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rt tt ts t strs tr trt s t r r s

  1. ❊♥✉♠❡r❛t✐♥❣ ❧❛tt✐❝❡ ♣❛t❤s ✇✐t❤ s②♠♠❡tr✐❡s t❤r♦✉❣❤ ♠✉❧t✐✈❛r✐❛t❡ ❞✐❛❣♦♥❛❧s✳ ❙t❡♣❤❡♥ ▼❡❧❝③❡r ✫ ▼❛r♥✐ ▼✐s❤♥❛ ❙✐♠♦♥ ❋r❛s❡r ❯♥✐✈❡rs✐t②

  2. ✷ ❋♦r ✐♥st❛♥❝❡✱ ❣✐✈❡♥ ❙ ❂ ❢ ✭✶ ❀ � ✶✮ ❀ ✭✶ ❀ ✶✮ ❀ ✭ � ✶ ❀ ✶✮ ❀ ✭ � ✶ ❀ � ✶✮ ❣ ✒ ✇❡ ❤❛✈❡✿ ❝ ✶ ❂ ✶ ▲❛tt✐❝❡ P❛t❤s ✐♥ ❛♥ ❖rt❤❛♥t Given: ❆ s❡t ♦❢ st❡♣s✴❞✐r❡❝t✐♦♥s ❙ ✒ Z ❞ Count: ❝ ♥ ❂ ★ ♦❢ ✇❛❧❦s st❛②✐♥❣ ✐♥ N ❞ ✱ st❛rt✐♥❣ ❛t ♦r✐❣✐♥

  3. ▲❛tt✐❝❡ P❛t❤s ✐♥ ❛♥ ❖rt❤❛♥t Given: ❆ s❡t ♦❢ st❡♣s✴❞✐r❡❝t✐♦♥s ❙ ✒ Z ❞ Count: ❝ ♥ ❂ ★ ♦❢ ✇❛❧❦s st❛②✐♥❣ ✐♥ N ❞ ✱ st❛rt✐♥❣ ❛t ♦r✐❣✐♥ ❋♦r ✐♥st❛♥❝❡✱ ❣✐✈❡♥ ❙ ❂ ❢ ✭✶ ❀ � ✶✮ ❀ ✭✶ ❀ ✶✮ ❀ ✭ � ✶ ❀ ✶✮ ❀ ✭ � ✶ ❀ � ✶✮ ❣ ✒ Z ✷ ✇❡ ❤❛✈❡✿ ❝ ✶ ❂ ✶

  4. ▲❛tt✐❝❡ P❛t❤s ✐♥ ❛♥ ❖rt❤❛♥t Given: ❆ s❡t ♦❢ st❡♣s✴❞✐r❡❝t✐♦♥s ❙ ✒ Z ❞ Count: ❝ ♥ ❂ ★ ♦❢ ✇❛❧❦s st❛②✐♥❣ ✐♥ N ❞ ✱ st❛rt✐♥❣ ❛t ♦r✐❣✐♥ ❋♦r ✐♥st❛♥❝❡✱ ❣✐✈❡♥ ❙ ❂ ❢ ✭✶ ❀ � ✶✮ ❀ ✭✶ ❀ ✶✮ ❀ ✭ � ✶ ❀ ✶✮ ❀ ✭ � ✶ ❀ � ✶✮ ❣ ✒ Z ✷ ✇❡ ❤❛✈❡✿ ❝ ✷ ❂ ✹

  5. ▲❛tt✐❝❡ P❛t❤s ✐♥ ❛♥ ❖rt❤❛♥t Given: ❆ s❡t ♦❢ st❡♣s✴❞✐r❡❝t✐♦♥s ❙ ✒ Z ❞ Count: ❝ ♥ ❂ ★ ♦❢ ✇❛❧❦s st❛②✐♥❣ ✐♥ N ❞ ✱ st❛rt✐♥❣ ❛t ♦r✐❣✐♥ ❋♦r ✐♥st❛♥❝❡✱ ❣✐✈❡♥ ❙ ❂ ❢ ✭✶ ❀ � ✶✮ ❀ ✭✶ ❀ ✶✮ ❀ ✭ � ✶ ❀ ✶✮ ❀ ✭ � ✶ ❀ � ✶✮ ❣ ✒ Z ✷ ✇❡ ❤❛✈❡✿ ❝ ✸ ❂ ✾

  6. ❆s②♠♣t♦t✐❝s ❆s②♠♣t♦t✐❝s ❙ ❙ ✙ ✁ ✹ ♥ ✙ ✁ ✹ ♥ ✹ ✁ ♥ � ✶ ✹ ♥ ❂ ✹ ✹ ✁ ♥ � ✶ ✹ ♥ ❂ ✷ ♣ ♣ ♥ ♥ ✶ ✁ ✶ ✷ ✁ ✷ ✙ ✙ ♣ ✙ ✁ ✻ ♥ ✸ ✙ ✁ ✽ ♥ ✻ ✻ ✽ ✁ ♥ � ✶ ✽ ♥ ❂ ✽ ✁ ♥ � ✶ ✻ ♥ ❂ ♣ ♣ ♥ ♥ ✙ ✸ ✁ ✷ ✙ ✸ ✁ ✸ ❚❤❡ ❍✐❣❤❧② ❙②♠♠❡tr✐❝ ▼♦❞❡❧s ❋✐① ❛ ❞✐♠❡♥s✐♦♥ ❞ ❛♥❞ st❡♣ s❡t ❙ ✒ ❢✝ ✶ ❀ ✵ ❣ ❞ ✇❤✐❝❤ ✐s s②♠♠❡tr✐❝ ♦✈❡r ❡✈❡r② ❛①✐s✿ ✭ ✐ ✶ ❀ ✿ ✿ ✿ ❀ ✐ ❦ ❀ ✿ ✿ ✿ ❀ ✐ ❞ ✮ ✷ ❙ ✐♠♣❧✐❡s ✭ ✐ ✶ ❀ ✿ ✿ ✿ ❀ � ✐ ❦ ❀ ✿ ✿ ✿ ❀ ✐ ❞ ✮ ✷ ❙ ✿

  7. ❚❤❡ ❍✐❣❤❧② ❙②♠♠❡tr✐❝ ▼♦❞❡❧s ❋✐① ❛ ❞✐♠❡♥s✐♦♥ ❞ ❛♥❞ st❡♣ s❡t ❙ ✒ ❢✝ ✶ ❀ ✵ ❣ ❞ ✇❤✐❝❤ ✐s s②♠♠❡tr✐❝ ♦✈❡r ❡✈❡r② ❛①✐s✿ ✭ ✐ ✶ ❀ ✿ ✿ ✿ ❀ ✐ ❦ ❀ ✿ ✿ ✿ ❀ ✐ ❞ ✮ ✷ ❙ ✐♠♣❧✐❡s ✭ ✐ ✶ ❀ ✿ ✿ ✿ ❀ � ✐ ❦ ❀ ✿ ✿ ✿ ❀ ✐ ❞ ✮ ✷ ❙ ✿ ❙ ❆s②♠♣t♦t✐❝s ❙ ❆s②♠♣t♦t✐❝s ✙ ✁ ✹ ♥ ✙ ✁ ✹ ♥ ✹ ✁ ♥ � ✶ ✹ ♥ ❂ ✹ ✹ ✁ ♥ � ✶ ✹ ♥ ❂ ✷ ♣ ♣ ♥ ♥ ✙ ✶ ✁ ✶ ✙ ✷ ✁ ✷ ♣ ✙ ✁ ✻ ♥ ✸ ✙ ✁ ✽ ♥ ✻ ✻ ✽ ✁ ♥ � ✶ ✽ ♥ ❂ ✽ ✁ ♥ � ✶ ✻ ♥ ❂ ♣ ♣ ♥ ♥ ✙ ✸ ✁ ✷ ✙ ✸ ✁ ✸

  8. ❚❤❡ ❍✐❣❤❧② ❙②♠♠❡tr✐❝ ▼♦❞❡❧s ▼❛✐♥ ❚❤❡♦r❡♠ ✭▼✳ ❛♥❞ ▼✐s❤♥❛ ✭✷✵✶✹✮✮ ▲❡t ❙ ✒ ❢� ✶ ❀ ✵ ❀ ✶ ❣ ❞ ♥ ❢ 0 ❣ ❜❡ s②♠♠❡tr✐❝ ✇✐t❤ r❡s♣❡❝t t♦ ❡❛❝❤ ❛①✐s ❛♥❞ t❛❦❡ ❛ ♣♦s✐t✐✈❡ st❡♣ ✐♥ ❡❛❝❤ ❞✐r❡❝t✐♦♥✳ ❚❤❡♥ ✔✏ ✕ s ✭✶✮ ✁ ✁ ✁ s ✭ ❞ ✮ ✑ � ✶ ❂ ✷ ✙ � ❞❂ ✷ ❥❙❥ ❞❂ ✷ ✁ ♥ � ❞❂ ✷ ✁ ❥❙❥ ♥ ❀ ❝ ♥ ✘ ✇❤❡r❡ s ✭ ❦ ✮ ❂ ★ ♦❢ st❡♣s ✇❤✐❝❤ ❤❛✈❡ ❦ t❤ ❝♦♦r❞✐♥❛t❡ ✶✳

  9. ❚❤❡ ❙tr❛t❡❣② ✶✳ ●❡t t❤❡ ❦❡r♥❡❧ ❡q✉❛t✐♦♥ ❢♦r ❈ ✭ z ❀ t ✮ ❈♦♥✈❡rt t♦ ❛ ❞✐❛❣♦♥❛❧ ✷✳ ❡①tr❛❝t✐♦♥ ♦❢ ● ✭ y ❀ t ✮ ❂❍ ✭ y ❀ t ✮ ✸✳ ❋✐♥❞ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥ts ♦❢ V ✭ ❍ ✮ ✹✳ ❘❡☞♥❡ t♦ ♠✐♥✐♠❛❧ ♣♦✐♥ts ♦❢ V ✭ ❍ ✮ ❋✐♥❞ ❛s②♠♣t♦t✐❝s ✈✐❛ ❢♦r♠✉❧❛s ✺✳ ♦❢ P❡♠❛♥t❧❡ ❛♥❞ ❲✐❧s♦♥ ✭✷✵✶✸✮

  10. ❚r❛♥s❧❛t✐♥❣ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ r❡❝✉rr❡♥❝❡ ✰ ■♥❝❧✉s✐♦♥✴❊①❝❧✉s✐♦♥✿ ❈ ✭ ❀ t ✮ ❂ ✶ ✰ tP ✭ ✮ ❈ ✭ ❀ t ✮ ☞ � � ✁ ❳ ✭ � ✶✮ ❥ ❯ ❥ P ✭ ✮ ❈ ✭ ❀ t ✮ ☞ ☞ ❢ ③ ❥ ❂✵✿ ❥ ✷ ❯ ❣ ❯ ✒❢ ✶ ❀✿✿✿❀❞ ❣ ✶✳ ❚❤❡ ❦❡r♥❡❧ ❡q✉❛t✐♦♥ ▲❡t z i ❂ ③ ✐ ✶ ✶ ✁ ✁ ✁ ③ ✐ ❞ z ❂ ✭ ③ ✶ ❀ ✿ ✿ ✿ ❀ ③ ❞ ✮ i ❂ ✭ ✐ ✶ ❀ ✿ ✿ ✿ ❀ ✐ ❞ ✮ ❞ ❀ ❛♥❞ ❝♦♥s✐❞❡r ❝ i ❀♥ ❂ ★ ♦❢ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ ♥ ❡♥❞✐♥❣ ❛t i ❳ ❳ ❝ i ❀♥ z i t ♥ z i ❈ ✭ z ❀ t ✮ ❂ P ✭ z ✮ ❂ i ❀♥ ✕ ✵ i ✷❙

  11. ✶✳ ❚❤❡ ❦❡r♥❡❧ ❡q✉❛t✐♦♥ ▲❡t z i ❂ ③ ✐ ✶ ✶ ✁ ✁ ✁ ③ ✐ ❞ z ❂ ✭ ③ ✶ ❀ ✿ ✿ ✿ ❀ ③ ❞ ✮ i ❂ ✭ ✐ ✶ ❀ ✿ ✿ ✿ ❀ ✐ ❞ ✮ ❞ ❀ ❛♥❞ ❝♦♥s✐❞❡r ❝ i ❀♥ ❂ ★ ♦❢ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ ♥ ❡♥❞✐♥❣ ❛t i ❳ ❳ ❝ i ❀♥ z i t ♥ z i ❈ ✭ z ❀ t ✮ ❂ P ✭ z ✮ ❂ i ❀♥ ✕ ✵ i ✷❙ ❚r❛♥s❧❛t✐♥❣ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ r❡❝✉rr❡♥❝❡ ✰ ■♥❝❧✉s✐♦♥✴❊①❝❧✉s✐♦♥✿ ❈ ✭ z ❀ t ✮ ❂ ✶ ✰ tP ✭ z ✮ ❈ ✭ z ❀ t ✮ ☞ � z � 1 ✁ ❳ ✭ � ✶✮ ❥ ❯ ❥ z 1 P ✭ z ✮ ❈ ✭ z ❀ t ✮ ☞ ☞ ❢ ③ ❥ ❂✵✿ ❥ ✷ ❯ ❣ ❯ ✒❢ ✶ ❀✿✿✿❀❞ ❣

  12. ✶✳ ❚❤❡ ❦❡r♥❡❧ ❡q✉❛t✐♦♥ ▲❡t z i ❂ ③ ✐ ✶ ✶ ✁ ✁ ✁ ③ ✐ ❞ z ❂ ✭ ③ ✶ ❀ ✿ ✿ ✿ ❀ ③ ❞ ✮ i ❂ ✭ ✐ ✶ ❀ ✿ ✿ ✿ ❀ ✐ ❞ ✮ ❞ ❀ ❛♥❞ ❝♦♥s✐❞❡r ❝ i ❀♥ ❂ ★ ♦❢ ✇❛❧❦s ♦❢ ❧❡♥❣t❤ ♥ ❡♥❞✐♥❣ ❛t i ❳ ❳ ❝ i ❀♥ z i t ♥ z i ❈ ✭ z ❀ t ✮ ❂ P ✭ z ✮ ❂ i ❀♥ ✕ ✵ i ✷❙ ❚r❛♥s❧❛t✐♥❣ ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ r❡❝✉rr❡♥❝❡ ✰ ■♥❝❧✉s✐♦♥✴❊①❝❧✉s✐♦♥✿ ❈ ✭ z ❀ t ✮ ❂ ✶ ✰ tP ✭ z ✮ ❈ ✭ z ❀ t ✮ ☞ � z � 1 ✁ ❳ ✭ � ✶✮ ❥ ❯ ❥ z 1 P ✭ z ✮ ❈ ✭ z ❀ t ✮ ☞ ☞ ❢ ③ ❥ ❂✵✿ ❥ ✷ ❯ ❣ ❯ ✒❢ ✶ ❀✿✿✿❀❞ ❣

  13. ❚❤❡ ❦❡r♥❡❧ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s ❈ ✭ ①❀ ②❀ ③❀ t ✮ ❂ ✶ ✰ tP ✭ ①❀ ②❀ ③ ✮ ❈ ✭ ①❀ ②❀ ③❀ t ✮ � t① ✭ ③ ✰ ③ ✮ ❈ ✭✵ ❀ ②❀ ③ ✮ � t② ✭ ③ ✰ ③ ✮ ❈ ✭ ①❀ ✵ ❀ ③ ✮ � t③ ✭ ① ✰ ② ✰ ① ✰ ② ✮ ❈ ✭ ①❀ ②❀ ✵✮ ✰ t①③❈ ✭✵ ❀ ②❀ ✵✮ ✰ t②③❈ ✭ ①❀ ✵ ❀ ✵✮ ✶✳ ❚❤❡ ❦❡r♥❡❧ ❡q✉❛t✐♦♥ ❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ✸❉ ♠♦❞❡❧ ✇✐t❤ ❡✐❣❤t st❡♣s ❙ ❂ ❢ ✭ � ✶ ❀ ✵ ❀ ✶✮ ❀ ✭✶ ❀ ✵ ❀ ✶✮ ❀ ✭✵ ❀ ✶ ❀ ✶✮ ❀ ✭✵ ❀ � ✶ ❀ ✶✮ ❀ ✭ � ✶ ❀ ✵ ❀ � ✶✮ ❀ ✭✶ ❀ ✵ ❀ � ✶✮ ❀ ✭✵ ❀ ✶ ❀ � ✶✮ ❀ ✭✵ ❀ � ✶ ❀ � ✶✮ ❣ s♦ t❤❛t P ✭ ①❀ ②❀ ③ ✮ ❂ ✭ ① ✰ ② ✰ ① ✰ ② ✮✭ ③ ✰ ③ ✮✳

  14. ✶✳ ❚❤❡ ❦❡r♥❡❧ ❡q✉❛t✐♦♥ ❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ✸❉ ♠♦❞❡❧ ✇✐t❤ ❡✐❣❤t st❡♣s ❙ ❂ ❢ ✭ � ✶ ❀ ✵ ❀ ✶✮ ❀ ✭✶ ❀ ✵ ❀ ✶✮ ❀ ✭✵ ❀ ✶ ❀ ✶✮ ❀ ✭✵ ❀ � ✶ ❀ ✶✮ ❀ ✭ � ✶ ❀ ✵ ❀ � ✶✮ ❀ ✭✶ ❀ ✵ ❀ � ✶✮ ❀ ✭✵ ❀ ✶ ❀ � ✶✮ ❀ ✭✵ ❀ � ✶ ❀ � ✶✮ ❣ s♦ t❤❛t P ✭ ①❀ ②❀ ③ ✮ ❂ ✭ ① ✰ ② ✰ ① ✰ ② ✮✭ ③ ✰ ③ ✮✳ ❚❤❡ ❦❡r♥❡❧ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s ❈ ✭ ①❀ ②❀ ③❀ t ✮ ❂ ✶ ✰ tP ✭ ①❀ ②❀ ③ ✮ ❈ ✭ ①❀ ②❀ ③❀ t ✮ � t① ✭ ③ ✰ ③ ✮ ❈ ✭✵ ❀ ②❀ ③ ✮ � t② ✭ ③ ✰ ③ ✮ ❈ ✭ ①❀ ✵ ❀ ③ ✮ � t③ ✭ ① ✰ ② ✰ ① ✰ ② ✮ ❈ ✭ ①❀ ②❀ ✵✮ ✰ t①③❈ ✭✵ ❀ ②❀ ✵✮ ✰ t②③❈ ✭ ①❀ ✵ ❀ ✵✮

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