เสริมคิดคณิตศาสตร์ม.ต้น 49-50

3. ¡’‰¢àÕ¬àŸ∑—ÈßÀ¡¥‰¡à‡°‘π 500 øÕß ∂â“𔉪„ à„πμ–°√â“„∫≈– 2, 3, 4, 5 À√◊Õ 6 øÕß ‡∑à“ Ê °—π ®–‡À≈◊Õ‰¢à 1 øÕ߇ ¡Õ ·μà∂â“„ àμ–°√â“„∫≈– 7 øÕß ‡∑à“ Ê °—π ®–‰¡à¡’‰¢à‡À≈◊Õ ®ßÀ“«à“ ¡’‰¢à∑—ÈßÀ¡¥°’ËøÕß ·π«§‘¥ «‘‡§√“–Àå n = 60k + 1 (§.√.π. ¢Õß 2, 3, 4, 5, 6 §◊Õ 60) ¥—ßπ—Èπ 7| (60k + 1) ≈Õß·∑π k ¥â«¬ 1 n = 61 ≈Õß·∑π k ¥â«¬ 2 n = 121 ≈Õß·∑π k ¥â«¬ 3 n = 181 ≈Õß·∑π k ¥â«¬ 4 n = 241 ≈Õß·∑π k ¥â«¬ 5 n = 301 À√◊ÕÀ“μ—«∑’Ë„°≈⇧’¬ß°—∫ 60k ·μàÀ“√¥â«¬ 7 ≈ßμ—« §◊Õ 56k ¥—ßπ—Èπ 7| (60k + 1) › 56k π—Ëπ§◊Õ 7| (4k + 1) ∂â“ k = 5 ®–∑”„Àâ 7| (4k + 1) ¥—ßπ—Èπ ®÷ß·∑π k ¥â«¬ 5 ≈ß„π 4k + 1 ®–‰¥â 7| ((60)(5) + 1) 7| (301) μÕ∫ 301 øÕß 98 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

4. ∂â“¡’‡ß‘π 3,440 ∫“∑ ‡ªìπ∏π∫—μ√©∫—∫≈– 20 ∫“∑ 50 ∫“∑ ·≈– 100 ∫“∑ √«¡ 72 ©∫—∫ ·≈â«®–¡’∏π∫—μ√∑—Èß “¡™π‘¥„π‡ß◊ËÕπ‰¢∑’Ë°”Àπ¥„À≥Ⱂ˫‘∏’ ·π«§‘¥ «‘∏’∑’Ë 1 ➀ ‡¡◊ËÕ¡’∏π∫—μ√©∫—∫≈– 20 ∫“∑ ®”π«π 2 ©∫—∫ ‡ªìπ‡ß‘π 40 ∫“∑ ‡À≈◊Õ©∫—∫≈– 50 ∫“∑ ·≈– 100 ∫“∑ √«¡ 70 ©∫—∫ ‡ªìπ‡ß‘π 3,400 ∫“∑ ©∫—∫≈– 50 ∫“∑ ®”π«π x ©∫—∫ ‡ªìπ‡ß‘π 50x ∫“∑ ©∫—∫≈– 100 ∫“∑ ®”π«π 70 › x ©∫—∫ ‡ªìπ‡ß‘π 7,000 › 100x ∫“∑ √«¡ Õß©∫—∫‡ªìπ‡ß‘π 50x + 7,000 › 100 x = 3,400 ∫“∑ 4,600 = 50 x x = 4,600 = 92 ©∫—∫ 50 ¡’∏π∫—μ√©∫—∫≈– 50 ∫“∑ 92 ©∫—∫ ‡ªìπ‰ª‰¡à‰¥â ➁ ‡¡◊ËÕ¡’∏π∫—μ√©∫—∫≈– 20 ∫“∑ ®”π«π 7 ©∫—∫ ‡ªìπ‡ß‘π 140 ∫“∑ ‡À≈◊Õ©∫—∫≈– 50 ∫“∑ ·≈– 100 ∫“∑ √«¡ 65 ©∫—∫ ‡ªìπ‡ß‘π 3,300 ∫“∑ ©∫—∫≈– 50 ∫“∑ ®”π«π x ©∫—∫ ‡ªìπ‡ß‘π 50x ∫“∑ ©∫—∫≈– 100 ∫“∑ ®”π«π 65 › x ©∫—∫ ‡ªìπ‡ß‘π 6,500 › 100x ∫“∑ √«¡ Õß©∫—∫‡ªìπ‡ß‘π 50 x + 6,500 › 100 x = 3,300 ∫“∑ 50 x = 3,200 x = 64 ®”π«π©∫—∫ ©∫—∫≈– 50 ∫“∑ ©∫—∫≈– 100 ∫“∑ À¡“¬‡Àμÿ ©∫—∫≈– 20 ∫“∑ (®”π«π‡ß‘π) (®”π«π‡ß‘π) (®”π«π‡ß‘π) (®”π«π‡ß‘π) §”μÕ∫∑’Ë... 1 7(140) 64(3,200) 1(100) 2 12(240) 56(2,800) 4(400) 3 17(340) 48(2,400) 7(700) 4 22(440) 40(2,000) 10(1,000) 5 27(540) 32(1,600) 13(1,300) μÕ∫ 8 «‘∏’ 6 32(640) 24(1,200) 16(1,600) 7 37(740) 16(800) 19(1,900) 8 42(840) 8(400) 22(2,200) 9 47(940) 0(0) 25(2,500) ‰¡à¡’∏π∫—μ√©∫—∫≈– 50 ∫“∑ À¡“¬‡Àμÿ ‡æ‘Ë¡∑’≈– 5 ©∫—∫ ≈¥∑’≈– 8 ©∫—∫ ‡æ‘Ë¡∑’≈– 3 ©∫—∫ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 99

·π«§‘¥ «‘∏’∑’Ë 2 °”Àπ¥„Àâ x ·∑π®”π«π∏π∫—μ√©∫—∫≈– 20 ∫“∑ y ·∑π®”π«π∏π∫—μ√©∫—∫≈– 50 ∫“∑ ·≈– z ·∑π®”π«π∏π∫—μ√©∫—∫≈– 100 ∫“∑ ®–‰¥â«à“ x+y+z = 72 ......................... ➊ 2x + 5y + 10z = 344 ......................... ➋ ®“° ➊ z = 72 › x › y ·∑π„π ➋ 2x + 5y + 10(72 › x › y) = 344 2x + 5y + 720 › 10x › 10y = 344 ›8x › 5y = 344 › 720 ›8x › 5y = ›376 8x + 5y = 376 8x = 376 › 5y x = 47 › 5y 8 §”μÕ∫∑’Ë 1 y = 8, x = 47 › 5 = 42, z = 22 §”μÕ∫∑’Ë 2 y = 16, x = 47 › 10 = 37, z = 19 §”μÕ∫∑’Ë 3 y = 24, x = 47 › 15 = 32, z = 16 §”μÕ∫∑’Ë 4 y = 32, x = 47 › 20 = 27, z = 13 §”μÕ∫∑’Ë 5 y = 40, x = 47 › 25 = 22, z = 10 §”μÕ∫∑’Ë 6 y = 48, x = 47 › 30 = 17, z = 7 §”μÕ∫∑’Ë 7 y = 56, x = 47 › 35 = 12, z = 4 §”μÕ∫∑’Ë 8 y = 64, x = 47 › 40 = 7, z = 1 ·μà y = 72, x = 47 › 5(72) = 2, z = ›2 ‡ªìπ‰ª‰¡à‰¥â 8 μÕ∫ 8 «‘∏’ 100 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

5. ∂â“ a ‡ªìπ À.√.¡. ¢Õß 234 °—∫ 324 ·≈– b, c ‡ªìπ®”π«π‡μÁ¡„π™à«ß ›10 ∂÷ß 10 ∑’Ë∑”„Àâ a = 234b + 324c ·≈â« a + b + c ¡’§à“‡∑à“„¥ ·π«§‘¥ ‡π◊ËÕß®“° 234 = 2 Ó 32 Ó 13 ·≈– 324 = 22 Ó 34 À.√.¡. ¢Õß 234 °—∫ 324 §◊Õ 2 Ó 32 = 18 = a ®“° a = 234b + 324c 18 = 2 Ó 32 (13b + 18c) 18 = 18(13(7) + 18(›5)) ®–‰¥â b = 7, c = ›5 ¥—ßπ—Èπ a + b + c = 18 + 7 › 5 = 20 μÕ∫ 20 6. (1 › 1 ) (1 › 1 ) (1 › 1 ) ... (1 › 1 ) ¡’§à“‡∑à“„¥ 22 32 42 (2007)2 ·π«§‘¥ (1 › 1 ) (1 › 1 ) (1 › 1 ) ... (1 › 1 ) 22 32 42 (2007)2 = ( 22 › 1)( 32 › 1 )( 42 › 1) ... (20(2007027›)21) 22 32 42 = (2 + 1)(2 › 1) Ó (3 + 1)(3 › 1)Ó (4 + 1)(4 › 1) Ó (5 + 1)(5 › 1)Ó... Ó(2007 + 1)(2007 › 1) 22 32 42 52 20072 = (3)(1)(4)(2)(5)(3)(6)(4) ... (2006 + 1)(2005)(2008)(2006) 22 32 42 52... (2007)2 = (2)(3)2(4)2(5)2... (2005)2(2006)2(2007)(2008) 22 32 42 52 ... (2007)2 = 1 Ó 2008 = 1004 2 2007 2007 μÕ∫ 1004 2007 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 101

7. ∂â“ A = 0.5(22550 + 2› 2550), B = 0.5(22550 › 2› 2550) ·≈â« A2 › B2 ¡’§à“‡∑à“„¥ ·π«§‘¥ A2 › B2 = (A › B)(A + B) = (2›2550)(22550) = 20 =1 μÕ∫ 1 8. ∂â“ x ‡∑à“°—∫ x% ¢Õß y ·≈– y ‡∑à“°—∫ y% ¢Õß z ‚¥¬∑’Ë x, y, z ‡ªìπ®”π«π®√‘ß∫«° ·≈â« y + z ¡’§à“‡∑à“„¥ ·π«§‘¥ ®“° y = y% ¢Õß z ®–‰¥â y = yz 100 100y › yz = 0 y(100 › z) = 0 ·μà y > 0 ¥—ßπ—Èπ 100 › z = 0 π—Ëπ§◊Õ z = 100 ®“° x = x% ¢Õß y ®–‰¥â x = x y 100 ¥—ßπ—Èπ y = 100 ®–‰¥â y + z = 100 + 100 = 200 μÕ∫ 200 9. §”μÕ∫¢Õß ¡°“√ x › x › x › x › 99 = 99 ¡’§à“‡∑à“„¥ ·π«§‘¥ ®“° x › x › x › x › 99 = 99 99 = x › x › x › x › 99 ®–‰¥â x › 99 = 99 x › 99 = 992 x = 992 + 99 = 99(99 + 1) = 99 Ó 100 μÕ∫ 9900 102 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

10. ®“°√Ÿª Δ ABC ¡’ M ‡ªìπ®¥ÿ °÷Ëß°≈“ߢÕߥâ“π BC ∂â“ AB = 4 Àπ૬, AM = 3 Àπ૬ ·≈– AC = 8 Àπ૬ ·≈â« BC ¬“«°’ËÀπ૬ A 8 43 C BM ·π«§‘¥ ®“°√Ÿª„π Δ ABO; AO2 = 42 › (x › a)2 ........ ➊ A 8 „π Δ AOM; AO2 = 32› a2 ........ ➋ 3 4 ➊ = ➋ ®–‰¥â 42 › (x › a)2 = 32› a2 x-a C 16 › x2 + 2ax ›a2 = 9 › a2 B Oa x C 2x x2 › 2ax › 7 = 0 ....... ➌ „π Δ AOC; AO2 = 82› (a+x)2 ....... ➍ ➊ = ➍ ®–‰¥â 16 › x2+ 2ax ›a2 = 64 ›a2› 2ax ›x2 4ax = 48 ax = 12 𔉪·∑π„π ➌ ‰¥â x2 ›2(12) › 7 = 0 x2 = 31 x = 31 ·μà BC = 2x π—Ëπ§◊Õ ¥â“π BC ¬“« 2 31 Àπ૬ μÕ∫ 2 31 Àπ૬ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 103

11. ∂â“ N = 10296 › 10259 + 10222 › 10185 + 10148 › 10111 + 1074 › 1037 + 1 ·≈– 1 = 0.d⋅ d d ... d⋅ ‡ªìπ∑»π‘¬¡´È” ´÷Ëß¡’‡≈¢‚¥¥´È”„π·μà≈–™¥ÿ πâÕ¬∑’Ë ÿ¥ m ®”π«π N 1 2 3 m ·≈â« m ¡’§à“‡∑à“„¥ ·π«§‘¥ 1 = 0.d d d ...d N 123 m 10m › 1 ®–‰¥â N|(10m › 1) (1037 + 1)N = 10333 + 1 (10333 › 1)(1037 + 1)N = 10666 › 1 ®–‰¥â N|(10666 › 1) ¥—ßπ—Èπ m = 666 μÕ∫ 666 12. ∂â“°”Àπ¥æÀÿπ“¡ (x + 5)3 + (3x › 2)4 › (2x + 1)5 ·≈â« —¡ª√– ‘∑∏‘Ï¢Õß x3 ¡’§à“‡∑à“„¥ ·π«§‘¥  —¡ª√– ‘∑∏‘Ï¢Õß x3 ¢Õß (x + 5)3 §◊Õ 1  —¡ª√– ‘∑∏‘Ï¢Õß x3 ¢Õß (3x › 2)4 §◊Õ 4 Ó 33 Ó (›2) = ›216  —¡ª√– ‘∑∏‘Ï¢Õß x3 ¢Õß (2x + 1)5 §◊Õ 10 Ó 23 Ó 1 = 80 ¥—ßπ—Èπ  —¡ª√– ‘∑∏‘Ï¢Õß x3 ®“° (x + 5)3 + (3x › 2)4› (2x + 1) §◊Õ 1 + (›216) › 80 = ›295 μÕ∫ ›295 104 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

13. ∂â“ x + y = 11 ·≈– x2 + y2 = 16 ·≈â« x4 + y4 ¡’§à“‡∑à“„¥ ·π«§‘¥ x4 + y4 = (x2 + y2)2 › 2x2y2 = (16)2 › 2x2y2 .................... ➊ ®“° x + y = 11 .................... ➋ ➋2 ®–‰¥â«à“ x2 + y2+ 2xy = 11 16 + 2xy = 11 2xy = ›5 xy = ›5 ·∑π„π ➊ 2 x4 + y4 = (16)2 › 2(›5 )2 2 = 256 › 2( 25 ) 4 = 256 › ( 25 ) 2 = 256 › 12.5 = 243.5 μÕ∫ 243.5 14. °”Àπ¥„Àâ x ‡ªìπ®”π«π‡μÁ¡∫«° ·≈– n(x) ‡ªìπº≈∫«°¢Õ߇≈¢‚¥¥∑’ˇ¢’¬π·∑π x ‡™àπ n(517) = 5 + 1 + 7 = 13 ·≈– n(3229) = 3 + 2 + 2 + 9 = 16 ‡ªìπμâπ ∂â“ y = (10k + 2 + 3.10k)2 ‡¡◊ËÕ k ‡ªìπ®”π«π‡μÁ¡∫«°·≈â« n(y) ¡’§à“‡∑à“„¥ ·π«§‘¥ y = (10k + 2 + 3.10k)2 ®“° ®–‰¥â y = (10k (10 2 + 3.))2 = (10k (100 + 3))2 = (10k(103))2 = 102k(10609) ¥—ßπ—Èπ n(y) = 1 + 0 + 6 + 0 + 9 = 16 μÕ∫ 16 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 105

15. ∂â“π”°√«¬‚≈À–ª≈“¬μ—¥∑’Ë¡’√—»¡’∞“π 2 ‡´π쑇¡μ√ ·≈– 5 ‡´π쑇¡μ√ ¡“À≈Õ¡‡ªìπ≈°Ÿ ∫“»°å ¬“«¥â“π≈– 2 ‡´π쑇¡μ√ ·≈â«®–‰¥â≈Ÿ°∫“»°å∑—ÈßÀ¡¥ 143 ≈Ÿ° ®ßÀ“«à“°√«¬‚≈À– ª≈“¬μ—¥π’È¡’§«“¡ ßŸ °’ˇ´π쑇¡μ√ (°”Àπ¥ π = 22) 7 ·π«§‘¥ A Δ ABC ~ Δ ADE 2x AB = 2x ¥—ßπ—Èπ BD = 3x BC ª√‘¡“μ√°√«¬ª≈“¬μ—¥ 3x 1 π(5)2(5x) › 1 π22(2x) = 1 πx (125 › 8) ≈∫.´¡. D 3 3 3 5 ´¡. E 117πx = ≈∫.´¡. 3 π”¡“À≈Õ¡‡ªìπ≈Ÿ°∫“»°å¬“«¥â“π≈– 2 ´¡. ‰¥â 143 ≈Ÿ° ª√‘¡“μ√°√«¬ª≈“¬μ—¥ 117πx = 23 Ó 143 ≈∫.´¡. 3 x = 8 Ó 143 Ó 3 Ó 7´¡. 117 Ó 22 3x = 28 Ó 3 ´¡. 3 °√«¬‚≈À–ª≈“¬μ—¥¡’§«“¡ ßŸ 28 ‡´π쑇¡μ√ μÕ∫ 28 ‡´π쑇¡μ√ 16. ∂â“ a + b + c = 4 ·≈– 1 + 1 + 1 = 0 ·≈â« a2 + b2 + c2 ¡’§à“‡∑à“„¥ ab c ·π«§‘¥ ®“° a + b + c = 4 ®–‰¥â a2 + b2 + c2 + 2ab + 2ac + 2bc = 16 ..................... ➊ ·≈– 1 + 1 + 1 = 0 ab c ®–‰¥â bc + ac + ab = 0 2bc + 2ac + 2ab = 0 𔉪·∑π„π ➊ ®–‰¥â a2 + b2+ c2 + 0 = 16 π—Ëπ§◊Õ a2 + b2 + c2 = 16 μÕ∫ 16 106 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

17. °”Àπ¥ P ‡ªìπæ“√“‚∫≈“ y = x2 › 3x ∂â“ P′ ‰¥â®“°°“√‡≈◊ËÕπ P ¢π“π‰ª¢â“ß∫π ( )4 Àπ૬ ·≈–‰ª∑“ߴ⓬ 3 Àπ૬ ·≈– P′ ºà“π®¥ÿ › 2, 2a › 4 ·≈â« a ¡’§à“‡∑à“„¥ 5 ·π«§‘¥ y = x2 › 3x ‡≈◊ËÕπ·°π‰ª‡ªìπ√–¬–∑“ß (›3, 4) ¥—ßπ—Èπ y › 4 = (x + 3)2 › 3(x + 3) = x2 + 6x + 9 › 3x › 9 y = x2 + 3x + 4 ºà“π®¥ÿ (›2, 2a › 4 ) ¥—ßπ—Èπ y = 2a › 4 = (›2)2 + 3(›2) + 4 55 2a › 4 = 5(4 + (›6) + 4) 2a = 5 Ó 2 + 4 a = 14 =7 2 μÕ∫ 7 18. §«“¡‡√«Á ¢Õßπ”È ∑‰Ë’ À≈º“à π°Õä °∑ÀË’ πßË÷ ·≈–°Õä °∑’ Ë Õß ‡ªπì Õμ— √“ «à π 2 : 3 ∂“â ‡ª¥î °Õä °∑ Ë’ Õß ·≈–°äÕ°∑’Ë “¡æ√âÕ¡°—π ·≈â«πÈ”®–‰À≈‡¢â“∂—ß≈Ÿ°∫“»°å´÷Ëß°«â“ߥâ“π≈– 1.2 ‡¡μ√ ‚¥¬„™â ‡«≈“ 20 π“∑’ ®÷ß®–‡μÁ¡∂—ß ·≈–∂Ⓡªî¥°äÕ°∑’ËÀπ÷Ëß°—∫°äÕ°∑’Ë “¡æ√âÕ¡°—π ·≈â«πÈ”®–‰À≈ ÕÕ°®“°∂—ߥ—ß°≈à“«‚¥¬„π‡«≈“ 5 π“∑’ ®–‡À≈◊ÕπÈ” 4 ¢Õß∂—ß ®ßÀ“«à“πÈ”‰À≈ºà“π°äÕ°∑’Ë “¡ 5 ¥â«¬§«“¡‡√Á«°’Ë≈‘μ√μàÕπ“∑’ ·π«§‘¥ °äÕ°∑’Ë 1 πÈ”‰À≈ 2x ≈‘μ√/π“∑’ °äÕ°∑’Ë 2 πÈ”‰À≈ 3x ≈‘μ√/π“∑’ °äÕ°∑’Ë 3 πÈ”‰À≈ y ≈‘μ√/π“∑’ ‡ªî¥°äÕ°∑’Ë 2 ·≈– 3 πÈ”‰À≈ 3x + y ≈‘μ√/π“∑’ „™â‡«≈“ 20 π“∑’ πÈ”‰À≈‡¢â“ 20(3x + y) = 1.2 Ó 1.2 Ó 1.2 Ó 1000 ≈‘μ√ 3x + y = 86.4 ...................... ➊ ‡ªî¥°äÕ°∑’Ë 1 ·≈– 3 æ√âÕ¡°—∫πÈ”‰À≈ 2x + y ≈‘μ√/π“∑’ „™â‡«≈“ 5 π“∑’ πÈ”‰À≈ 5(2x + y) = 1 Ó 1.2 Ó 1.2 Ó 1.2 Ó 1000 ≈‘μ√ 5 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 107

2x + y = 69.12 ..................... ➋ x = 17.28 ≈‘μ√ y = 69.12 › 34.56 ≈‘μ√ = 34.56 ≈‘μ√ μ√«®§”μÕ∫ ‡ªî¥°äÕ° 2 ·≈– 3 πÈ”‰À≈ 3 Ó 17.28 + 34.56 = 86.4 ≈‘μ√/π“∑’ ‡«≈“ 20 π“∑’ πÈ”‰À≈ 86.4 Ó 20 = 1728.0 = 12 Ó 12 Ó 12 ≈‘μ√ ‡ªî¥°äÕ°∑’Ë 1 ·≈– 3 æ√âÕ¡°—∫πÈ”‰À≈ 2 Ó 17.28 + 34.56 = 69.12 ≈‘μ√/π“∑’ ‡«≈“ 5 π“∑’ πÈ”‰À≈ 5 Ó 69.12 = 345.6 ≈‘μ√ ‡ªìπ 12 345.6 12 = 0.2 ¢Õß∂—ß Ó 12 Ó ‡À≈◊ÕπÈ” 1 › 0.2 = 0.8 = 4 ¢Õß∂—ß 5 ¥—ßπ—Èπ °äÕ°∑’Ë 3 πÈ”‰À≈¥â«¬§«“¡‡√Á« 34.56 ≈‘μ√/π“∑’ μÕ∫ 34.56 ≈‘μ√/π“∑’ 19. °√–¥“πÀ¡“°√ÿ°°√–¥“πÀπ÷Ë߉¥â√—∫°“√√–∫“¬ ’μ“¡™àÕ߬àÕ¬ Ê ¥â«¬ ’¢“«À√◊Õ ’πÈ”‡ß‘π ™àÕß≈–Àπ÷Ëß ’‡∑à“π—Èπ ‚¥¬æ∫«à“„π∑ÿ°√Ÿª ’ˇÀ≈’ˬ¡º◊πºâ“∑’Ë¡’ 6 ™àÕ߬àÕ¬ (¢π“¥ 2 Ó 3 À√◊Õ 3 Ó 2) ®–¡’ 2 ™àÕ߬àÕ¬‡ªìπ ’πÈ”‡ß‘π  à«π∑’ˇÀ≈◊Õ‡ªìπ ’¢“«‡ ¡Õ ∂â“°√–¥“ππ’È¡’¢π“¥ 9 Ó 11 ·≈â«®–¡’™àÕ߬àÕ¬∑’ˇªìπ ’πÈ”‡ß‘π∑—ÈßÀ¡¥°’Ë™àÕß ·π«§‘¥ μÕ∫ 33 ™àÕß 108 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

20. ®“°√ªŸ «ß°≈¡ O ·≈–«ß°≈¡ P μ¥— °π— ∑®’Ë ¥ÿ C ·≈– D ‚¥¬¡‡’  πâ μ√ß AB ‡ªπì ‡ πâ  ¡— º — √«à ¡ ∂â“ AC^B = 107 Ì ·≈â« A^DB ¡’¢π“¥°’ËÕß»“ A B C OP ·π«§‘¥ D ≈“° CD ®–‰¥â Δ ACD ·≈– Δ BCD ‡ªìπ√ªŸ  “¡‡À≈’ˬ¡∑’Ë·π∫„π«ß°≈¡ }1. B^AC = A^DC ¡¡ÿ ∑’ˇ âπ —¡º— ®√¥°—∫§Õ√奡’¢π“¥‡∑à“°—∫¡¡ÿ „π A 107 Ì 2. A^BC = B^DC  à«π‚§âßμ√ߢⓡ CB 3. AC^B = 107 Ì °”Àπ¥„Àâ 4. BA^C + A^BC + A^CB = 180 Ì º≈∫«°¡ÿ¡¿“¬„π√ªŸ  “¡‡À≈’ˬ¡ O P 5. BA^C + A^BC = 73 Ì ®“°¢âÕ 3, 4 D 6. ·μà BA^C + A^BC = A^DC + B^DC ®“°¢âÕ 1, 2  ¡∫—μ‘°“√‡∑à“°—π 7. ¥—ßπ—Èπ A^DC + BD^C = 73 Ì ®“°¢âÕ 5, 6 μÕ∫ 73 Õß»“ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 109

μÕπ∑’Ë 2 21. ∂â“ ABC ‡ªìπ√ªŸ  “¡‡À≈’ˬ¡Àπâ“®—Ë«∑’Ë¡’ AB = AC, B^AC = 66 Ì, D ‡ªìπ®ÿ¥∫π AB, E ‡ªìπ®¥ÿ ∫π AC ∑”„Àâ DE = BD + CE ·≈– I ‡ªìπ®¥ÿ ∫π BC ∑’Ë∑”„Àâ DI ·∫àߧ√÷Ëß B^DE ·≈â« D^IE ¡’¢π“¥°’ËÕß»“ ·π«§‘¥ A 66 Ì ( ( DM ( E (57 Ì ( (57 Ì C 57 Ì BI „Àâ M ‡ªìπ®ÿ¥∫π DE ∑”„Àâ BD = MD ≈“° IM ®–‰¥â Δ DBI ≅ Δ DMI (¥.¡.¥.) ¥—ßπ—Èπ DM^I = 57 Ì ·≈– B^ID = M^ID ·≈– EM = EC IM^E + EC^I = 180 Ì ¥—ßπ—Èπ IMEC ·π∫„π«ß°≈¡ E^IM = E^IC ‡æ√“–©–π—Èπ D^IE = M^ID + E^IM = 180 Ì = 90 Ì 2 μÕ∫ 90 Õß»“ 110 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

22. °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡Àπâ“®—Ë«∑’Ë¡’ AB = AC, P ‡ªìπ®¥ÿ ∫π¥â“π AC ·≈– Q ‡ªìπ®ÿ¥∫π¥â“π AB ∑”„Àâ AP = PQ = QB = BC ·≈â« BA^C ¡’¢π“¥°’ËÕß»“ ·π«§‘¥ A ≈“° QR æ∫ AC ∑’Ë R 4θ „Àâ QP = QR ·≈– Q^AP = 4θ P ®–‰¥â Q^PR = Q^RP = 8θ BQ^R = 180 Ì › (4θ + 180 Ì › 16θ) = 12θ ‰¥â Q^BR = Q^RB = 90 Ì › 6θ Q ¥—ßπ—Èπ B^RC = 180 Ì › (90 Ì › 6θ) › 8θ = 90 Ì › 2θ R ·≈– B^CR = 180 Ì › 4θ = 90 Ì › 2θ 2 ®–‰¥â BR = BC B C ¥—ßπ—Èπ Δ BQR ‡ªìπ√ªŸ  “¡‡À≈’ˬ¡¥â“π‡∑à“ 12θ = 60 Ì, 4θ = 20 Ì ‡æ√“–©–π—Èπ BA^C = 20 Ì μÕ∫ 20 Õß»“ 23. ∂â“ a, b, c ·≈– d ‡ªìπ®”π«π®√‘ß ´÷Ëß Õ¥§≈âÕß°—∫ ¡°“√ a = 82 › 58 › a b = 82 + 58 › b c = 82 › 58 + c ·≈– d = 82 + 58 + d ·≈â« abcd ¡’§à“‡∑à“„¥ ·π«§‘¥ a2 = 82 › 58 › a (a2 › 82)2 = 58 › a a4 › 164a2 + a + 6666 = 0 „π∑”πÕ߇¥’¬«°—π®–‰¥â b4 › 164b2 + b + 6666 = 0 c4 › 164c2 › c + 6666 = 0 d4 › 164d2 › d + 6666 = 0 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 111

®–‰¥â a, b, ›c, ›d ‡ªìπ§”μÕ∫¢Õß ¡°“√ x4 › 164x2 › x + 6666 = 0 ¥—ßπ—Èπ abcd = 6666 μÕ∫ 6666 24. ∂â“ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡Àπâ“®—Ë«∑’Ë¡’ BA^C = 100 Ì, M ‡ªìπ®ÿ¥¿“¬„π∑”„Àâ M^BA = 10 Ì ·≈– M^CA = 5 Ì ·≈â« BM^A ¡’¢π“¥°’ËÕß»“ ·π«§‘¥ A EM 10 Ì 5Ì 5Ì B 30 Ì 30 Ì DC ≈“° AD ⊥ BC ∑’Ë D μ—¥ BM ∑’Ë E ≈“° CE ®–‰¥â C^AD = 50 Ì AE^M = CE^M = 60 Ì (¡ÿ¡¿“¬πÕ°√Ÿª “¡‡À≈’ˬ¡¡’¢π“¥‡∑à“°—∫º≈∫«° ¢Õß¡ÿ¡¿“¬„π∑’ˉ¡à„™à¡¡ÿ ª√–™‘¥) AC^M = EC^M = 5 Ì ¥—ßπ—Èπ M ‡ªìπ®ÿ¥»πŸ ¬å°≈“ߢÕß«ß°≈¡·π∫„π√ªŸ  “¡‡À≈’ˬ¡ AEC ©–π—Èπ EA^M = CA^M = 25 ‡æ√“– AM ·∫àߧ√÷Ëß EA^C ·≈– AM^E = 180 Ì › 60 Ì › 25 Ì = 95 Ì μÕ∫ 95 Õß»“ 112 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

25. ∂â“ √Ÿª ’ˇÀ≈’ˬ¡ ABCD ¡’ BA^C = 81 Ì, C^AD = 27 Ì, AB^D = 36 Ì ·≈– CB^D = 30 Ì ·≈â« A^DC ¡’¢π“¥°’ËÕß»“ ·π«§‘¥ D 36 Ì 24 Ì C A 54 Ì B^DA = 36 Ì = AB^D 66 Ì 81 Ì ∴ AB = AD 66 Ì ≈“° AE æ∫ BC ∑’Ë®ÿ¥ E „Àâ AE^B = 66 Ì 36 Ì 30 Ì ≈“° ED ®–‰¥â AB = AE ·≈– BA^E = 48 Ì E B EA^C = EC^A = 33 Ì AE = CE EA^D = 60 Ì ·≈– AE = AD ®–‰¥â Δ ADE ‡ªìπ√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“ ‰¥â AD = AE = ED ∴ E^DC = EC^D = 180 Ì › 54 Ì = 63 Ì 2 A^DC = 123 Ì μÕ∫ 123 Õß»“ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 113

26. √Ÿª “¡‡À≈’ˬ¡ ABC ·π∫„π«ß°≈¡ O ‚¥¬¡’ P ‡ªìπ®ÿ¥°÷Ëß°≈“ß OA ·≈– Q ‡ªìπ ®ÿ¥°÷Ëß°≈“ß BC ∂â“ A^BC ¡’¢π“¥‡ªìπ 4 ‡∑à“¢Õß¢π“¥¢Õß O^PQ ·≈– A^CB ¡’¢π“¥‡ªìπ 6 ‡∑à“¢Õß¢π“¥¢Õß O^PQ ·≈â« O^PQ ¡’¢π“¥°’ËÕß»“ ·π«§‘¥ A P C O BQ „Àâ O^PQ = x ®–‰¥â A^BC = 4x ·≈– AC^B = 6x ≈“° OC ®–‰¥â AO^C = 8x OA^C = OC^A = 90 Ì › 4x ¥—ßπ—Èπ OC^Q = 10x › 90 Ì QO^C = 180 Ì › 10x ¥—ßπ—Èπ P^OQ = 180 Ì › 2x OQ^P = x ∴ OP = OQ = OC 2 ∴ OC^Q = 30 Ì = 10x › 90 Ì x = 12 Ì μÕ∫ 12 Õß»“ 114 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

27. °”Àπ¥„Àâ AB ‡ªìπ‡ âπºà“π»πŸ ¬å°≈“ß«ß°≈¡∑’Ë¡’√—»¡’ 99 Àπ૬ ·≈– P ‡ªìπ®¥ÿ ¿“¬„π ®ßÀ“§«“¡πà“®–‡ªìπ∑’Ë A^PB ≤ 135 Ì (μÕ∫„π√ªŸ ¢Õß π) ·π«§‘¥ A P D B11111111111111111111111111111111111112222222222222222222222222222222222222333333333333333333333333333333333333344444444444444444444444444444444444445555555555555555555555555555555555555666666666666666666666666666666666666677777777777777777777777777777777777778888888888888888888888888888888888888999999999999999999999999999999999999900000000000000000000000000000000000001111111111111111111111111111111111111222222222222222222222222222222222222233333333333333333333333333333333333334444444444444444444444444444444444444555555555555555555555555555555555555566666666666666666666666666666666666667777777777777777777777777777777777777888888888888888888888888888888888888899999999999999999999999999999999999990000000000000000000000000000000000000111111111111111111111111111111111111122222222222222222222222222222222222223333333333333333333333333333333333333444444444444444444444444444444444444455555555555555555555555555555555555556666666666666666666666666666666666666777777777777777777777777777777777777788888888888888888888888888888888888889999999999999999999999999999999999999000000000000000000000000000000000000011111111111111111111111111111111111112222222222222222222222222222222222222111111111111111111111111111111111111122222222222222222222222222222222222223333333333333333333333333333333333333444444444444444444444444444444444444455555555555555555555555555555555555556666666666666666666666666666666666666( „Àâ C ‡ªìπ®¥ÿ »πŸ ¬å°≈“ß√—»¡’ BC ‡¢’¬π à«π‚§âß ‡≈◊Õ°®ÿ¥ P ∫π à«π‚§âß ∑’Ë∑”„Àâ A^PB = 135 Ì BC = 99 2 C æ◊Èπ∑’ˇ´°‡¡πμå AB = 1 π(99 2)2 › 1 Ó (99 2)2 4 2 æ◊Èπ∑’Ë·√‡ß“ = π(99)2 › 2 æ◊Èπ∑’ˇ´°‡¡πμå AB = 2 Ó 992 n(S) = π Ó 992 n(E) = 2 Ó 992 ∴ p(E) = 2 2 π μÕ∫ π 28. °”Àπ¥„Àâ (x , y ), (x , y ) ·≈– (x , y ) ‡ªìπ§”μÕ∫¢Õß√–∫∫ ¡°“√ 11 22 33 x3 › 3xy2 = 1999 y3 › 3x2y = 1998 ·≈â« 1 ¡’§à“‡∑à“„¥ (1 › x ) (1 › x ) (1 › x ) 1 2 3 y y y 123 ·π«§‘¥ x3 › 3xy2 › 1999 (y3 › 3x2y) = 0 1998 x3 + 1999 .3x2y › 3xy2 ›1999 y3 = 0 1998 1998 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 115

π” y3 À“√∑—Èß ¡°“√ ( x )3 + 5997 (x )2 › 3(x ) ›1999 = 0 y 1998 y y 1998 x xx ®–¡’ 1 , 2 , 3 ‡ªìπ§”μÕ∫¢Õß ¡°“√ y y y 1 23 „Àâ P(a) = a3 + 5997 a2 › 3a › 1999 1998 1998 xx x y1, 2 3 ¡’ a = ·≈– y y 12 3 P(1 › a) = (1 › a)3 + 5997 (1 › a)2 › 3(1 › a) › 1999 1998 1998 x x x 1 y2, 1 › ¡’§”μÕ∫¢Õß P(1 › a) = 0 §◊Õ 1 › , 1 › 3 y 2 1 y 3 ®–‰¥â (1 › x )(1 › x )(1 › x ) = 1 + 5997 › 3 › 1999 1998 1998 1 2 3 y y y 12 3 =1 999 ¥—ßπ—Èπ 1 x = 999 xx y1)(1 2 y3) (1 › › )(1 › y 12 3 μÕ∫ 999 116 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

29. ∂â“°”Àπ¥√–∫∫ ¡°“√ 10x2 + 5y2 › 2xy › 38x › 6y + 41 = 0 3x2 › 2y2 + 5xy › 17x › 6y + 20 = 0 ·≈â«§à“¢Õß x3 + y3 ‡ªìπ‡∑à“„¥ ·π«§‘¥ ®“°√–∫∫ ¡°“√®–‰¥â 7x2 + 7y2 › 7xy › 21x + 21 = 0 x2 › xy + y2 › 3x + 3 = 0 (x2 › 2xy + y)2 + ( 3x2 › 6x + 3) = 0 4 2 4 2 ( x › y)2 + 3( x › 1)2 = 0 22 ∴ ®”π«π®√‘ß a , a2 ≥ 0 ∴ x › y = 0 ·≈– x › 1 = 0 22 x = 2, y = 1 ¥—ßπ—Èπ x3 + y3 = 9 μÕ∫ 9 30. ∂â“°”Àπ¥√–∫∫ ¡°“√ a + a + a + ... + a = 96 123 n a 2 + a 2 + a 2 + ... + a 2 = 144 12 3 n a 3 + a 3 + a 3 + ... + a 3 = 216 12 3 n ‡¡◊ËÕ a ‡ªìπ®”π«π®√‘ß∫«°  ”À√—∫∑ÿ° i = 1, 2, 3, ..., n ·≈â« a 4 + a 4 + a 4 + ... + a 4 i 123 n ¡’§à“‡∑à“„¥ ·π«§‘¥ (a + a + a + ... + a )(a 3 + a 3 + a 3 + ... + a 3) = 96 Ó 216 = (144)2 123 n1 2 3 n ∴(a + a + a + ... + a )(a 3 + a 3 + a 3 + ... + a 3) = (a 2 + a 2 + 123 n1 2 3 n 12 a 2 + ... + a 2)2 3n ®–‰¥â a = a = a = ... = a 123 n ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 117

¥—ßπ—Èπ na = 96 1 na 2 = 144 1 na 3 = 216 1 (na 2 )(na 3) 144 Ó 216 1 1= na 96 1 na 4 = 324 1 ∴ a 4 + a 4 + a 4 + ... + a 4 = 324 123 n μÕ∫ 324 118 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

μ—«Õ¬à“ß ·∫∫∑¥ Õ∫§≥μ‘ »“ μ√å √–¥∫— ™«à ߙȗπ∑’Ë 3 (©∫—∫∑’Ë 2) ‡æ◊ÕË °“√§¥— ‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑» ªï æ.». 2550 §”™’È·®ß 1. ·∫∫∑¥ Õ∫©∫—∫π’ȇªìπ·∫∫‡μ‘¡§”μÕ∫ 2. ·∫∫∑¥ Õ∫©∫—∫π’È¡’®”π«π 30 ¢âÕ ·∫à߇ªìπ 2 μÕπ §–·ππ‡μÁ¡ 100 §–·ππ μÕπ∑’Ë 1 ®”π«π 20 ¢âÕ Ê ≈– 3 §–·ππ √«¡ 60 §–·ππ μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ Ê ≈– 4 §–·ππ √«¡ 40 §–·ππ 3. ‡«≈“∑’Ë„™â„π°“√ Õ∫ „™â‡«≈“ 1 ™—Ë«‚¡ß 30 π“∑’ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 119

μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 (©∫—∫∑’Ë 2) ‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑» ªï æ.». 2550 §”™’È·®ß ·∫∫∑¥ Õ∫©∫—∫π’ȇªìπ·∫∫‡μ‘¡§”μÕ∫ §–·ππ‡μÁ¡ 100 §–·ππ ¡’ 2 μÕπ ¥—ßπ’È μÕπ∑’Ë 1 ®”π«π 20 ¢âÕ ¢âÕ≈– 3 §–·ππ μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ ¢âÕ≈– 4 §–·ππ μÕπ∑’Ë 1 1. ( 22 )( 32 32 ) ( 42 ) ... ( 20072 ) ¡’§à“‡∑à“„¥ 22 › › 42 › 20072 › 1 1 1 1 2. °”Àπ¥ x ‡ªìπ®”π«π‡μÁ¡´÷Ëß Õ¥§≈âÕß°—∫ ¡°“√ x + 4 x › 4 + x › 4 x › 4 = 4 º≈∫«°¢Õß§à“ x ∑—ÈßÀ¡¥‡ªìπ‡∑à“„¥ 3. ∂â“ m ∗ n = m + n ·≈â« ((...(2550 ∗ 2549) ∗ 2548) ∗ ... ∗ 1) ∗ 0 ¡’§à“‡∑à“„¥ mn + 4 4. ®ß‡¢’¬π§ŸàÕ—π¥—∫ (x, y) ∑’Ë ¡“™‘°·μà≈–μ—«Õ¬àŸ„π√Ÿª‡≈¢¬°°”≈—ß ‚¥¬∑’Ë x ‡ªìπ®”π«π ∑’Ë¡“° ÿ¥¢Õß 235, 515 ·≈– 614 ·≈– y ‡ªìπ®”π«π∑’ËπâÕ¬∑’Ë ¥ÿ ¢Õß “¡®”π«ππ’È 5. ‡≈¢‚¥¥À≈—ß®ÿ¥∑»π‘¬¡μ”·Àπàß∑’Ë 8884 ¢Õß 88 ‡ªìπ‡∑à“„¥ 97 6. ∂â“æÀπÿ “¡ P(x) ∑’Ë¡’¥’°√’ 4 ·≈–¡’ ¡∫—μ‘¥—ßμàÕ‰ªπ’È P(0) = 0, P(1) = P(›1) = P(2) = P(›2) = 1 ·≈â« P(5) ¡’§à“‡∑à“„¥ 7. ‡¢’¬π®”π«π 365 ®”π«π√Õ∫«ß°≈¡ ®”π«π∑’Ë 88 §◊Õ 7 ®”π«π∑’Ë 111 §◊Õ 2 ®”π«π∑’Ë 224 §◊Õ 1 ®”π«π∑’Ë 365 §◊Õ 5 ∂⓺≈∫«°¢Õß∑ÿ° 60 ®”π«π∑’ˇ√’¬ßμ‘¥μàÕ°—π¡’§à“‡ªìπ 216 ‡ ¡Õ ·≈â«®”π«π∑’Ë 122 ¡’§à“‡∑à“„¥ 120 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

8. ∂â“ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡Àπâ“®—Ë«∑’Ë¡’ AB = AC, B^AC = 80 Ì, M ‡ªìπ®ÿ¥¿“¬„π ∑’Ë∑”„Àâ MA^C = 20 Ì ·≈– MC^A = 30 Ì ·≈â« M^BA ¡’¢π“¥°’ËÕß»“ 9. °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“∑’Ë¡’ P ‡ªìπ®ÿ¥¿“¬„π∑”„Àâ PA = 6 Àπ૬ PB = 8 Àπ૬ ·≈– PC = 10 Àπ૬ √ªŸ  “¡‡À≈’ˬ¡ ABC ¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπ૬ 10. ®“°√Ÿª BO ·∫àߧ√÷Ëß¡ÿ¡ CBA, CO ·∫àߧ√÷Ëß¡ÿ¡ ACB ·≈– MN ºà“π O ¢π“π°—∫ BC ∂â“ AB = 12 Àπ૬ BC = 24 Àπ૬ ·≈– AC = 18 Àπ૬ ·≈⫧«“¡¬“«‡ âπ√Õ∫√Ÿª √Ÿª “¡‡À≈’ˬ¡ AMN ¡’§à“°’ËÀπ૬ A O N M BC 11. °”Àπ¥ P(x) ‡ªìπæÀÿπ“¡ ‚¥¬∑’Ë P(x) = x4 + ax3 + bx2 + cx + d ∂â“ P(1) = 6, P(›2) = 3, P(3) = ›2 ·≈– P(›4) = ›9 ·≈â« |P(4) + 4)| ¡’§à“‡∑à“„¥ 12. °”Àπ¥ x , x , x , ..., x ‡ªìπ®”π«π®√‘ß∫«° ´÷Ëß∑”„Àâ 1 2 3 100 x + 1 = 1, x + 1 = 4, x + 1 = 1, ... , x + 1 = 1 ·≈– x + 1 = 4 1x 2 x 3x 99 x 100 x 2 34 100 1 ·≈â« x + x + x + ... + x ¡’§à“‡∑à“„¥ 12 3 100 13. ∂â“ x + 4x + 9x + 16x + 25x + 36x + 49x =4 123 4 5 6 7 4x + 9x + 16x + 25x + 36x + 49x + 64x = 44 12 3 4 5 6 7 9x + 16x + 25x + 36x + 49x + 64x + 81x = 444 1234567 ·≈â« 16x + 25x + 36x + 49x + 64x + 81x + 100x ¡’§à“‡∑à“„¥ 123456 7 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 121

14. ®“°√ªŸ ABC, DEF ·≈– PQR ‡ªìπ√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“ ‚¥¬∑’Ë AD = BD ·≈â«æ◊Èπ∑’Ë √ªŸ  “¡‡À≈’ˬ¡ ABC ‡ªìπ°’ˇ∑à“¢Õßæ◊Èπ∑’Ë√ªŸ  “¡‡À≈’ˬ¡ PQR A D PF Q R BEC 15. ∂â“√Ÿª ’ˇÀ≈’ˬ¡ ABCD ¡’®ÿ¥¬Õ¥Õ¬àŸ∫π‡ âπ√Õ∫«ß¢Õß«ß°≈¡∑’Ë¡’ O ‡ªìπ®ÿ¥»Ÿπ¬å°≈“ß ‚¥¬∑’Ë AC ‡ªìπ‡ âπºà“π»πŸ ¬å°≈“ߢÕß«ß°≈¡ ·≈– CA^D = 18 Ì, B^DC = 50 Ì ·≈â« OB^D ¡’¢π“¥°’ËÕß»“ 16. ∂â“ 7x + y = 21 ·≈– 32x + y = 1 ·≈â« 3(7x + 1 + 7y › 2) ¡’§à“‡∑à“„¥ 17. °”Àπ¥ ¡°“√ 15x + 14y = 7 ∂â“ x ∑’ˇªìπ®”π«π‡μÁ¡∑’Ë¡“°∑’Ë ¥ÿ ´÷Ëߪ√–°Õ∫¥â«¬ ¡“™‘° 4 À≈—° ∑’Ë∑”„Àâ (x, y) ‡ªìπ§”μÕ∫¢Õß ¡°“√π’È ‡¡◊ËÕ y ‡ªìπ®”π«π‡μÁ¡ ·≈â« x ¡’§à“‡∑à“„¥ 18. °”Àπ¥„Àâ n ·≈– d ‡ªìπ®”π«π‡μÁ¡∫«° ∂â“ d À“√ 13n + 6 ·≈– 12n + 5 ≈ßμ—« ·≈⫺≈∫«°¢Õß§à“ d ∑’ˇªìπ‰ª‰¥â∑—ÈßÀ¡¥ ¡’§à“‡∑à“„¥ 19. √ŸªÀ≈“¬‡À≈’ˬ¡¥â“π‡∑à“¡ÿ¡‡∑à“√ŸªÀπ÷Ëß ¡’‡ âπ∑·¬ß¡ÿ¡∑—ÈßÀ¡¥ 20 ‡ âπ ·≈–·π∫„π «ß°≈¡√—»¡’ 2 Àπ૬ √ŸªÀ≈“¬‡À≈’ˬ¡π’È¡’æ◊Èπ∑’ˇ∑à“„¥ 20. √Ÿª ’ˇÀ≈’ˬ¡ ABCD ¡’æ◊Èπ∑’Ë 2,550 μ“√“ßÀπ૬ ‚¥¬∑’Ë P, Q, R ·≈– S ‡ªìπ®ÿ¥°÷Ëß°≈“ߥâ“π ∂â“√Ÿª ’ˇÀ≈’ˬ¡ BNMP ·≈–√Ÿª ’ˇÀ≈’ˬ¡ DTOR ¡’æ◊Èπ∑’Ë 123 ·≈– 456 μ“√“ßÀπ૬ μ“¡≈”¥—∫ ·≈â«æ◊Èπ∑’Ë¢Õß√Ÿª ’ˇÀ≈’ˬ¡ MNOT ‡∑à“°—∫°’Ëμ“√“ßÀπ૬ 122 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

μÕπ∑’Ë 2 21. ·∫∫®”≈Õß CH ·π∫„π∑√߇À≈’ˬ¡ ’ËÀπâ“ ‚¥¬∑’Ë·μà≈–ÀπⓇªìπ√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“ 4 ∑’ˬ“«¥â“π≈– 6 6 Àπ૬ ·μà≈–ÀπⓇ∑à“°—π∑ÿ°ª√–°“√ ·≈–¡’ H Õ¬Ÿà∑’Ë®ÿ¥¡ÿ¡ ·≈â«·¢π C › H ¬“«‡∑à“„¥ 22. ∂â“ (x2 › x + 1)3(x3 + 2x2 + 2x + 1)5 = a x21 + a x20 + a x19+ ... + a x2 + a x + a 21 20 19 2 10 ‡¡◊ËÕ a ‡ªìπ§à“§ß∑’Ë ·≈â« a + a + a + + a ¡’§à“‡∑à“„¥ i 1 2 3 ... 10 23. ∂â“ 2 + x + 2 › x = 2 ·≈â« x ¡’§à“‡∑à“„¥ 2+ 2+x 2› 2›x 24. ∂â“ x, y ·≈– z ‡ªìπ®”π«π®√‘ß ∑’Ë Õ¥§≈âÕß°—∫√–∫∫ ¡°“√ x + 2y + 3z = 13 x2 + 4y2 + 9z2 + 3x › 2y + 15z = 82 ·≈â« xyz + x + y + z ¡’§à“‡∑à“„¥ 25. ∂â“ a2 › 2a = ›1, b2 › 3b = 1 ·≈– c2 › 4c = ›1 ·≈â« 3a3 › b3 + c3 + 3 + 1 + 200 ¡’§à“‡∑à“„¥ a3 b3 1 c3 + 26. °”Àπ¥„Àâ a, b, c ‡ªìπ®”π«π‡μÁ¡∫«° ∂â“ a + b + c = 20 = ab + bc › ca › b2 ·≈â« º≈∫«°¢Õß§à“ abc ∑’ˇªìπ‰ª‰¥â∑—ÈßÀ¡¥ ¡’§à“‡∑à“„¥ 27. √Ÿª “¡‡À≈’ˬ¡ ABC ¡’ D, E ·≈– F ‡ªìπ®¥ÿ ∫π¥â“π BC ∑”„Àâ AD ⊥ BC, AE ·∫àߧ√÷Ëß BA^C ·≈– BF = CF ∂â“ B^AD = DA^E = EA^F = FA^C ·≈â« B^AC + 2A^BC + 4AC^B ¡’§à“‡∑à“„¥ 28. ∂â“°”Àπ¥√–∫∫ ¡°“√ = 71 abc + ab + bc + ca + a + b + c = 191 bcd + bc + cd + db + b + c + d = 95 cda + cd + da + ac + c + d + a = 143 dab + da + ab + bd + d + a + b ·≈â« abcd + a + b + c + d ¡’§à“‡∑à“„¥ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 123

29. ∂â“°”Àπ¥√–∫∫ ¡°“√ x + x + x = 6, x + x + x = 8, x + x + x = 4, 234 345 123 x + x + x = 15, x + x + x = 19, x + x + x = 12, 567 678 456 x + x + x = 27, x + x + x = 30, x + x + x = 23, 8 9 10 9 10 1 789 x + x + x = 36 10 1 2 ·≈â« 3x + 4x ¡’§à“‡∑à“„¥ 1 10 30. ∂â“ ABCD ‡ªìπ√ªŸ  ’ˇÀ≈’ˬ¡ ¡’ BA^C = C^AD = 66 Ì, B^CA = 15 Ì ·≈– A^CD = 9 Ì ·≈â« 2A^BD + A^DB ¡’§à“‡∑à“„¥ 124 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

μ—«Õ¬à“ß ·π«§‘¥·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥∫— ™à«ß™È—π∑’Ë 3 (©∫—∫∑Ë’ 2) ‡æÕ◊Ë °“√§¥— ‡≈Õ◊ °π°— ‡√¬’ π√–¥∫— ª√–‡∑» ªï æ.». 2550 §”™’È·®ß 1. ·∫∫∑¥ Õ∫©∫—∫π’ȇªìπ·∫∫‡μ‘¡§”μÕ∫ 2. ·∫∫∑¥ Õ∫©∫—∫π’È¡’®”π«π 30 ¢âÕ ·∫à߇ªìπ 2 μÕπ §–·ππ‡μÁ¡ 100 §–·ππ μÕπ∑’Ë 1 ®”π«π 20 ¢âÕ Ê ≈– 3 §–·ππ √«¡ 60 §–·ππ μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ Ê ≈– 4 §–·ππ √«¡ 40 §–·ππ 3. ‡«≈“∑’Ë„™â„π°“√ Õ∫ „™â‡«≈“ 1 ™—Ë«‚¡ß 30 π“∑’ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 125

μ—«Õ¬à“ß·π«§‘¥·∫∫∑¥ Õ∫§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 (©∫—∫∑’Ë 2) ‡æ◊ËÕ°“√§—¥‡≈◊Õ°π—°‡√’¬π√–¥—∫ª√–‡∑» ªï æ.». 2550 §”™’È·®ß ·∫∫∑¥ Õ∫©∫—∫π’ȇªìπ·∫∫‡μ‘¡§”μÕ∫ §–·ππ‡μÁ¡ 100 §–·ππ ¡’ 2 μÕπ ¥—ßπ’È μÕπ∑’Ë 1 ®”π«π 20 ¢âÕ ¢âÕ≈– 3 §–·ππ μÕπ∑’Ë 2 ®”π«π 10 ¢âÕ ¢âÕ≈– 4 §–·ππ μÕπ∑’Ë 1 1. ( 22 ) ( 32 )( 42 ) ... ( 2,0072 ) ¡’§à“‡∑à“„¥ 22 › 32 › 42 › 2,0072 › 1 1 1 1 ·π«§‘¥ ‡π◊ËÕß®“° 22 = ( 2 )( 2) ®–‰¥â 22 › 1 2›1 2+1 ( 22 )( 32 )( 42 ) ... (2,020,07027›2 ) = (2 ⋅ 2)(3 ⋅ 3)(4 ⋅ 4) ... (2,007 ⋅ 2,007) 22 › 32 › 42 › 1 1 1 1 1 32 43 5 2,006 2,008 = ( 2 )( 22,,000087 ) 1 μÕ∫ 2,007 2,007 1,004 = 1,004 126 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

2. °”Àπ¥ x ‡ªìπ®”π«π‡μÁ¡´÷Ëß Õ¥§≈âÕß°—∫ ¡°“√ x + 4 x › 4 + x › 4 x › 4 = 4 º≈∫«°¢Õß§à“ x ∑—ÈßÀ¡¥‡ªìπ‡∑à“„¥ ·π«§‘¥ (x › 4) + 4 x › 4 + 4 + (x › 4) › 4 x › 4 + 4 = 4 ⏐ x › 4 + 2⏐+⏐ x › 4 › 2⏐= 4 ®–‰¥â x ≥ 4 ·≈– x › 4 + 2 + 2 › x › 4 = 4 ®√‘ß ¥—ßπ—Èπ 2 › x › 4 ≥ 0 ®–‰¥â x ≤ 8 ©–π—Èπ x = 4, 5, 6, 7, 8 º≈∫«°§◊Õ 4 + 5 + 6 + 7 + 8 = 30 μÕ∫ 30 3. ∂â“ m ∗ n = m + n ·≈â« (...(2550 ∗ 2549) ∗ 2548) ∗ ... ∗ 1) ∗ 0 ¡’§à“‡∑à“„¥ mn + 4 ·π«§‘¥ m∗2= m+2 = 1 2m + 4 2 ( )1 + 1 (...(2550 ∗ 2549) ∗ 2548) ∗ ... ∗ 1) ∗ 0 = 2 ∗ 0 1 +4 2 = 1 ∗ 0 3 1 =3 4 =1 12 μÕ∫ 1 12 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 127

4. ®ß‡¢’¬π§ŸàÕ—π¥—∫ (x, y) ∑’Ë ¡“™‘°·μà≈–μ—«Õ¬àŸ„π√Ÿª‡≈¢¬°°”≈—ß ‚¥¬∑’Ë x ‡ªìπ®”π«π ∑’Ë¡“° ¥ÿ ¢Õß 235, 515 ·≈– 614 ·≈– y ‡ªìπ®”π«π∑’ËπâÕ¬∑’Ë ÿ¥¢Õß “¡®”π«ππ’È ·π«§‘¥ ‡æ√“–«à“ 235 = (25)7 = 327 614 = (62)7 = 367 ®–‰¥â 614 > 235 ‡æ√“–«à“ 515 = (53)5 = 1255 235 = (27)5 = 1285 ®–‰¥â 235 > 515 π—Ëπ§◊Õ 614 ¡’§à“¡“°∑’Ë ÿ¥ ·≈ 515 ¡’§à“πâÕ¬∑’Ë ÿ¥ μÕ∫ (614, 515) 5. ‡≈¢‚¥¥À≈—ß®ÿ¥∑»π‘¬¡μ”·Àπàß∑’Ë 8,884 ¢Õß 88 ‡ªìπ‡∑à“„¥ 97 ·π«§‘¥ ®–‰¥â 88 ‡ªìπ∑»π‘¬¡´È” ´÷Ëß¡’μ—«‡≈¢‚¥¥´È”‡ªìπ™¥ÿ Ê ≈– 96 μ—« 97 ‡π◊ËÕß®“° 8,884 = 96(92) + 52 ¥—ßπ—Èπ μ—«‡≈¢‚¥¥„πμ”·Àπàß∑’Ë 8,884 ¢Õß 88 ®–μ√ß°—∫μ—«‡≈¢‚¥¥„πμ”·Àπàß 97 ∑’Ë 52 ¢Õß 88 97 „Àâ 88 = 0.d⋅ d d ... d⋅ 97 1 2 3 96 ®–‰¥â d +d = 9 ¥—ßπ—Èπ 1 49 μÕ∫ 7 d +d = 9 2 50 d +d = 9 3 51 d +d = 9 4 52 d =2 4 d =7 52 128 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

6. ∂â“æÀÿπ“¡ P(x) ∑’Ë¡’¥’°√’ 4 ·≈–¡’ ¡∫—μ‘¥—ßμàÕ‰ªπ’È P(0) = 0, P(1) = P(›1) = P(2) = P(›2) = 1 ·≈â« P(5) ¡’§à“‡∑à“„¥ ·π«§‘¥ ‡π◊ËÕß®“°æÀπÿ “¡ P(x) ¡’¥’°√’ 4 ¥—ßπ—Èπ P(X) Õ¬Ÿà„π√ªŸ P(X) = ax4 + bx3 + cx2 + dx + e „Àâ Q(x) ‡ªìπæÀπÿ “¡∑’Ë°”À𥂥¬ Q(1) = Q(›1) = Q(2) = Q(›2) = 0 ¥—ßπ—Èπ Q(x) = c(x › 1)(x + 1)(x › 2)(x + 2) ‡¡◊ËÕ c ‡ªìπ§à“§ßμ—« ®“°‡ß◊ËÕπ‰¢ ®–‰¥â«à“ P(x) = Q(x) + 1 ·≈– P(0) = 0 = c(›1)(1)(›2)(2) + 1 ®–‰¥â«à“ 4c + 1 = 0 ¥—ßπ—Èπ c = › 1 4 ©–π—Èπ P(x) = › 1 (x › 1)(x + 1)(x › 2)(x + 2) + 1 4 ¥—ßπ—Èπ P(5) = › 1 (5 › 1)(5 + 1)(5 › 2)(5 + 2) + 1 4 = › 1 (4)(6)(3)(7) + 1 4 = ›126 + 1 = ›125 μÕ∫ ›125 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 129

7. ‡¢’¬π®”π«π 365 ®”π«π√Õ∫«ß°≈¡ ®”π«π∑’Ë 88 §◊Õ 7 ®”π«π∑’Ë 111 §◊Õ 2 ®”π«π∑’Ë 224 §◊Õ 1 ®”π«π∑’Ë 365 §◊Õ 5 ∂⓺≈∫«°¢Õß∑°ÿ 60 ®”π«π∑’ˇ√’¬ßμ‘¥μàÕ°—π¡’§à“‡ªìπ 216 ‡ ¡Õ ·≈â«®”π«π∑’Ë 122 ¡’§à“‡∑à“„¥ ·π«§‘¥ À.√.¡. ¢Õß 60 °—∫ 365 ‡ªìπ 5 μ“¡∑ƒ…Æ’√—ßπ°æ‘√“∫ ®–‰¥âº≈∫«°¢Õß®”π«π 5 ®”π«π∑’ˇ√’¬ßμ‘¥°—π∑°ÿ ™ÿ¥¡’§à“§ß∑’Ë ®”π«π∑’Ë 88 ‡∑à“°—∫®”π«π∑’Ë 3 §◊Õ 7 ®”π«π∑’Ë 111 ‡∑à“°—∫®”π«π∑’Ë 1 §◊Õ 2 ®”π«π∑’Ë 224 ‡∑à“°—∫®”π«π∑’Ë 4 §◊Õ 1 ®”π«π∑’Ë 365 ‡∑à“°—∫®”π«π∑’Ë 5 §◊Õ 5 º≈∫«°¢Õß®”π«π 5 ®”π«π∑’ˇ√’¬ßμ‘¥μàÕ°—π‡ªìπ 216 = 18 12 ¥—ßπ—Èπ ®”π«π∑’Ë 2 §◊Õ 3 ·≈–®”π«π∑’Ë 122 §◊Õ 3 μÕ∫ 3 8. ∂â“ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡Àπâ“®—Ë«∑’Ë¡’ AB = AC, B^AC = 80 Ì, M ‡ªìπ®ÿ¥¿“¬„π ∑’Ë∑”„Àâ MA^C = 20 Ì ·≈– MC^A = 30 Ì ·≈â« M^BA ¡’¢π“¥°’ËÕß»“ ·π«§‘¥  √â“ß ≈“° AO ⊥ BC ∑’Ë X ∑”„Àâ BC^O = C^BO = 10 Ì A M B XC O «‘∏’À“ 1. BA^O = CA^O = 40 Ì 2. A^BC = AC^B = 180 Ì› 80 Ì = 50 Ì 2 3. AC^M = MC^O = 30 Ì ·≈– MA^C = MA^O = 20 Ì 4. M ‡ªìπ®ÿ¥»πŸ ¬å°≈“ß«ß°≈¡·π∫„π Δ AOC 5. MO ·∫àߧ√÷Ëß A^OC ·≈– A^OC = 180 Ì › 40 Ì › 60 Ì = 80 Ì 6. A^OM = 40 Ì ·≈– BA^M = BA^O + MA^O = 40 Ì + 20 Ì = 60 Ì 7. ΔOXB ≅ ΔOXC ∑”„Àâ AO^B = A^OC = 80 Ì 8. MO^B = M^OA + A^OB = 40 Ì + 80 Ì 9. ABOM ·π∫„π«ß°≈¡∑”„Àâ MB^A = AO^M = 40 Ì μÕ∫ 40 Ì 130 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

9. °”Àπ¥ ABC ‡ªìπ√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“∑’Ë¡’ P ‡ªìπ®ÿ¥¿“¬„π∑”„Àâ PA = 6 Àπ૬ PB = 8 Àπ૬ ·≈– PC = 10 Àπ૬ √ªŸ  “¡‡À≈’ˬ¡ ABC ¡’æ◊Èπ∑’Ë°’Ëμ“√“ßÀπ૬ ·π«§‘¥ A E D Δ ADC ‰¥â®“°°“√À¡ÿπ Δ APB ∑«π‡¢Á¡π“Ãî°“ 60 Õß»“ √Õ∫®ÿ¥ A Δ BEA ‰¥â®“°°“√À¡ÿπ Δ BPC ∑«π‡¢Á¡π“Ãî°“ 60 Õß»“ √Õ∫®ÿ¥ B P Δ APC ‰¥â®“°°“√À¡ÿπ Δ BFC ∑«π‡¢Á¡π“Ãî°“ 60 Õß»“ √Õ∫®ÿ¥ C B C 2 (æ◊Èπ∑’Ë Δ ABC) = 3 (62 + 82 + 102) + 3 Ó 24 F 4 = 50 3 + 72 æ◊Èπ∑’Ë Δ ABC = 36 + 25 3 μ“√“ßÀπ૬ μÕ∫ 36 + 25 3 μ“√“ßÀπ૬ 10. ®“°√ªŸ BO ·∫àߧ√÷Ëß¡ÿ¡ CBA, CO ·∫àߧ√÷Ëß¡ÿ¡ ACB ·≈– MN ºà“π O ¢π“π°—∫ BC ∂â“ AB = 12 Àπ૬ BC = 24 Àπ૬ ·≈– AC = 18 Àπ૬ ·≈⫧«“¡¬“«‡ âπ√Õ∫√Ÿª √Ÿª “¡‡À≈’ˬ¡ AMN ¡’§à“°’ËÀπ૬ A O N M BC (‚®∑¬å°”Àπ¥) (®“°¢âÕ 1) ·π«§‘¥ (‚®∑¬å°”Àπ¥) 1. BO ·∫àߧ√÷Ëß A^BC (®“°¢âÕ 3 ¡ÿ¡·¬âß) 2. MB^O = OB^C (®“°¢âÕ 4) (®“°¢âÕ 5) 3. NM // BC (‚®∑¬å°”Àπ¥) 4. MO^B = OB^C 5. Δ BMO ‡ªìπ Δ Àπâ“®—Ë« 6. MB = MO 7. OC ·∫àߧ√÷Ëß BC^A ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 131

8. ®–‰¥â NC^O = OC^B (®“°¢âÕ 7) 9. NO^C = O^CB (®“°¢âÕ 3 ¡¡ÿ ·¬âß) 10. Δ ONC ‡ªìπ√ªŸ  “¡‡À≈’ˬ¡Àπâ“®—Ë« (®“°¢âÕ 9) 11. NO = NC (®“°¢âÕ 3  ¡∫—μ‘¢Õß√ªŸ  “¡‡À≈’ˬ¡Àπâ“®—Ë«) 12. ‡ âπ√Õ∫√Ÿª Δ AMN = AM + MN + NA = AM + MO + ON + AN = AM + MB + NC + AN ∴ ‡ âπ√Õ∫√ªŸ Δ AMN = AB + AC = 12 + 18 Àπ૬ μÕ∫ 30 Àπ૬ 11. °”Àπ¥ P(x) ‡ªìπæÀÿπ“¡‚¥¬∑’Ë P(x) = x4 + ax3 + bx2 + cx + d ∂â“ P(1) = 6, P(›2) = 3, P(3) = ›2 ·≈– P(›4) = ›9 ·≈â« |P(4) + 4| ¡’§à“‡∑à“„¥ ·π«§‘¥ „Àâ P(x) = Q(x) + 7 › x2 ®–‰¥â P(1) = Q(1) + 7 › 12 6 = Q(1) + 6 ¥—ßπ—Èπ Q(1) = 0 P(›2) = Q(›2) + 7 › (›2)2 3 = Q(›2) + 3 ¥—ßπ—Èπ Q(›2) = 0 P(3) = Q(3) + 7 › 32 ›2 = Q(3) › 2 ¥—ßπ—Èπ Q(3) = 0 P(›4) = Q(›4) + 7 › (›4)2 ›9 = Q(›4) › 9 ¥—ßπ—Èπ Q(›4) = 0 · ¥ß«à“ (x › 1), (x + 2), (x › 3), (x + 4) ‡ªìπμ—«ª√–°Õ∫¢Õß Q(x) π—Ëπ§◊Õ P(x) = (x › 1)(x + 2)(x › 3)(x + 4) + 7 › x2 P(4) = (4 › 1)(4 + 2)(4 › 3)(4 + 4) + 7 › 42 = (3)(6)(1)(8) + 7 › (4)2 = 144 + 7 › 16 = 135 ¥—ßπ—Èπ |P(4) + 4| = 135 + 4 = 139 μÕ∫ 139 132 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

12. °”Àπ¥ x , x , x , ..., x ‡ªìπ®”π«π®√‘ß∫«° ´÷Ëß∑”„Àâ 1 2 3 100 x + 1 = 1, x + 1 = 4, x + 1 = 1, ..., x + 1 = 1 ·≈– x + 1 = 4 1x 2 x 3x 99 x 100 x 2 34 100 1 ·≈â« x + x + x + ... + x ¡’§à“‡∑à“„¥ 12 3 100 ·π«§‘¥ x+1 ≥ 2 x 1 1x x 22 x+1 ≥ 2 x 2 2x ... 2 x 3 ≥ 3 x+ 1 x 99 x 99 100 x 100 x +1 ≥ 2 x 100 x 100 1 x 1 ®–‰¥â (x1 + 1 ) (x2 + 1 ) ... ( x + 1 )(x100 + x1) ≥ 2100 x x x 2 99 1 3 100 ®–‰¥â 450 ≥ 2100 2100 = 2100 ∴ ®–‰¥â x + 1 = x ; x= 1 21 1x 1x x 22 2 x+1 x x=1 2x = 2 2 ; 2x ... 3 ; x +1 x 3 100 x 3 1 = 2 x 100 x= 1 x x 100 1 1 ·∑π§à“·≈â«·°â ¡°“√®–‰¥â x = x = x = ... = x = 1 135 99 2 x = x = x = ... = x = 2 246 100 x + x + x + ... + x = 125 123 100 μÕ∫ 125 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 133

13. ∂â“ x + 4x + 9x + 16x + 25x + 36x + 49x = 4 123 4 5 6 7 4x + 9x + 16x + 25x + 36x + 49x + 64x = 44 12 3 4 5 6 7 9x + 16x + 25x + 36x + 49x + 64x + 81x = 444 1234567 ·≈â« 16x + 25x + 36x + 49x + 64x + 81x + 100x ¡’§à“‡∑à“„¥ 123456 7 ·π«§‘¥ x + 4x + 9x + 16x + 25x + 36x + 49x = 4.........................➊ 123 4 5 6 7 4x + 9x + 16x + 25x + 36x + 49x + 64x = 44...................... ➋ 12 3 4 5 6 7 9x + 16x + 25x + 36x + 49x + 64x + 81x = 444.................... ➌ 1234567 ➋ › ➊ 3x + 5x + 7x + 9x + 11x + 13x + 15x = 40...................... ➍ 1234 5 6 7 ➌ › ➋ 5x + 7x + 9x + 11x + 13x + 15x + 17x = 400.................... ➎ 123 4 5 6 7 ➎ › ➊ 2x + 2x + 2x + 2x + 2x + 2x + 2x = 360.................... ➏ 1234567 ➏ + ➎ 7x + 9x + 11x + 13x + 15x + 17x + 19x = 760 ................... ➐ 12 3 4 5 6 7 ➐ › ➌ 16x + 25x + 36x + 49x + 64x + 81x + 100x = 1,204 123456 7 μÕ∫ 1,204 134 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

14. ®“°√Ÿª ABC, DEF ·≈– PQR ‡ªìπ√Ÿª “¡‡À≈’ˬ¡¥â“π‡∑à“ ‚¥¬∑’Ë AD = BD æ◊Èπ∑’Ë √Ÿª “¡‡À≈’ˬ¡ ABC ‡ªìπ°’ˇ∑à“¢Õßæ◊Èπ∑’Ë√ªŸ  “¡‡À≈’ˬ¡ PQR A D PF Q C R R BE ·π«§‘¥ F 2 z x+y x A RA 4 C x+y „Àâ AC = 4 ®–‰¥â AF = 2 „Àâ AP = x, PR = y ·≈– FR = z ¥—ßπ—Èπ Δ AFR ∼ Δ ARC 1. Δ AFR ∼ Δ ARC 2 = z = x+y ®–‰¥â x + y = 2 2 x+y x 4 2. Δ QRE ∼ Δ CRF ¥—ßπ—Èπ z = 2 π—Ëπ§◊Õ 2 z=x x 22 QE = ER = RQ CF FR RC z = 2 › z = y 2 z x z2 = 4 › 2z z2 + 2z › 4 = 0 z = ›2 ± 4 + 16 = 1 + 5 (§«“¡¬“«¥â“π‡ªìπ®”π«π®√‘ß∫«°) 2 z =y 2 2z ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 135

y = 2z2 = 2 (6 › 2 5) = 2(3 › 5) 2 2 æ◊Èπ∑’Ë Δ ABC = 16 = 4 = 7 + 3 5 æ◊Èπ∑’Ë Δ PQR 2(14 › 6 5) 7 › 3 5 μÕ∫ 7 + 3 5 15. ∂â“√Ÿª ’ˇÀ≈’ˬ¡ ABCD ¡’®ÿ¥¬Õ¥Õ¬àŸ∫π‡ âπ√Õ∫«ß¢Õß«ß°≈¡∑’Ë¡’ O ‡ªìπ®ÿ¥»Ÿπ¬å°≈“ß ‚¥¬∑’Ë AC ‡ªìπ‡ âπºà“π»πŸ ¬å°≈“ߢÕß«ß°≈¡ ·≈– C^AD = 18 Ì, B^DC = 50 Ì ·≈â« O^BD ¡’¢π“¥°’ËÕß»“ ·π«§‘¥ D ‡æ√“–«à“ AD^C = 90 Ì ·≈– B^DC = BA^C = 50 Ì P C ®–‰¥â A^DB = 40 Ì ∑”„Àâ A^OB = 80 Ì ( O ‡æ√“–«à“ DA^C = CB^D = 18 Ì 18 Ì A B ·μà A^BC = 90 Ì ¥—ßπ—Èπ O^BD = 90 Ì › 50 Ì › 18 Ì = 22 Ì μÕ∫ 22 Ì 16. ∂â“ 7x + y = 21 ·≈– 32x + y = 1 ·≈â« 3(7x + 1 + 7y › 2) ¡’§à“‡∑à“„¥ ·π«§‘¥ =1 ®“° 32x + y = 30 ®–‰¥â 32x + y ¥—ßπ—Èπ 2x + y = 0 y = ›2x ®“° 7x + y = 21 21 ∂â“ y = ›2x ®–‰¥â 7(x + (›2x)) = 21 7›x = 21›1 = 1 7x = 21 7y = ( )›2 7 ›2x = 1 = (21)2 21 136 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

π—Ëπ§◊Õ [ ]3(7x + 1 + 7y › 2) = 3 (7x . 7) + (7y . 712) μÕ∫ 28 = 3[(211 Ó 7)+ (212 Ó 1 )] = 28 72 17. °”Àπ¥ ¡°“√ 15x + 14y = 7 ∂â“ x ∑’ˇªìπ®”π«π‡μÁ¡∑’Ë¡“°∑’Ë ¥ÿ ´÷Ëߪ√–°Õ∫¥â«¬ ¡“™‘° 4 À≈—° ∑’Ë∑”„Àâ (x, y) ‡ªìπ§”μÕ∫¢Õß ¡°“√π’È ‡¡◊ËÕ y ‡ªìπ®”π«π‡μÁ¡ ·≈â« x ¡’§à“‡∑à“„¥ ·π«§‘¥ ‡æ√“–«à“ 15(7) + 14(›7) = 7 ¥—ßπ—Èπ (7, ›7) ‡ªìπ§”μÕ∫Àπ÷ËߢÕß ¡°“√π’È ‡π◊ËÕß®“° 15x + 14y = 7 ®–‰¥â y = 7 › 15x 14 ∂â“„Àâ x = 7 + a ®–‰¥â y= 7 › 15(7 + a) = ›7 › 15a 14 14 ‡æ√“–«à“ y ‡ªìπ®”π«π‡μÁ¡ ¥—ßπ—Èπ 14 À“√ 15a ≈ßμ—« ·μà À.√.¡. ¢Õß 15 °—∫ 14 ‡∑à“°—∫ 1 ¥—ßπ—Èπ a μâÕ߇ªìπæÀ§ÿ ≥Ÿ ¢Õß 14 ¥—ßπ—Èπ ¡’®”π«π‡μÁ¡ b ∑’Ë∑”„Àâ a = 14b π—Ëπ§◊Õ x = 7 + 14b ≥ 9,999 ·μà x = 7 + 14b ≥ 9,992 14b b ≥ 713.7 ®–‰¥â b = 713 π—Ëπ§◊Õ x = 7 + 14(713) = 9,989 μÕ∫ 9,989 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 137

18. °”Àπ¥„Àâ n ·≈– d ‡ªìπ®”π«π‡μÁ¡∫«° ∂â“ d À“√ 13n + 6 ·≈– 12n + 5 ≈ßμ—« ·≈⫺≈∫«°¢Õß§à“ d ∑’ˇªìπ‰ª‰¥â∑—ÈßÀ¡¥ ¡’§à“‡∑à“„¥ ·π«§‘¥ ‡æ√“–«à“ d À“√ 13n + 6 ≈ßμ—« ·≈– d À“√ 12n + 5 ≈ßμ—« ¥—ßπ—Èπ d À“√ 12(13n + 6) ≈ßμ—« ·≈– d À“√ 13(12n + 5) ≈ßμ—« ®–‰¥â d À“√ 12(13n + 6) › 13(12n + 5 ≈ßμ—«¥â«¬ ‡æ√“–«à“ 12(13n + 6) › 13(12n + 5) = 156n + 72 › 156n › 65 = 7 ®–‰¥â d À“√ 7 ≈ßμ—« ®“° d ‡ªìπ®”π«π‡μÁ¡ ¥—ßπ—Èπ d = 1 À√◊Õ d = 7 º≈∫«°‡∑à“°—∫ 1 + 7 = 8 μÕ∫ 8 19. √ŸªÀ≈“¬‡À≈’ˬ¡¥â“π‡∑à“¡ÿ¡‡∑à“√ŸªÀπ÷Ëß ¡’‡ âπ∑·¬ß¡ÿ¡∑—ÈßÀ¡¥ 20 ‡ âπ ·≈–·π∫„π «ß°≈¡√—»¡’ 2 Àπ૬ √ªŸ À≈“¬‡À≈’ˬ¡π’È¡’æ◊Èπ∑’ˇ∑à“„¥ ·π«§‘¥ √ªŸ 4 ‡À≈’ˬ¡¡’‡ âπ∑·¬ß¡¡ÿ 2 ‡ âπ √Ÿª 5 ‡À≈’ˬ¡¡’‡ âπ∑·¬ß¡ÿ¡ 5 ‡ âπ √Ÿª 6 ‡À≈’ˬ¡¡’‡ âπ∑·¬ß¡ÿ¡ 9 ‡ âπ √Ÿª 7 ‡À≈’ˬ¡¡’‡ âπ∑·¬ß¡ÿ¡ 14 ‡ âπ √ªŸ 8 ‡À≈’ˬ¡¡’‡ âπ∑·¬ß¡ÿ¡ 20 ‡ âπ √ªŸ  “¡‡À≈’ˬ¡ 1 √Ÿª ¡’¡ÿ¡¬Õ¥°“ß 360 Ì = 45 Ì 8 æ◊Èπ∑’Ë√ªŸ  “¡‡À≈’ˬ¡ = 1 ( 2)( 2) sin 45 Ì = 2 μ“√“ßÀπ૬ 22 ( )¥—ßπ—Èπ √Ÿª 8 ‡À≈’ˬ¡®–¡’æ◊Èπ∑’Ë 82 = 4 2 μ“√“ßÀπ૬ 2 μÕ∫ 4 2 μ“√“ßÀπ૬ 138 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

20. √Ÿª ’ˇÀ≈’ˬ¡ ABCD ¡’æ◊Èπ∑’Ë 2,550 μ“√“ßÀπ૬ ‚¥¬∑’Ë P, Q, R ·≈– S ‡ªìπ®ÿ¥°÷Ëß°≈“ߥâ“π ∂â“√Ÿª ’ˇÀ≈’ˬ¡ BNMP ·≈–√Ÿª ’ˇÀ≈’ˬ¡ DTOR ¡’æ◊Èπ∑’Ë 123 ·≈– 456 μ“√“ßÀπ૬ μ“¡≈”¥—∫ ·≈â«æ◊Èπ∑’Ë¢Õß√ªŸ  ’ˇÀ≈’ˬ¡ MNOT ‡∑à“°—∫°’Ëμ“√“ßÀπ૬ AP B SM T NQ O DR C ·π«§‘¥ ≈“° BD ®–‰¥â æ◊Èπ∑’Ë Δ DAP = æ◊Èπ∑’Ë Δ DBP ·≈–æ◊Èπ∑’Ë Δ BCR = æ◊Èπ∑’Ë Δ BDR æ◊Èπ∑’Ë BPDR = 1 æ◊Èπ∑’Ë ABCD = 1,275 μ“√“ßÀπ૬ 2 æ◊Èπ∑’Ë MNOT = 1,275 › (123 + 456) = 696 μ“√“ßÀπ૬ μÕ∫ 696 μ“√“ßÀπ૬ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 139

μÕπ∑’Ë 2 21. ·∫∫®”≈Õß CH ·π∫„π∑√߇À≈¬’Ë ¡ À’Ë π“â ‚¥¬∑·’Ë μ≈à –Àπ“â ‡ªπì √ªŸ  “¡‡À≈¬’Ë ¡¥“â π‡∑“à ∑¬’Ë “« 4 ¥â“π≈– 6 6 Àπ૬ ·μà≈–ÀπⓇ∑à“°—π∑ÿ°ª√–°“√ ·≈–¡’ H Õ¬àŸ∑’Ë®ÿ¥¡¡ÿ ·≈â« ·¢π C › H ¬“«‡∑à“„¥ H 1 C H 4 T HH PH 2 3 ·π«§‘¥ = (6 6)2 › (3 6)2 HP = 3 6⋅ 3 4 = 92 ∴H T = 6 2= HT 4 3 „Àâ·¢π C › H ¬“« x ®–‰¥â CH = x 3 CH =x = (6 6)2 › (6 2)2 1 Δ H TH , H T 131 HT = 12 = 12 › x 1 Δ CTH , CT 3 CH = x 3 x2 = (12 › x)2 + (6 2)2 x2 = 144 › 24x + x2 + 72 24x = 216 x = 9 Àπ૬ μÕ∫ 9 Àπ૬ 140 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

22. ∂â“ (x2 › x + 1)3(x3 + 2x2 + 2x + 1)5 = a x21 + a x20 + a x19+ ... + a x2 + a x + a 21 20 19 2 10 ‡¡◊ËÕ a ‡ªìπ§à“§ß∑’Ë ·≈â« a + a + a + + a ¡’§à“‡∑à“„¥ 1 1 2 3 ... 10 ·π«§‘¥ ®“° x2 › x + 1 ·≈– x3 + 2x2 + 2x + 1 ‡ªìπ palindrome ®–‰¥âº≈§Ÿ≥‡ªìπ palindrome ¥â«¬ ¥—ßπ—Èπ a + a + a + + a = 1 (a + a + a + + a ) 1 2 3 ... 10 2 1 2 3 ... 20 = 1 (a + a + a + + a › a › a ) 2 0 1 2 ... 21 0 21 = 1 (a + a + a + + a › 2) 2 0 1 2 ... 21 ®“° (x2 › x + 1)3(x3 + 2x2 + 2x + 1)5 = a x21 + a x20 + ... + a x + a 21 20 10 „Àâ x = 1 ®–‰¥â (13)(65) = a + a + ... + a + a 21 20 10 a + a + a + + a = 65 = 7,776 0 1 2 ... 21 a + a + a + + a = 1 (7,776 › 2) 1 2 3 ... 10 2 = 3,887 μÕ∫ 3,887 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 141

23. ∂â“ 2 + x + 2 › x = 2 ·≈â« x ¡’§à“‡∑à“„¥ 2+ 2+x 2› 2›x ·π«§‘¥ a = 2 + x, b = 2 › x = 2( 2 + a)( 2 › b) a2 + b2 = 4 = 2(2 + 2a › 2b › ab) a2( 2 › b) + b2( 2 + a) = 2 2 + 2a › 2b › 2ab 2a2 › a2b + 2b2 + ab2 = a2b › ab2 + 2a › 2b › 2ab › 2 2 4 2 › a2b + ab2 0 0 = ab(a › b) + 2(a › b) › 2(ab + 2) 0 = (ab + 2)(a › b › 2) ‡π◊ËÕß®“° ab + 2 ≠ 0 ·≈– a › b › 2 = 0 ¥—ßπ—Èπ 2 + x › 2 › x = 2 ¬°°”≈—ß Õß 2 + x › 2 (2 + x)(2 › x) + 2 › x = 2 2 = 2 4 › x2 1 = 4 › x2 1 = 4 › x2 x2 = 3 x = ±3 μ√«®§”μÕ∫®–‰¥â x = ± 3 μÕ∫ ± 3 142 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

24. ∂â“ x, y ·≈– z ‡ªìπ®”π«π®√‘ß ∑’Ë Õ¥§≈âÕß°—∫√–∫∫ ¡°“√ x + 2y + 3z = 13 x2 + 4y2 + 9z2 + 3x › 2y + 15z = 82 ·≈â« xyz + x + y + z ¡’§à“‡∑à“„¥ ·π«§‘¥ x2 + 4y2 + 9z2 + 3x + 15z = 82 + 2y 82 + x + 2y + 3z x2 + 4y2 + 9z2 + 4x + 18z = 82 + 13 + 4 + 9 108 x2 + 4x + 4 + 4y2 + 9z2 + 18z + 9 = (x + 2)2 + (2y)2 + (3z + 3)2 = ®“° x + 2y + 3z = 13 (x + 2) + 2y + (3z + 3) = 18 „Àâ a = x + 2, b = 2y, c = 3z + 3 ®–‰¥â a2 + b2 + c2 = 108 ........................... ➊ a+b+c = 18 (a + b + c)2 = 324 ®–‰¥â ab + bc + ca = 108 ........................... ➋ 2 Ó ➊ › 2 Ó ➋, 2a2 + 2b2 + 2c2 › 2ab › 2bc › 2ca = 0 (a › b)2 + (b › c)2 + (c › a)2 = 0 a=b=c=6 ¥—ßπ—Èπ x = 4, y = 3 ·≈– z = 1 xyz + x + y + z = 20 μÕ∫ 20 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 143

25. ∂â“ a2 › 2a = ›1, b2 › 3b = 1 ·≈– c2 › 4c = ›1 ·≈â« 3a3 › b3 + c3 + 3 + 1 + + 200 ¡’§à“‡∑à“„¥ a3 b3 1 c3 ·π«§‘¥ a + 1 = 2, b › 1 = 3, c + 1 = 4 abc a3 + 1 = (a + 1)3 › 3(a + 1 ) a3 a a = 8›6 b3 › 1 =2 b3 = (b › 1)3 + 3(b › 1 ) bb = 27 + 9 c3 + 1 = 36 c3 = (c + 1 )3 › 3(c + 1 ) cc = 64 › 12 = 52 ¥—ßπ—Èπ 3a3 › b3 + c3 + 3 + 1 + 1 + 200 = 222 a3 b3 c3 μÕ∫ 222 144 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

26. °”Àπ¥„Àâ a, b, c ‡ªìπ®”π«π‡μÁ¡∫«° ∂â“ a + b + c = 20 = ab + bc › ca › b2 ·≈â« º≈∫«°¢Õß§à“ abc ∑’ˇªìπ‰ª‰¥â∑—ÈßÀ¡¥ ¡’§à“‡∑à“„¥ ·π«§‘¥ ®“° ab + bc › ca › b2 = 20 ®–‰¥â (a › b)(b › c) = 20 ‚¥¬‰¡à‡ ’¬π—¬∑—Ë«‰ª „Àâ a ≥ b ≥ c æ‘®“√≥“§à“∑’ˇªìπ‰ª‰¥â a › b b › c a b c a + b + c = 20 abc 1 20 22 21 1 - - 2 10 13 11 1 - - 4 5 11 7 2 20 154 5 4 12 7 3 - - 11 6 2 10 2 14 4 2 20 112 20 1 22 2 1 - - §à“ abc ∑’ˇªìπ‰ª‰¥â§◊Õ 112 °—∫ 154 º≈∫«°®–‡∑à“°—∫ 112 + 154 = 266 μÕ∫ 266 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 145

27. √ªŸ  “¡‡À≈’ˬ¡ ABC ¡’ D, E ·≈– F ‡ªìπ®ÿ¥∫π¥â“π BC ∑”„Àâ AD ⊥ BC, AE ·∫àߧ√÷Ëß BA^C ·≈– BF = CF ∂â“ B^AD = DA^E = EA^F = FA^C ·≈â« B^AC + 2A^BC + 4AC^B ¡’§à“‡∑à“„¥ A x xx x ) ) ) ) 2x Ì ) T 3x 3x ) )B 90 Ì x 2x )90 Ì › 3x C D E F ·π«§‘¥ „Àâ BA^D = x ®–‰¥â A^BD = 90 Ì › x ·≈– AC^B = 90 Ì › 3x  √â“ß ≈“° FT „Àâ TF ⊥ BC æ∫ AC ∑’Ë T ≈“° BT 1. Δ BFT ≅ Δ CFT (¥.¡.¥.) 2. BT^F = CT^F = 3x Ì 3. TB^F = TC^F = 90 Ì › 3x Ì 4. ®–‰¥â T^BA = T^FA = 2x Ì ¥—ßπ—Èπ ∴ ABFT ·π∫„π«ß°≈¡ 4x = 90 Ì x = 22.5 Ì BA^C = 90 Ì, AB^C = 67.5 Ì ·≈– A^CB = 22.5 Ì ∴ BA^C + 2A^BC + 4AC^B = 315 Ì μÕ∫ 315 Ì 146 ✎ ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550)

28. ∂â“°”Àπ¥√–∫∫ ¡°“√ abc + ab + bc + ca + a + b + c = 71 bcd + bc + cd + db + b + c + d = 191 cda + cd + da + ac + c + d + a = 95 dab + da + ab + bd + d + a + b = 143 ·≈â« abcd + a + b + c + d ¡’§à“‡∑à“„¥ ·π«§‘¥ π” 1 ∫«° ∑°ÿ  ¡°“√ ®–‰¥â (a + 1)(b + 1)(c + 1) = 72 = 23 Ó 32 ......................................➊ (b + 1)(c + 1)(d + 1) = 192 = 26 Ó 3 ......................................➋ (c + 1)(d + 1)(a + 1) = 96 = 25 Ó 3 ......................................➌ (d + 1)(a + 1)(b + 1) = 144 = 24 Ó 32 ......................................➍ ➊ Ó ➋ Ó ➌ Ó ➍; (a + 1)3(b + 1)3(c + 1)3(d + 1)3 = 218 Ó 36 (a + 1)(b + 1)(c + 1)(d + 1) = 26 Ó 32................➎ ➎÷➊ d + 1 = 23 ➎÷➋ d =7 a+1 = 3 ➎÷➌ a =2 b+1 = 2Ó3 ➎÷➍ b =5 c + 1 = 22 c =3 §à“¢Õß abcd + a + b + c + d = (2 Ó 5 Ó 3 Ó 7) + (2 + 5 + 3 + 7) = 210 + 17 = 227 μÕ∫ 227 ‡ √‘¡§‘¥§≥‘μ»“ μ√å √–¥—∫™à«ß™—Èπ∑’Ë 3 μ—«Õ¬à“ß·∫∫∑¥ Õ∫§≥‘μ»“ μ√å (ªï æ.». 2549-2550) ✎ 147