‫ﺗﻤﺮﻳﻦ ﺳﺮﻱ ﭼﻬﺎﺭﻡ‬
‫ﺳﺎﺧﺘﻤﺎﻥﻫﺎﻱ ﮔﺴﺴﺘﻪ‬
‫ﻣﺪﺭﺱ‪ :‬ﺩﻛﺘﺮ ﺁﺑﺎﻡ‬
‫ﺩﺳﺘﻴﺎﺭ ﺁﻣﻮﺯﺷﻲ‪ :‬ﻫﺎﺩﻱ ﻳﺎﻣﻲ‬
‫‪ - 1‬ﺩﺭﺳﺘﻲ ﺭﻭﺍﺑﻂ ﺯﻳﺮ ﺭﺍ ﺑﻪ ﻭﺳﻴﻠﻪﻱ ﺍﺳﺘﻘﺮﺍﻱ ﺭﻳﺎﺿﻲ ﺑﺮﺭﺳﻲ ﻛﻨﻴﺪ‪.‬‬
‫ﺍﻟﻒ(‬
‫ﺏ(‬
‫ﺝ(‬
‫)‪n(2n-1)(2n+1‬‬
‫𝑛‬
‫)𝑛‪𝑎(𝑎+‬‬
‫=‬
‫‪1‬‬
‫‪3‬‬
‫‪1‬‬
‫)‪𝑘(𝑘−1‬‬
‫= )‪∑nk=1 k(k + 1‬‬
‫𝑎‪∑𝑛+‬‬
‫‪𝑘=𝑎+1‬‬
‫‪2n-1(3n + 5n) ≥ 8n , n ∈ N‬‬
‫‪ - 2‬ﺍﻟﻒ( ﺛﺎﺑﺖ ﻛﻨﻴﺪ ﺑﻪ ﺍﺯﺍﻱ ﻫﺮ ﻋﺪﺩ ﻃﺒﻴﻌﻲ ﻣﺎﻧﻨﺪ ‪ ،n‬ﻋﺪﺩ ‪ 11n+1 + 122n-1‬ﺑﺮ ‪ 133‬ﺑﺨﺶﭘﺬﻳﺮ ﺍﺳﺖ‪.‬‬
‫ﺏ( ﺛﺎﺑﺖ ﻛﻨﻴﺪ ﺗﻤﺎﻡ ﺍﻋﺪﺍﺩ ﺑﻪ ﺷﻜﻞ ‪ . . . ،100117 ،10017 ،1007‬ﺑﺮ ‪ 53‬ﺑﺨﺶﭘﺬﻳﺮ ﻫﺴﺘﻨﺪ‪.‬‬
‫‪ - 3‬ﺛﺎﺑﺖ ﻛﻨﻴﺪ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺳﻜﻪﻫﺎﻱ ‪ 3‬ﻭ ‪ 5‬ﺗﻮﻣﺎﻧﻲ ﻣﻲﺗﻮﺍﻥ ‪ n‬ﺗﻮﻣﺎﻥ ﭘﻮﻝ ﺭﺍ ﺧﺮﺩ ﻛﺮﺩ‪) .‬ﺑﻪ ﺍﺯﺍﻱ ‪(n > 7‬‬
‫‪ - 4‬ﺩﺭ ﺍﻧﺒﺎﺭﻱ ‪ n‬ﺗﺎ ﺻﻨﺪﻭﻕ ﺑﻪ ﺷﻤﺎﺭﻩﻫﺎﻱ ‪ 1‬ﺗﺎ ‪ n‬ﺩﺭ ﺩﻭ ﺳﺘﻮﻥ ﭼﻴﺪﻩ ﺷﺪﻩﺍﻧﺪ‪ .‬ﺑﺎ ﻳﻚ ﻟﻴﻔﺖﺗﺮﺍﻙ ﻫﺮ ﺑﺎﺭ ﻣﻲﺗﻮﺍﻥ ﭼﻨﺪ ﺗﺎ ﺻﻨﺪﻭﻕ‬
‫ﺭﺍ ﺍﺯ ﺑﺎﻻﻱ ﻳﻜﻲ ﺍﺯ ﺍﻳﻦ ﺳﺘﻮﻥﻫﺎ ﺑﺮﺩﺍﺷﺖ ﻭ ﺑﺎﻻﻱ ﺳﺘﻮﻥ ﺩﻳﮕﺮ ﮔﺬﺍﺷﺖ‪ .‬ﺛﺎﺑﺖ ﻛﻨﻴﺪ ﻛﻪ ﻫﻤﻪﻱ ﺍﻳﻦ ﺻﻨﺪﻭﻕﻫﺎ ﺭﺍ ﻣﻲﺗﻮﺍﻥ ﺩﺭ‬
‫ﻳﻚ ﺳﺘﻮﻥ ﺑﻪ ﺗﺮﺗﻴﺐ ﺍﻓﺰﺍﻳﺶ ﺷﻤﺎﺭﻩﻫﺎﻳﺸﺎﻥ ﺑﺎ ﺣﺪﺍﻛﺜﺮ ‪ 2n - 1‬ﺑﺎﺭ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﻟﻴﻔﺖﺗﺮﺍﻙ ﭼﻴﺪ‪.‬‬
‫‪ - 5‬ﺻﻔﺤﻪﺍﻱ ﺩﺍﺭﻳﻢ ﻛﻪ ﺩﺭ ﺁﻥ ﭼﻨﺪ ﺧﻂ ﻏﻴﺮﻣﻮﺍﺯﻱ ﻣﻮﺟﻮﺩ ﺍﺳﺖ‪ ،‬ﺑﻪ ﻃﻮﺭﻱ ﻛﻪ ﻫﻴﭻ ﺳﻪ ﺧﻄﻲ ﺍﺯ ﻳﻚ ﻧﻘﻄﻪ ﻧﻤﻲﮔﺬﺭﺩ‪ .‬ﺛﺎﺑﺖ‬
‫ﻛﻨﻴﺪ ﻣﻲﺗﻮﺍﻥ ﺍﻳﻦ ﺻﻔﺤﻪ ﺭﺍ ﻃﻮﺭﻱ ﺑﺎ ‪ 2‬ﺭﻧﮓ‪ ،‬ﺭﻧﮓ ﻛﺮﺩ ﺑﻪ ﻃﻮﺭﻱ ﻛﻪ ﺧﺎﻧﻪﻫﺎﻱ ﻫﻤﺴﺎﻳﻪ ﺭﻧﮓ ﻣﺨﺎﻟﻒ ﻫﻢ ﺩﺍﺷﺘﻪ ﺑﺎﺷﻨﺪ‪) .‬ﺩﻭ‬
‫ﺧﺎﻧﻪ ﻣﺠﺎﻭﺭ ﻫﻢ ﻫﺴﺘﻨﺪ ﺍﮔﺮ ﻭ ﺗﻨﻬﺎ ﺍﮔﺮ ﻣﺠﺎﻭﺭ ﺿﻠﻌﻲ ﺑﺎﺷﻨﺪ‪(.‬‬
‫‪ - 6‬ﻳﻚ ﻣﻬﻤﺎﻧﻲ ﺭﺍ ﺑﺎ ﺷﺮﻛﺖ ‪ n‬ﺯﻭﺝ ﺍﺯﺩﻭﺍﺝ ﻛﺮﺩﻩ‪ ،‬ﺩﺭ ﻧﻈﺮ ﻣﻲﮔﻴﺮﻳﻢ‪ .‬ﻓﺮﺽ ﻛﻨﻴﻢ ﻫﻴﭻ ﺷﺨﺼﻲ ﺑﺎ ﻫﻤﺴﺮﺵ ﺩﺳﺖ ﻧﺪﻫﺪ ﻭ ‪2n-1‬‬
‫ﻧﻔﺮ ﺑﻪ ﺟﺰ ﺁﻗﺎﻱ ﻣﻴﺰﺑﺎﻥ ﺑﺎ ﺗﻌﺪﺍﺩ ﻣﺨﺘﻠﻔﻲ ﺍﺯ ﺍﻓﺮﺍﺩ ﺩﺳﺖ ﺑﺪﻫﻨﺪ‪ .‬ﺧﺎﻧﻢ ﻣﻴﺰﺑﺎﻥ ﺑﺎ ﭼﻨﺪ ﻧﻔﺮ ﺩﺳﺖ ﺩﺍﺩﻩ ﺍﺳﺖ؟ )ﻧﮕﺮﺍﻥ ﻧﺒﺎﺷﻴﺪ‪،‬‬
‫ﺍﻃﻼﻋﺎﺕ ﻣﺴﺌﻠﻪ ﻛﺎﻣﻞ ﺍﺳﺖ!(‬
‫‪ - 7‬ﺩﺭ ﻛﺸﻮﺭﻱ ‪ n‬ﺷﻬﺮ ﻣﻬﻢ ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﻛﻪ ﺩﻭﺑﻪﺩﻭ ﺑﺎ ﺟﺎﺩﻩﻫﺎﻳﻲ ﺑﻪ ﻫﻢ ﻣﺘﺼﻞ ﻫﺴﺘﻨﺪ‪ .‬ﻣﻲﺧﻮﺍﻫﻴﻢ ﻫﻤﻪﻱ ﺍﻳﻦ ﺟﺎﺩﻩﻫﺎ ﺭﺍ ﻳﻚ‬
‫ﻃﺮﻓﻪ ﻛﻨﻴﻢ‪ ،‬ﺑﻪ ﻃﻮﺭﻱ ﻛﻪ ﺍﺯ ﻫﺮ ﺷﻬﺮ ﺑﻪ ﺷﻬﺮ ﺩﻳﮕﺮ ﺑﺘﻮﺍﻥ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﻳﻚ ﻳﺎ ﺩﻭ ﺟﺎﺩﻩ ﺭﻓﺖ‪ .‬ﺛﺎﺑﺖ ﻛﻨﻴﺪ ﺑﻪ ﺍﺯﺍﻱ ‪ n > 4‬ﺍﻳﻦ‬
‫ﻛﺎﺭ ﻫﻤﻮﺍﺭﻩ ﻣﻤﻜﻦ ﺍﺳﺖ‪.‬‬
‫‪ - 8‬ﺛﺎﺑﺖ ﻛﻨﻴﺪ ﺑﺎ ﺭﻗﻢﻫﺎﻱ ‪ 1‬ﻭ ‪ 2‬ﻣﻲﺗﻮﺍﻥ ‪ 2n+1‬ﻋﺪﺩ ﺳﺎﺧﺖ‪ ،‬ﺑﻪ ﻧﺤﻮﻱ ﻛﻪ ﻫﺮ ﻛﺪﺍﻡ ﺍﺯ ﺁﻥﻫﺎ ‪ 2n‬ﺭﻗﻢ ﺩﺍﺷﺘﻪ ﺑﺎﺷﻨﺪ ﻭ ﺩﺭ ﺿﻤﻦ‬
‫ﻫﺮ ﺩﻭ ﻋﺪﺩ ﺩﺳﺖﻛﻢ ﺩﺭ ‪ 2n-1‬ﻣﺮﺗﺒﻪ ﺑﺎ ﻳﻜﺪﻳﮕﺮ ﺍﺧﺘﻼﻑ ﺩﺍﺷﺘﻪ ﺑﺎﺷﻨﺪ‪.‬‬
‫‪ 2n - 9‬ﺟﻌﺒﻪ ﺩﺭ ﻳﻚ ﺭﺩﻳﻒ ﺑﻪ ﺗﺮﺗﻴﺐ ﺑﺎ ﺷﻤﺎﺭﻩﻫﺎﻱ ‪ 1‬ﺗﺎ ‪ 2n‬ﻗﺮﺍﺭ ﺩﺍﺩﻩ ﺷﺪﻩﺍﻧﺪ‪ .‬ﺩﺭ ﻫﺮ ﻳﻚ ﺍﺯ ﺟﻌﺒﻪﻫﺎﻱ ‪ 2n – 1‬ﻭ ‪ 2n‬ﻳﻚ‬
‫ﻣﻬﺮﻩ ﻗﺮﺍﺭ ﺩﺍﺭﺩ‪ .‬ﺩﻭ ﻧﻔﺮ ﺑﻪ ﻧﻮﺑﺖ ﺑﻪ ﺍﻳﻦ ﺻﻮﺭﺕ ﺑﺎﺯﻱ ﻣﻲﻛﻨﻨﺪ ﻛﻪ ﻫﺮ ﻓﺮﺩ ﺩﺭ ﻧﻮﺑﺖ ﺧﻮﺩ ﻳﻜﻲ ﺍﺯ ﻣﻬﺮﻩﻫﺎ ﺭﺍ ﺍﺯ ﺩﺍﺧﻞ ﻳﻚ ﺟﻌﺒﻪ‬
‫ﺑﺮﻣﻲﺩﺍﺭﺩ ﻭ ﺁﻥ ﺭﺍ ﺩﺍﺧﻞ ﺟﻌﺒﻪﺍﻱ ﺧﺎﻟﻲ ﺑﺎ ﺷﻤﺎﺭﻩﻱ ﻛﻮﭼﻜﺘﺮ ﻗﺮﺍﺭ ﻣﻲﺩﻫﺪ ﻓﺮﺩﻱ ﻛﻪ ﺩﺭ ﻧﻮﺑﺖ ﺧﻮﺩ ﻧﺘﻮﺍﻧﺪ ﺣﺮﻛﺘﻲ ﺍﻧﺠﺎﻡ ﺩﻫﺪ‬
‫ﺑﺎﺯﻧﺪﻩﻱ ﺑﺎﺯﻱ ﺍﺳﺖ‪ .‬ﺛﺎﺑﺖ ﻛﻨﻴﺪ ﻧﻔﺮ ﺩﻭﻡ ﻣﻲﺗﻮﺍﻧﺪ ﻃﻮﺭﻱ ﺑﺎﺯﻱ ﻛﻨﺪ ﻛﻪ ﺑﺮﻧﺪﻩﻱ ﺑﺎﺯﻱ ﺷﻮﺩ‪.‬‬
‫‪- 10‬ﻓﺮﺽ ﻛﻨﻴﺪ ‪ n‬ﺗﻴﻢ ﺩﺭ ﻳﻚ ﺗﻮﺭﻧﻤﻨﺖ ﺑﺎ ﻳﻜﺪﻳﮕﺮ ﺑﺎﺯﻱ ﻛﺮﺩﻩﺍﻧﺪ‪ .‬ﺍﮔﺮ ﻫﻴﭻﻳﻚ ﺍﺯ ﺩﻭ ﺗﻴﻢ ﻣﺴﺎﻭﻱ ﻧﻜﺮﺩﻩ ﺑﺎﺷﻨﺪ‪ ،‬ﺛﺎﺑﺖ ﻛﻨﻴﺪ‬
‫ﺩﻧﺒﺎﻟﻪﻱ ‪ t1, t2, . . ., tn‬ﺍﺯ ﺗﻴﻢﻫﺎ ﻭﺟﻮﺩ ﺩﺍﺭﺩ ﺑﻪ ﻃﻮﺭﻱ ﻛﻪ ﺗﻴﻢ ‪ t1‬ﺍﺯ ﺗﻴﻢ ‪ t2‬ﺑﺮﺩﻩ‪ ،‬ﺗﻴﻢ ‪ t2‬ﺍﺯ ﺗﻴﻢ ‪ t3‬ﺑﺮﺩﻩ‪ . . . ،‬ﻭ ﺗﻴﻢ ‪ tn-1‬ﺍﺯ‬
‫ﺗﻴﻢ ‪ tn‬ﺑﺮﺩﻩ ﺍﺳﺖ‪.‬‬
‫‪- 11‬ﻓﺮﺽ ﻛﻨﻴﺪ ‪ α‬ﻋﺪﺩﻱ ﺣﻘﻴﻘﻲ ﺑﺎﺷﺪ ﺑﻪ ﻃﻮﺭﻱ ﻛﻪ‬
‫𝑍∈‬
‫‪1‬‬
‫𝛼‬
‫‪ . α +‬ﺛﺎﺑﺖ ﻛﻨﻴﺪ 𝑍 ∈‬
‫‪1‬‬
‫𝑛‪α‬‬
‫‪ αn +‬ﺑﺮﺍﻱ ﻫﺮ 𝑁 ∈ 𝑛 ‪.‬‬
‫‪- 12‬ﻳﻚ ﭘﺎﺭﻩﺧﻂ ﺑﻪ ﻃﻮﻝ ﻭﺍﺣﺪ ﺩﺍﺭﻳﻢ‪ .‬ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺧﻂﻛﺶ ﻭ ﭘﺮﮔﺎﺭ‪ ،‬ﭘﺎﺭﻩﺧﻄﻲ ﺑﻪ ﻃﻮﻝ 𝑛√ ﺭﺳﻢ ﻛﻨﻴﺪ‪(𝑛 ∈ 𝑁) .‬‬
‫‪ n- 13‬ﭘﻴﭻ ﻭ ‪ n‬ﻣﻬﺮﻩ ﺍﺯ ﻧﻈﺮ ﻇﺎﻫﺮﻱ ﺷﺒﻴﻪ ﺑﻪ ﻫﻢ ﻫﺴﺘﻨﺪ‪ ،‬ﺩﺍﺩﻩ ﺷﺪﻩ ﺍﺳﺖ‪ .‬ﻣﻲﺩﺍﻧﻴﻢ ﻛﻪ ﻫﺮ ﭘﻴﭻ ﺗﻨﻬﺎ ﺑﻪ ﻳﻚ ﻣﻬﺮﻩ ﻣﻲﺧﻮﺭﺩ )ﺑﺎ‬
‫ﺁﻥ ﻫﻢﺍﻧﺪﺍﺯﻩ ﺍﺳﺖ( ﻭ ﻫﻴﭻ ﺩﻭ ﭘﻴﭽﻲ ﻫﻢﺍﻧﺪﺍﺯﻩ ﻧﻴﺴﺘﻨﺪ‪.‬‬
‫ﻋﻤﻞ »ﺁﺯﻣﻮﻥ« ﻳﻌﻨﻲ ﺑﺮﺩﺍﺷﺘﻦ ﻳﻚ ﭘﻴﭻ ﻭ ﻳﻚ ﻣﻬﺮﻩ ﻭ ﺍﻣﺘﺤﺎﻥ ﻛﺮﺩﻥ ﺁﻥﻫﺎ‪ .‬ﺑﺎ ﺍﻳﻦ ﻛﺎﺭ ﺗﺸﺨﻴﺺ ﻣﻲﺩﻫﻴﻢ ﻛﻪ ﭘﻴﭻ ﺍﺯ ﻣﻬﺮﻩ‬
‫ﺑﺰﺭﮔﺘﺮ ﺍﺳﺖ ﻳﺎ ﻫﺮ ﺩﻭ ﻫﻢﺍﻧﺪﺍﺯﻩ ﻫﺴﺘﻨﺪ‪.‬‬
‫ﻣﻲﺧﻮﺍﻫﻴﻢ ﺑﺎ ﺍﻧﺠﺎﻡ ﺗﻌﺪﺍﺩﻱ ﻋﻤﻞ »ﺁﺯﻣﻮﻥ«‪ ،‬ﻛﻮﭼﻜﺘﺮﻳﻦ ﭘﻴﭻ ﻭ ﻛﻮﭼﻜﺘﺮﻳﻦ ﻣﻬﺮﻩ )ﻛﻪ ﻣﺴﻠﻤﺎ ﺑﻪ ﻫﻢ ﻣﻲﺧﻮﺭﻧﺪ( ﺭﺍ ﭘﻴﺪﺍ ﻛﻨﻴﻢ‪.‬‬
‫ﺗﻮﺟﻪ ﻛﻨﻴﺪ ﻛﻪ ﻧﻤﻲﺗﻮﺍﻥ ﺩﻭ ﻣﻬﺮﻩ ﻳﺎ ﺩﻭ ﭘﻴﭻ ﺭﺍ ﻣﺴﺘﻘﻴﻤﺎ ﺑﺎ ﻫﻢ ﻣﻘﺎﻳﺴﻪ ﻛﺮﺩ‪.‬‬
‫ﺭﻭﺷﻲ ﺍﺭﺍﺋﻪ ﺩﻫﻴﺪ ﺗﺎ ﺑﺘﻮﺍﻥ ﻣﺴﺌﻠﻪ ﺭﺍ ﺩﺭ ﺣﺎﻟﺖ ﻛﻠﻲ ﺑﺎ ‪ 2n – 2‬ﺁﺯﻣﻮﻥ ﺣﻞ ﻛﺮﺩ‪.‬‬
‫‪- 14‬ﺩﺭ ﺭﻭﺯﮔﺎﺭﺍﻥ ﻗﺪﻳﻢ‪ ،‬ﺍﺯ ﻃﺮﻑ ﻓﺮﻣﺎﻧﺮﻭﺍﻱ ﺷﻬﺮﻱ ﺍﻳﻦ ﺍﻃﻼﻋﻴﻪ ﺻﺎﺩﺭ ﺷﺪﻩ ﺍﺳﺖ‪» :‬ﻣﺴﻠﻢ ﺷﺪﻩ ﺍﺳﺖ ﻛﻪ ﺩﺭ ﺷﻬﺮ ﺯﻧﺎﻧﻲ ﻭﺟﻮﺩ‬
‫ﺩﺍﺭﻧﺪ ﻛﻪ ﺑﻪ ﺷﻮﻫﺮﺍﻥ ﺧﻮﺩ ﺧﻴﺎﻧﺖ ﻣﻲﻛﻨﻨﺪ‪ ،‬ﻭ ﻣﺴﻠﻢ ﺷﺪﻩ ﺍﺳﺖ ﻛﻪ ﺑﺮ ﺧﻴﺎﻧﺖ ﻫﺮ ﻳﻚ ﺍﺯ ﺍﻳﻦ ﺯﻧﺎﻥ‪ ،‬ﻫﻤﻪﻱ ﻣﺮﺩﻡ ﺷﻬﺮ ﺑﻪ ﻏﻴﺮ‬
‫ﺍﺯ ﺷﻮﻫﺮ ﺍﻳﺸﺎﻥ ﺍﻃﻼﻉ ﺩﺍﺭﻧﺪ‪ .‬ﺍﺯ ﻣﺮﺩﺍﻥ ﺧﻮﺍﺳﺘﻪ ﻣﻲﺷﻮﺩ ﻛﻪ ﻓﻘﻂ ﺍﺯ ﺗﻌﻘﻞ ﻭ ﺑﺪﻭﻥ ﺗﻔﺤﺺ ﺍﺯ ﺩﻳﮕﺮﺍﻥ ﺩﺭﺑﺎﺭﻩﻱ ﺯﻧﺎﻥ ﺧﻮﺩ‬
‫ﺑﻴﻨﺪﻳﺸﻨﺪ ﻭ ﻫﺮ ﻣﺮﺩ ﻛﻪ ﺩﺭﻳﺎﻓﺖ ﺯﻧﺶ ﺑﻪ ﻭﻱ ﺧﻴﺎﻧﺖ ﻣﻲﻛﻨﺪ ﺑﺎﻳﺪ ﻛﻪ ﺑﺎﻣﺪﺍﺩ ﺭﻭﺯ ﺑﻌﺪ ﺁﻥ ﺯﻥ ﺭﺍ ﻣﻜﺎﻓﺎﺕ ﻗﺘﻞ ﺑﺮﺳﺎﻧﺪ!« ﺛﺎﺑﺖ‬
‫ﻛﻨﻴﺪ ﺑﻌﺪ ﺍﺯ ﭼﻨﺪ ﺭﻭﺯ ﺑﺎﻻﺧﺮﻩ ﺭﻭﺯﻱ ﻓﺮﺍ ﻣﻲﺭﺳﺪ ﻛﻪ ﺗﻤﺎﻡ ﺯﻧﺎﻥ ﺧﻴﺎﻧﺖﻛﺎﺭ ﻛﺸﺘﻪ ﺷﺪﻩ ﺑﺎﺷﻨﺪ‪.‬‬
‫‪- 15‬ﻓﺮﺽ ﻛﻨﻴﺪ ﻛﻪ ﻳﻚ ﻣﺎﺷﻴﻦ ﺩﺭ ﺍﺧﺘﻴﺎﺭ ﺩﺍﺭﻳﻢ ﻛﻪ ﻣﻲﺗﻮﺍﻧﺪ ﺍﻳﻦ ﺳﻪ ﻛﺎﺭ ﺭﺍ ﺑﺮ ﺭﻭﻱ ﻛﺎﺭﺕﻫﺎﻳﻲ ﻛﻪ ﺑﺮ ﺭﻭﻱ ﻫﺮ ﻳﻚ ﺍﺯ ﺁﻥﻫﺎ ﻳﻚ‬
‫ﻛﻠﻤﻪ ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺍﺳﺖ ﺍﻧﺠﺎﻡ ﺩﻫﺪ‪:‬‬
‫• ﺩﻭ ﻛﺎﺭﺕ ﻛﻪ ﺑﺮ ﺭﻭﻱ ﺁﻥﻫﺎ ﺩﻭ ﻛﻠﻤﻪ ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺍﺳﺖ ﺭﺍ ﺑﮕﻴﺮﺩ ﻭ ﻳﻚ ﻛﺎﺭﺕ ﺗﻮﻟﻴﺪ ﻛﻨﺪ ﻛﻪ ﺑﺮ ﺭﻭﻱ ﺁﻥ ﺍﻳﻦ ﺩﻭ ﻛﻠﻤﻪ‬
‫ﭘﺸﺖ ﺳﺮﻫﻢ ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺑﺎﺷﻨﺪ‪ ) .‬ﺑﺮﺍﻱ ﻣﺜﺎﻝ ﺍﮔﺮ ﺑﺮ ﺭﻭﻱ ﻛﺎﺭﺕ ﺍﻭﻝ ﺭﺷﺘﻪﻱ ‪ aab‬ﻭ ﺑﺮ ﺭﻭﻱ ﻛﺎﺭﺕ ﺩﻭﻡ ﺭﺷﺘﻪﻱ ‪bab‬‬
‫ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺑﺎﺷﺪ‪ ،‬ﺧﺮﻭﺟﻲ ﻣﺎﺷﻴﻦ ﻛﺎﺭﺗﻲ ﺧﻮﺍﻫﺪ ﺑﻮﺩ ﻛﻪ ﺑﺮ ﺭﻭﻱ ﺁﻥ ‪ aabbab‬ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺍﺳﺖ‪(.‬‬
‫• ﻳﻚ ﻛﺎﺭﺕ ﻛﻪ ﺑﺮ ﺭﻭﻱ ﺁﻥ ﻛﻠﻤﻪﻱ ‪ S‬ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺍﺳﺖ ﺭﺍ ﺩﺭﻳﺎﻓﺖ ﻛﻨﺪ ﻭ ﺩﺭ ﺧﺮﻭﺟﻲ ﻛﺎﺭﺗﻲ ﺍﻳﺠﺎﺩ ﻛﻨﺪ ﻛﻪ ﺑﺮ ﺭﻭﻱ ﺁﻥ‬
‫‪ aSb‬ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺍﺳﺖ‪) .‬ﺑﺮﺍﻱ ﻣﺜﺎﻝ ﺍﮔﺮ ﺑﺮ ﺭﻭﻱ ﻛﺎﺭﺕ ﻭﺭﻭﺩﻱ ﻛﻠﻤﻪﻱ ‪ aba‬ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺑﺎﺷﺪ‪ ،‬ﺧﺮﻭﺟﻲ ﻣﺎﺷﻴﻦ‬
‫ﻛﺎﺭﺗﻲ ﺧﻮﺍﻫﺪ ﺑﻮﺩ ﻛﻪ ﺑﺮ ﺭﻭﻱ ﺁﻥ ﻛﻠﻤﻪﻱ ‪ aabab‬ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺍﺳﺖ‪(.‬‬
‫• ﻳﻚ ﻛﺎﺭﺕ ﻛﻪ ﺑﺮ ﺭﻭﻱ ﺁﻥ ﻛﻠﻤﻪﻱ ‪ S‬ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺍﺳﺖ ﺭﺍ ﺩﺭﻳﺎﻓﺖ ﻛﻨﺪ ﻭ ﺩﺭ ﺧﺮﻭﺟﻲ ﻛﺎﺭﺗﻲ ﺍﻳﺠﺎﺩ ﻛﻨﺪ ﻛﻪ ﺑﺮ ﺭﻭﻱ ﺁﻥ‬
‫‪ bSa‬ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺍﺳﺖ‪) .‬ﺑﺮﺍﻱ ﻣﺜﺎﻝ ﺍﮔﺮ ﺑﺮ ﺭﻭﻱ ﻛﺎﺭﺕ ﻭﺭﻭﺩﻱ ﻫﻴﭻ ﻛﻠﻤﻪﺍﻱ ﻧﻮﺷﺘﻪ ﻧﺸﺪﻩ ﺑﺎﺷﺪ‪ ،‬ﺧﺮﻭﺟﻲ ﻣﺎﺷﻴﻦ‬
‫ﻛﺎﺭﺗﻲ ﺧﻮﺍﻫﺪ ﺑﻮﺩ ﻛﻪ ﺑﺮ ﺭﻭﻱ ﺁﻥ ﻛﻠﻤﻪﻱ ‪ ba‬ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺍﺳﺖ‪(.‬‬
‫ﺩﺭ ﺍﺑﺘﺪﺍ ﺗﻌﺪﺍﺩ ﺯﻳﺎﺩﻱ ﻛﺎﺭﺕ ﻛﻪ ﺑﺮ ﺭﻭﻱ ﺁﻥﻫﺎ ﻫﻴﭻ ﻛﻠﻤﻪﺍﻱ ﻧﻮﺷﺘﻪ ﻧﺸﺪﻩ ﺍﺳﺖ ﺩﺭ ﺍﺧﺘﻴﺎﺭ ﻣﺎ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪ ﺍﺳﺖ‪.‬‬
‫ﺍﻟﻒ( ﻧﺸﺎﻥ ﺩﻫﻴﺪ ﻛﻪ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺍﻳﻦ ﻛﺎﺭﺕﻫﺎ ﻭ ﺑﺎ ﺍﻳﻦ ﻣﺎﺷﻴﻦ ﻣﻲﺗﻮﺍﻥ ﻛﺎﺭﺗﻲ ﺭﺍ ﺍﻳﺠﺎﺩ ﻛﺮﺩ ﻛﻪ ﺑﺮ ﺭﻭﻱ ﺁﻥ ﻛﻠﻤﻪﻱ‬
‫‪ abbaba‬ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺑﺎﺷﺪ‪.‬‬
‫ﺏ( ﺛﺎﺑﺖ ﻛﻨﻴﺪ ﻛﻪ ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ ﺍﺯ ﺍﻳﻦ ﻣﺎﺷﻴﻦ ﻣﻲﺗﻮﺍﻥ ﻫﺮ ﻛﺎﺭﺗﻲ ﻛﻪ ﺑﺮ ﺭﻭﻱ ﺁﻥ ﻳﻚ ﻛﻠﻤﻪ ﻧﻮﺷﺘﻪ ﺷﺪﻩ ﺍﺳﺖ ﺭﺍ ﺗﻮﻟﻴﺪ ﻛﺮﺩ‪ ،‬ﺍﮔﺮ‬
‫ﻭ ﻓﻘﻂ ﺍﮔﺮ ﺍﻳﻦ ﻛﻠﻤﻪ ﺗﻨﻬﺎ ﺍﺯ ‪ a‬ﻭ ‪ b‬ﺗﺸﻜﻴﻞ ﺷﺪﻩ ﺑﺎﺷﺪ ﻭ ﺗﻌﺪﺍﺩ ‪a‬ﻫﺎﻱ ﺁﻥ ﺑﺮﺍﺑﺮ ﺗﻌﺪﺍﺩ ‪b‬ﻫﺎﻱ ﺁﻥ ﺑﺎﺷﺪ‪.‬‬
‫******************************************‬
‫ﺍﮔﺮ ﻫﺮﮔﻮﻧﻪ ﺍﺑﻬﺎﻡ ﻳﺎ ﻣﺸﻜﻞ ﺩﺭ ﻣﻮﺭﺩ ﺗﻤﺮﻳﻦ ﺩﺍﺷﺘﻴﺪ ﺑﻪ ﺁﺩﺭﺱ ﺯﻳﺮ ﺑﻔﺮﺳﺘﻴﺪ‪.‬‬
‫‪hadi.yami93@gmail.com‬‬