1. . . 2 : mboudour@upatras.gr 2014 . : .

2. . : . 3. (Pigeon-Hole Principle) 1 ( ) n () k () , n k n >k. , . : , , (2) . (0) (1) . m , 0 m k. , , (1) k m, k m. , k m k rm, n,m,r . , r + 1 . : , , r . , , , , rm , , rm k >0 (n)k k X. , k , k (, k X) . , 9, k k! (, k X k! , ). , k ( ) (, k) k!, (n)k = n k k! , (n)k = n (n 1) (n k + 1), . , k = 0 k = n, k [n] ( [n], ), n k = 1.9 9 , k , k Z, , k < 0 k > n, k [n], n k = 0. . : . 35. 18 X . |X| = n. A (, A X), A |A| n . : 52 , 13 . ( ) , , , ; 5 , 52 5 = 2 598 960 . ( ), 13 1 4 3 12 2 4 1 4 1 = 54 912, , 54 912/2 598 960 = 0.021128. ( ), 13 2 4 2 4 2 44 1 = 123 552, , 123 552/2 598 960 = 0.047539. (, ,10 ), 10 45 = 10 240, , 10240/2 598 960 = 0.003940. (, ), 4 13 5 = 5148, , 5148/2598960 = 0.001981. 10 . . : . 36. : , 90 - 5 , . , , ; , , 5 [90], 90 5 = 43 949 268. : , . - ; a1, a2, a3, a4, a5 . 5 31, , 1 a1 < a2 1 < a3 2 < a4 3 < a5 4 27. , , 5 [31], , 5 [27]. , 5 [31] , 5 [27] 27 5 = 80 730 . . : . 37. 19 N2 R, . (x1,y1) (xn,yn) N2 ( n) x1