(a) If S Z and S 3, prove that there exist distinct x, y e S where x y is even (b) Let S Z Z Find the minimal value of S that guarantees the existence of distinct ordered pairs (x1, x2), (y1, y2) S such that x1 y1 and x2 y2 are both even (c) Extending the ideas in parts (a) and (b), consider S Z Z Z What size must 5 be to guarantee the existence of distinct ordered triples (x1, x2, x3), (y1, y2, y3) S where x1 y1, x2 y2 and pj yj are all even d) Generalize the results of parts (a), (b), and (c) e) A point P(x, y) in the Cartesian plane is called a lattice point if x, y Z Given distinct lattice points P1(x1 y1), P2(x2, y2),, Pn(xn, yn), determine the smallest value of n that guarantees the existence of Pi(xt, yi), Pj(Xj, yj), 1 i j n, such that the midpoint of the line segment connecting Pi(xt, yt) and Pj (xj, yj) is also a lattice point

Question

(a) If S  Z  and  S   3, prove that there exist distinct x, y e S where x   y is even  (b) Let S  Z   Z   Find the minimal value of  S  that guarantees the existence of distinct ordered pairs (x1, x2), (y1, y2)  S such that x1   y1 and x2   y2 are both even  (c) Extending the ideas in parts (a) and (b), consider S  Z   Z   Z   What size must  5  be to guarantee the existence of distinct ordered triples (x1, x2, x3), (y1, y2, y3)  S where x1   y1, x2   y2 and pj   yj are all even  d) Generalize the results of parts (a), (b), and (c)  e) A point P(x, y) in the Cartesian plane is called a lattice point if x, y  Z  Given distinct lattice points P1(x1 y1), P2(x2, y2),, Pn(xn, yn), determine the smallest value of n that guarantees the existence of Pi(xt, yi), Pj(Xj, yj), 1   i   j   n, such that the midpoint of the line segment connecting Pi(xt, yt) and Pj (xj, yj) is also a lattice point

SolutionInn · Accepted Answer

a Since S 3 3x y S where xy are both even or both odd In ei...

Question: (a) If S Z+ and |S| >3, prove that there exist distinct x, y e S where x + y is even. (b) Let

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