not meet each other in the next party.
inductioncombinatorics unsolvedcombinatorics
Smallest n....Rationals

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12/9/2010
Determine the smallest odd integer n≥3n \ge 3, for which there exist nn rational numbers x1,x2,...,xnx_1 , x_2 , . . . , x_n with the following properties:
(a)(a) ∑i=1nxi=0, ∑i=1nxi2=1.\sum_{i=1}^{n} x_i =0 , \ \sum_{i=1}^{n} x_i^2 = 1.
(b)(b) xi⋅xj≥−1n ∀ 1≤i,j≤n.x_i \cdot x_j \ge - \frac 1n \ \forall \ 1 \le i,j \le n.
algebra unsolvedalgebra