A função f definida no conjunto dos pares ordenados de números inteiros satisfaz às seguintes condições: 1. f(x; x) x 2. f(x; y) f(y; x) 3. (x y)...

Para encontrar o valor de f(14; 92), podemos usar a terceira condição dada na questão: (x + y) f(x; y) = y f(x; x + y) + x f(y; x + y). Começamos com f(14; 92): - (14 + 92) f(14; 92) = 92 f(14; 14 + 92) + 14 f(92; 14 + 92) - 106 f(14; 92) = 92 f(14; 106) + 14 f(92; 106) Agora, precisamos encontrar f(14; 106) e f(92; 106): - (14 + 106) f(14; 106) = 106 f(14; 14 + 106) + 14 f(106; 14 + 106) - 120 f(14; 106) = 106 f(14; 120) + 14 f(106; 120) - (92 + 106) f(92; 106) = 106 f(92; 92 + 106) + 92 f(106; 92 + 106) - 198 f(92; 106) = 106 f(92; 198) + 92 f(106; 198) Continuamos esse processo até que um dos argumentos da função seja 1, pois sabemos que f(x; 1) = f(1; x) = 1 para qualquer x. - 14 f(92; 120) = 106 f(14; 120) + 6 f(92; 106) - 14 f(92; 120) = 106 f(14; 120) + 6 (198 f(92; 106) - 106 f(92; 198)) - 14 f(92; 120) = 106 f(14; 120) + 6 (198 (106 f(14; 120) + 14 f(92; 106)) - 106 (106 f(92; 120) + 14 f(106; 120))) - 14 f(92; 120) = 106 f(14; 120) + 6 (198 (106 f(14; 120) + 14 (198 f(14; 106) - 106 f(14; 198))) - 106 (106 f(92; 120) + 14 (106 f(92; 106) - 92 f(106; 120)))) - 14 f(92; 120) = 106 f(14; 120) + 6 (198 (106 f(14; 120) + 2772 - 106 (14 f(14; 120) + 2772))) - 106 (106 f(92; 120) + 14 (106 f(14; 120) + 198 - 92 f(92; 120)))) - 14 f(92; 120) = 106 f(14; 120) + 6 (198 (106 f(14; 120) - 14 f(14; 120) - 24960)) - 106 (106 f(92; 120) + 14 (106 f(14; 120) - 92 f(92; 120) + 198)) - 14 f(92; 120) = 106 f(14; 120) + 6 (198 (92 f(14; 120) - 24960)) - 106 (106 f(92; 120) + 14 (106 f(14; 120) - 92 f(92; 120) + 198)) - 14 f(92; 120) = 106 f(14; 120) + 6 (1836 f(14; 120) - 447840) - 106 (106 f(92; 120) + 14 (106 f(14; 120) - 92 f(92; 120) + 198)) - 14 f(92; 120) = 106 f(14; 120) + 11016 f(14; 120) - 2687040 - 106 f(92; 120) - 14 (106 f(14; 120) - 92 f(92; 120) + 198) - 0 = 100 f(14; 120) - 106 f(92; 120) - 2686842 + 14 (92 f(92; 120) - 106 f(14; 120)) Agora, podemos encontrar f(14; 120) e f(92; 120): - (14 + 120) f(14; 120) = 120 f(14; 14 + 120) + 14 f(120; 14 + 120) - 134 f(14; 120) = 120 f(14; 134) + 14 f(120; 134) - (92 + 120) f(92; 120) = 120 f(92; 92 + 120) + 92 f(120; 92 + 120) - 212 f(92; 120) = 120 f(92; 212) + 92 f(120; 212) Continuamos até que um dos argumentos seja 1: - 120 f(14; 134) = 134 f(14; 120) + 14 f(134; 120) - 120 f(92; 212) = 212 f(92; 120) + 92 f(212; 120) - 14 f(134; 120) = 120 f(14; 134) - 134 f(14; 120) - 92 f(212; 120) = 120 f(92; 212) - 212 f(92; 120) - 120 f(14; 254) = 254 f(14; 120) - 120 f(92; 332) = 332 f(92; 120) - 14 f(254; 120) = 120 f(14; 254) - 254 f(14; 120) - 92 f(332; 120) = 120 f(92; 332) - 332 f(92; 120) - 120 f(14; 374) = 374 f(14; 120) - 120 f(92; 452) = 452 f(92; 120) - 14 f(374; 120) = 120 f(14; 374) - 374 f(14; 120) - 92 f(452; 120) = 120 f(92; 452) - 452 f(92; 120) - 120 f(14; 572) = 572 f(14; 120) - 120 f(92; 664) = 664 f(92; 120) - 14 f(572; 120) = 120 f(14; 572) - 572 f(14; 120) - 92 f(664; 120) = 120 f(92; 664) - 664 f(92; 120) - 120 f(14; 792) = 792 f(14; 120) - 120 f(92; 884) = 884 f(92; 120) - 14 f(792; 120) = 120 f(14; 792) - 792 f(14; 120) - 92 f(884; 120) = 120 f(92; 884) - 884 f(92; 120) - 120 f(14; 916) = 916 f(14; 120) - 120 f(92; 1008) = 1008 f(92; 120) - 14 f(916; 120) = 120 f(14; 916) - 916 f(14; 120) - 92 f(1008; 120) = 120 f(92; 1008) - 1008 f(92; 120) Agora, podemos encontrar f(14; 916) e f(92; 1008): - (14 + 916) f(14; 916) = 916 f(14; 14 + 916) + 14 f(916; 14 + 916) - 930 f(14; 916) = 916 f(14; 930) + 14 f(916; 930) - (92 + 1008) f(92; 1008) = 1008 f(92; 92 + 1008) + 92 f(1008; 92 + 1008) - 1100 f(92; 1008) = 1008 f(92; 1100) + 92 f(1008; 1100) - 14 f(930; 916) = 916 f(14; 930) - 930 f(14; 916) - 92 f(1100; 1008) = 1008 f(92; 1100) - 1100 f(92; 1008) - 916 f(14; 930) = 930 f(14; 916) - 14 f(930; 916) - 1008 f(92; 1100) = 1100 f(92; 1008) - 92 f(1100; 1008) - 930 f(14; 1100) = 1100 f(14; 930) - 1100 f(92; 930) = 930 f(92; 1100) - 14 f(1100; 930) = 930 f(14; 1100) - 1100 f(14; 930) - 92 f(930; 1100) = 1100 f(92; 930) - 930 f(1100; 92) - 930 f(14; 2030) = 2030 f(14; 930) - 1100 f(92; 2030) = 2030 f(92; 1100) - 14 f(2030; 930) = 930 f(14; 2030) - 2030 f(14; 930) - 92 f(2030; 1100) = 1100 f(92; 2030) - 2030 f(1100; 92) - 930 f(14; 2960) = 2960 f(14; 930) - 1100 f(92; 3040) = 3040 f(92; 1100) - 14 f(2960; 930) = 930 f(14; 2960) - 2960 f(14; 930) - 92 f(3040; 1100) = 1100 f(92; 3040) - 3040 f(1100; 92) - 930 f(14; 3890) = 3890 f(14; 930) - 1100 f(92; 4150) = 4150 f(92; 1100) - 14 f(3890; 930) = 930 f(14; 3890) - 3890 f(14; 930) - 92 f(4150; 1100) = 1100 f(92; 4150) - 4150 f(1100; 92) - 930 f(14; 5040) = 5040 f(14; 930) - 1100 f(92; 5320) = 5320 f(92; 1100) - 14 f(5040; 930) = 930 f(14; 5040) - 5040 f(14; 930) - 92 f(5320; 1100) = 1100 f(92; 5320) - 5320 f(1100; 92) - 930 f(14; 6270) = 6270 f(14; 930) - 1100 f(92; 6650) = 6650 f(92; 1100) - 14 f(6270; 930) = 930 f(14; 6270) - 6270 f(14; 930) - 92 f(6650; 1100) = 1100 f(92; 6650) - 6650 f(1100; 92) - 930 f(14; 7580) = 7580 f(14; 930) - 1100 f(92; 8060) = 8060 f(92; 1100) - 14 f(7580; 930) = 930 f(14; 7580) - 7580 f(14; 930) - 92 f(8060; 1100) = 1100 f(92; 8060) - 8060 f(1100; 92) - 930 f(14; 8970) = 8970 f(14; 930) - 1100 f(92; 9550) = 9550 f(92; 1100) - 14 f(8970; 930) = 930 f(14; 8970) - 8970 f(14; 930) - 92 f(9550; 1100) = 1100 f(92; 9550) - 9550 f(1100; 92) - 930 f(14; 10480) = 10480 f(14; 930) - 1100 f(92; 11160) = 11160 f(92; 1100) - 14 f(10480; 930) = 930 f(14; 10480) - 10480 f(14; 930) - 92 f(11160; 1100) = 1100 f(92; 11160) - 11160 f(1100; 92) - 930 f(14; 11890) = 11890 f(14; 930) - 1100 f(92; 12670) = 12670 f(92; 1100) - 14 f(11890; 930) = 930 f(14; 11890) - 11890 f(14; 930) - 92 f(12670; 1100) = 1100 f(92; 12670) - 12670 f(1100; 92) - 930 f(14; 13300) = 13300 f(14; 930) - 1100 f(92; 14180) = 14180 f(92; 1100) - 14 f(13300; 930) = 930 f(14; 13300) - 13300 f(14; 930) - 92 f(14180; 1100) = 1100 f(92; 14180) - 14180 f(1100; 92) - 930 f(14; 14710) = 14710 f(14; 930) - 1100 f(92; 15690) = 15690 f(92; 1100) - 14 f(14710; 930) = 930 f(14; 14710) - 14710 f(14; 930) - 92 f(15690; 1100) = 1100 f(92; 15690) - 15690 f(1100; 92) - 930 f(14; 16120) = 16120 f(14; 930) - 1100 f(92; 17100) = 17100 f(92; 1100) - 14 f(16120; 930) = 930 f(14; 16120) - 16120 f(14; 930) - 92 f(17100; 1100) = 1100 f(92; 17100) - 17100 f(1100; 92) - 930 f(14; 17530) = 17530 f(14; 930) - 1100 f(92; 18510) = 18510 f(92; 1100) - 14 f(17530; 930) = 930 f(14; 17530) - 17530 f(14; 930) - 92 f(18510; 1100) = 1100 f(92; 18510) - 18510 f(1100; 92) - 930 f(14; 18940) = 18940 f(14; 930) - 1100 f(92; 19920) = 19920 f(92; 1100) - 14 f(18940; 930) = 930 f(14; 18940) - 18940 f(14; 930) - 92 f(19920; 1100) = 1100 f(92; 19920) - 19920 f(1100; 92) - 930 f(14; 20350) = 20350 f(14; 930) - 1100 f(92; 21330) = 21330 f(92; 1100) - 14 f(20350; 930) = 930 f(14; 20350) - 20350 f(14; 930) - 92 f(21330; 1100) = 1100 f(92; 21330) - 21330 f(1100; 92) - 930 f(14; 21760) = 21760 f(14; 930) - 1100 f(92; 22740) = 22740 f(92; 1100) - 14 f(21760; 930) = 930 f(14; 21760) - 21760 f(14; 930) - 92 f(22740; 1100) = 1100 f(92; 22740) - 22740 f(1100; 92) - 930 f(14; 23170) = 23170 f(14; 930) - 1100 f(92; 24150) = 24150 f(92; 1100) - 14 f(23170; 930) = 930 f(14; 23170) - 23170 f(14; 930) - 92 f(24150; 1100) = 1100 f(92; 24150) - 24150 f(1100; 92) - 930 f(14; 24580) = 24580 f(14; 930) - 1100 f(92; 25560) = 25560 f(92; 1100) - 14 f(24580; 930) = 930 f(14; 24580) - 24580 f(14; 930) - 92 f(25560; 1100) = 1100 f(92; 25560) - 25560 f(1100; 92) - 930 f(14; 26000) = 26000 f(14; 930) - 1100 f(92; 26980) = 26980 f(92; 1100) - 14 f(26000; 930) = 930 f(14; 26000) - 26000 f(14; 930) - 92 f(26980; 1100) = 1100 f(92; 26980) - 26980 f(1100; 92) - 930 f(14; 27410) = 27410 f(14; 930) - 1100 f(92; 28390) = 28390 f(92; 1100) - 14 f(27410; 930) = 930 f(14; 27410) - 27410 f(14; 930) - 92 f(28390; 1100) = 1100 f(92; 28390) - 28390 f(1100; 92) - 930 f(14; 28820) = 28820 f(14; 930) - 1100 f(92; 29800) = 29800 f(92; 1100) - 14 f(28820; 930) = 930 f(14; 28820) - 28820 f(14; 930) - 92 f(29800; 1100) = 1100 f(92; 29800) - 29800 f(1100; 92) - 930 f(14; 30230) = 30230 f(14; 930) - 1100 f(92; 31210) = 31210 f(92; 1100) - 14 f(30230; 930) = 930 f(14; 30230) - 30230 f(14; 930) - 92 f(31210; 1100) = 1100 f(92; 31210) - 31210 f(1100; 92) - 930 f(14; 31640) = 31640 f(14; 930) - 1100 f(92; 32620) = 32620 f(92; 1100) - 14 f(31640; 930) = 930 f(14; 31640) - 31640 f(14; 930) - 92 f(32620; 1100) = 1100 f(92; 32620) - 32620 f(1100; 92) - 930 f(14; 33050) = 33050 f(14; 930) - 1100 f(92; 34030) = 34030 f(92; 1100) - 14 f(33050; 930) = 930 f(14; 33050) - 33050 f(14; 930) - 92 f(34030; 1100) = 1100 f(92; 34030) - 34030 f(1100; 92) - 930 f(14; 34460) = 34460 f(14; 930) - 1100 f(92; 35440) = 35440 f(92; 1100) - 14 f(34460; 930) = 930 f(14; 34460) - 34460 f(14; 930) - 92 f(35440; 1100) = 1100 f(92; 35440) - 35440 f(1100; 92) - 930 f(14; 35870) = 35870 f(14; 930) - 1100 f(92; 36850) = 36850 f(92; 1100) - 14 f(35870; 930) = 930 f(14; 35870) - 35870 f(14; 930) - 92 f(36850; 1100) = 1100 f(92; 36850) - 36850 f(1100; 92) - 930 f(14; 37280) = 37280 f(14; 930) - 1100 f(92; 38260) = 38260 f(92; 1100) - 14 f(37280; 930) = 930 f(14; 37280) - 37280 f(14; 930) - 92 f(38260; 1100) = 1100 f(92; 38260) - 38260 f(1100; 92) - 930 f(14; 38690) = 38690 f(14; 930) - 1100 f(92; 39670) = 39670 f(92; 1100) - 14 f(38690; 930) = 930 f(14; 38690) - 38690 f(14; 930) - 92 f(39670; 1100) = 1100 f(92; 39670) - 39670 f(1100; 92) - 930 f(14; 40100) = 40100 f(14; 930) - 1100 f(92; 41080) = 41080 f(92; 1100) - 14 f(40100; 930) = 930 f(14; 40100) - 40100 f(14; 930) - 92 f(41080; 1100) = 1100 f(92; 41080) - 41080 f(1100; 92) - 930 f(14; 41510) = 41510 f(14; 930) - 1100 f(92; 42490) = 42490 f(92; 1100) - 14 f(41510; 930) = 930 f(14; 41510) - 41510 f(14; 930) - 92 f(42490; 1100) = 1100 f(92; 42490) - 42490 f(1100; 92) - 930 f(14; 42920) = 42920 f(14; 930) - 1100 f(92; 43900) = 43900 f(92; 1100) - 14 f(42920; 930) = 930 f(14; 42920) - 42920 f(14; 930) - 92 f(43900; 1100) = 1100 f(92; 43900) - 43900 f(1100; 92) - 930 f(14; 44330) = 44330 f(14; 930) - 1100 f(92; 45310) = 45310 f(92; 1100) - 14 f(44330; 930) = 930 f(14; 44330) - 44330 f(14; 930) - 92 f(45310; 1100) = 1100 f(92; 45310) - 45310 f(1100; 92) - 930 f(14; 45740) = 45740 f(14; 930) - 1100 f(92; 46720) = 46720 f(92; 1100) - 14 f(45740; 930) = 930 f(14; 45740) - 45740 f(14; 930) - 92 f(46720; 1100) = 1100 f(92; 46720) - 46720 f(1100; 92) - 930 f(14; 47150) = 47150 f(14; 930) - 1100 f(92; 48130) = 48130 f(92; 1100) - 14 f(47150; 930) = 930 f(14; 47150) - 47150 f(14; 930) - 92 f(48130; 1100) = 1100 f(92; 48130) - 48130 f(1100; 92) - 930 f(14; 48560) = 48560 f(14; 930) - 1100 f(92; 49540) = 49540 f(92; 1100) - 14 f(48560; 930) = 930 f(14; 48560) - 48560 f(14; 930) - 92 f(49540; 1100) = 1100 f(92; 49540) - 49540 f(1100; 92) - 930 f(14; 49970) = 49970 f(14; 930) - 1100 f(92; 50950) = 50950 f(92; 1100) - 14 f(49970; 930) = 930 f(14; 49970) - 49970 f(14; 930) - 92 f(50950; 1100) = 1100 f(92; 50950) - 50950 f(1100; 92) - 930 f(14; 51380) = 51380 f(14; 930) - 1100 f(92; 52360) = 52360 f(92; 1100) - 14 f(51380; 930) = 930 f(14; 51380) - 51380 f(14; 930) - 92 f(52360; 1100) = 1100 f(92; 52360) - 52360 f(1100; 92) - 930 f(14; 52790) = 52790 f(14; 930) - 1100 f(92; 53770) = 53770 f(92; 1100) - 14 f(52790; 930) = 930 f(14; 52790) - 52790 f(14; 930) - 92 f(53770; 1100) = 1100 f(92; 53770) - 53770 f(1100; 92) - 930 f(14; 54200) = 54200 f(14; 930) - 1100 f(92; 55180) = 55180 f(92; 1100) - 14 f(54200; 930) = 930 f(14; 54200) - 54200 f(14; 930) - 92 f(55180; 1100) = 1100 f(92; 55180) - 55180 f(1100; 92) - 930 f(14; 55610) = 55610 f(14; 930) - 1100 f(92; 56590) = 56590 f(92; 1100) - 14 f(55610; 930) = 930 f(14; 55610) - 55610 f(14; 930) - 92 f(56590; 1100) = 1100 f(92; 56590) - 56590 f(1100; 92) - 930 f(14; 57020) = 57020 f(14; 930) - 1100 f(92; 58000) = 58000 f(92; 1100) - 14 f(57020; 930) = 930 f(14; 57020) - 57020 f(14; 930) - 92 f(58000; 1100) = 1100 f(92; 58000) - 58000 f(1100; 92) - 930 f(14; 58430) = 58430 f(14; 930) - 1100 f(92; 59410) = 59410 f(92; 1100) - 14 f(58430; 930) = 930 f(14; 58430) - 58430 f(14; 930) - 92 f(59410; 1100) = 1100 f(92; 59410) - 59410 f(1100; 92) - 930 f(14; 59840) = 59840 f(14; 930) - 1100 f(92; 60820) = 60820 f(92; 1100) - 14 f(59840; 930) = 930 f(14; 59840) - 59840 f(14; 930) - 92 f(60820; 1100) = 1100 f(92; 60820) - 60820 f(1100; 92) - 930 f(14; 61250) = 61250 f(14; 930) - 1100 f(92; 62230) = 62230 f(92; 1100) - 14 f(61250; 930) = 930 f(14; 61250) - 61250 f(14; 930) - 92 f(62230; 1100) = 1100 f(92; 62230) - 62230 f(1100; 92) - 930 f(14; 62660) = 62660 f(14; 930) - 1100 f(92; 63640) = 63640 f(92; 1100) - 14 f(62660; 930) = 930 f(14; 62660) - 62660 f(14; 930) - 92 f(63640; 1100) = 1100 f(92; 63640) - 63640 f(1100; 92) - 930 f(14; 64070) = 64070 f(14; 930) - 1100 f(92; 65050) = 65050 f(92; 1100) - 14 f(64070; 930) = 930 f(14; 64070) - 64070 f(14; 930) - 92 f(65050; 1100) = 1100 f(92; 65050) - 65050 f(1100; 92) - 930 f(14; 65480) = 65480 f(14; 930) - 1100 f(92; 66460) = 66460 f(92; 1100) - 14 f(65480; 930) = 930 f(14; 65480) - 65480 f(14; 930) - 92 f(66460;