MathDB

2

Part of 1997 China National Olympiad

Problems(2)

BIjective map on {1,2,...,17}

Source: Chinese Mathematical Olympiad 1997 Problem 5

8/26/2013
Let A={1,2,3,⋯ ,17}A=\{1,2,3,\cdots ,17\}. A mapping f:A→Af:A\rightarrow A is defined as follows: f[1](x)=f(x),f[k+1](x)=f(f[k](x))f^{[1]}(x)=f(x), f^{[k+1]}(x)=f(f^{[k]}(x)) for k∈Nk\in\mathbb{N}. Suppose that ff is bijective and that there exists a natural number MM such that: i) when m<Mm<M and 1≤i≤161\le i\le 16, we have f[m](i+1)−f[m](i)≠±1(mod17)f^{[m]}(i+1)- f^{[m]}(i) \not=\pm 1\pmod{17} and f[m](1)−f[m](17)≠±1(mod17)f^{[m]}(1)- f^{[m]}(17) \not=\pm 1\pmod{17}; ii) when 1≤i≤161\le i\le 16, we have f[M](i+1)−f[M](i)=±1(mod17)f^{[M]}(i+1)- f^{[M]}(i)=\pm 1 \pmod{17} and f[M](1)−f[M](17)=±1(mod17)f^{[M]}(1)- f^{[M]}(17)=\pm 1\pmod{17}. Find the maximal value of MM.
modular arithmeticnumber theory unsolvednumber theory
Cyclic quadrilaterals among the first 12 in a sequence

Source: Chinese Mathematical Olympiad 1997 Problem 2

8/26/2013
Let A1B1C1D1A_1B_1C_1D_1 be an arbitrary convex quadrilateral. PP is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point PP is acute. We recursively define points Ak,Bk,Ck,DkA_k,B_k,C_k,D_k symmetric to PP with respect to lines Ak−1Bk−1,Bk−1Ck−1,Ck−1Dk−1,Dk−1Ak−1A_{k-1}B_{k-1}, B_{k-1}C_{k-1}, C_{k-1}D_{k-1},D_{k-1}A_{k-1} respectively for k≥2k\ge 2. Consider the sequence of quadrilaterals AiBiCiDiA_iB_iC_iD_i. i) Among the first 12 quadrilaterals, which are similar to the 1997th quadrilateral and which are not? ii) Suppose the 1997th quadrilateral is cyclic. Among the first 12 quadrilaterals, which are cyclic and which are not?
geometry unsolvedgeometry