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t r st s s t r t

  1. ❙❝❛❧✐♥❣ ❧✐♠✐t ♦❢ ❛ ❧❛②❡r ♦❢ ✉♥st❛❜❧❡ ♣❤❛s❡ ❨✈❛♥ ❱❡❧❡♥✐❦ ❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❉✐♠❛ ■♦✛❡ ❛♥❞ ❙❡♥②❛ ❙❤❧♦s♠❛♥

  2. ❙tr✉❝t✉r❡ ♦❢ t❤❡ t❛❧❦ ▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡ ✶ ●❡♥❡r❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧ ✷ ❘❡s✉❧t ✸ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢ ✹

  3. ▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡ ❈r✐t✐❝❛❧ ♣r❡✇❡tt✐♥❣ ✐♥ t❤❡ ■s✐♥❣ ♠♦❞❡❧ ◮ ✷❞ ■s✐♥❣ ♠♦❞❡❧ ✐♥ ❛ sq✉❛r❡ ❜♦① ◮ ❇♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✿ ✰ ✰ ✰ � ◮ ▼❛❣♥❡t✐❝ ✜❡❧❞✿ ❤ ❃ ✵ ❆s ❤ ★ ✵ ✱ t❤❡ t❤✐❝❦♥❡ss ♦❢ t❤❡ ❧❛②❡r ♦❢ ✉♥st❛❜❧❡ � ♣❤❛s❡ ✐♥❝r❡❛s❡s ❛s ❤ � ✶ ❂ ✸✰ ♦ ✭✶✮ ✭❛s ❧♦♥❣ ❛s ◆ ✢ ❤ � ✷ ❂ ✸ ✮ ❬❱✳✱ P❚❘❋ ✷✵✵✹❪

  4. ▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡ ✶ ✰ ✶ ✲❞✐♠❡♥s✐♦♥❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧ ❯♥st❛❜❧❡ ♣❤❛s❡ X ��������������������������������������������������������������������� ��������������������������������������������������������������������� � ◆ ��������������������������������������������������������������������� ��������������������������������������������������������������������� ◆ Pr♦❜❛❜✐❧✐t② ♦❢ ❛ ♥♦♥♥❡❣❛t✐✈❡ tr❛❥❡❝t♦r② X ❂ ✭ ❳ � ◆ ❀ ✿ ✿ ✿ ❀ ❳ ◆ ✮ ✿ ✏❛r❡❛✑ ③ ⑥⑤ ④ ♥ ◆ ♦ ❳ ✶ P ◆❀ ✰ ❀✕ ✭ X ✮ ❂ ❡①♣ � ✕ ❳ ✐ p ❘❲ ✭ X ✮ ❀ ❩ ◆❀ ✰ ❀✕ ✐ ❂ � ◆ ✇❤❡r❡ ✕ ❃ ✵ ✳

  5. ▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡ ✶ ✰ ✶ ✲❞✐♠❡♥s✐♦♥❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧ ❯♥st❛❜❧❡ ♣❤❛s❡ X ��������������������������������������������������������������������� ��������������������������������������������������������������������� � ◆ ��������������������������������������������������������������������� ��������������������������������������������������������������������� ◆ Pr♦❜❛❜✐❧✐t② ♦❢ ❛ ♥♦♥♥❡❣❛t✐✈❡ tr❛❥❡❝t♦r② X ❂ ✭ ❳ � ◆ ❀ ✿ ✿ ✿ ❀ ❳ ◆ ✮ ✿ ✏❛r❡❛✑ ③ ⑥⑤ ④ ♥ ◆ ♦ ❳ ✶ P ◆❀ ✰ ❀✕ ✭ X ✮ ❂ ❡①♣ � ✕ ❳ ✐ p ❘❲ ✭ X ✮ ❀ ❩ ◆❀ ✰ ❀✕ ✐ ❂ � ◆ ✇❤❡r❡ ✕ ❃ ✵ ✳

  6. ❲❤❛t ✐s t❤❡ s❝❛❧✐♥❣ ❧✐♠✐t ♦❢ ① ✕ ✭ t ✮ ❂ ✕ ✶ ❂ ✸ ❳ ❬ ✕ � ✷ ❂ ✸ t ❪ ❄ ❯s❡❢✉❧ t♦ ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❣❡♥❡r❛❧ s✐t✉❛t✐♦♥✳✳✳ ▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡ ❇❡❤❛✈✐♦r ❛s ✕ ★ ✵ ✿ ◮ ❋r❡❡ ❡♥❡r❣② ✘ ✕ ✷ ❂ ✸ ◮ ❚❤✐❝❦♥❡ss ✘ ✕ � ✶ ❂ ✸ ◮ ❈♦rr❡❧❛t✐♦♥ ❧❡♥❣t❤ ✘ ✕ � ✷ ❂ ✸ ❬❍r②♥✐✈ ❛♥❞ ❱✳✱ P❚❘❋ ✷✵✵✹❪ ✭s❡❡ ❛❧s♦ ❬❆❜r❛❤❛♠✫ ❙♠✐t❤✱ ❏❙P ✶✾✽✻❪ ✮

  7. ❯s❡❢✉❧ t♦ ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❣❡♥❡r❛❧ s✐t✉❛t✐♦♥✳✳✳ ▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡ ❇❡❤❛✈✐♦r ❛s ✕ ★ ✵ ✿ ◮ ❋r❡❡ ❡♥❡r❣② ✘ ✕ ✷ ❂ ✸ ◮ ❚❤✐❝❦♥❡ss ✘ ✕ � ✶ ❂ ✸ ◮ ❈♦rr❡❧❛t✐♦♥ ❧❡♥❣t❤ ✘ ✕ � ✷ ❂ ✸ ❬❍r②♥✐✈ ❛♥❞ ❱✳✱ P❚❘❋ ✷✵✵✹❪ ✭s❡❡ ❛❧s♦ ❬❆❜r❛❤❛♠✫ ❙♠✐t❤✱ ❏❙P ✶✾✽✻❪ ✮ ❲❤❛t ✐s t❤❡ s❝❛❧✐♥❣ ❧✐♠✐t ♦❢ ① ✕ ✭ t ✮ ❂ ✕ ✶ ❂ ✸ ❳ ❬ ✕ � ✷ ❂ ✸ t ❪ ❄

  8. ▼♦t✐✈❛t✐♥❣ ❡①❛♠♣❧❡ ❇❡❤❛✈✐♦r ❛s ✕ ★ ✵ ✿ ◮ ❋r❡❡ ❡♥❡r❣② ✘ ✕ ✷ ❂ ✸ ◮ ❚❤✐❝❦♥❡ss ✘ ✕ � ✶ ❂ ✸ ◮ ❈♦rr❡❧❛t✐♦♥ ❧❡♥❣t❤ ✘ ✕ � ✷ ❂ ✸ ❬❍r②♥✐✈ ❛♥❞ ❱✳✱ P❚❘❋ ✷✵✵✹❪ ✭s❡❡ ❛❧s♦ ❬❆❜r❛❤❛♠✫ ❙♠✐t❤✱ ❏❙P ✶✾✽✻❪ ✮ ❲❤❛t ✐s t❤❡ s❝❛❧✐♥❣ ❧✐♠✐t ♦❢ ① ✕ ✭ t ✮ ❂ ✕ ✶ ❂ ✸ ❳ ❬ ✕ � ✷ ❂ ✸ t ❪ ❄ ❯s❡❢✉❧ t♦ ❝♦♥s✐❞❡r ❛ ♠♦r❡ ❣❡♥❡r❛❧ s✐t✉❛t✐♦♥✳✳✳

  9. ●❡♥❡r❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧ ✶✳ ❚❤❡ ✉♥❞❡r❧②✐♥❣ r❛♥❞♦♠ ✇❛❧❦ ✭ ♣ ① ✮ ① ✷ Z ✿ tr❛♥s✳ ♣r♦❜❛❜✳ ♦❢ ❛♥ ❛♣❡r✐♦❞✐❝✱ ✐rr❡❞✉❝✐❜❧❡ ❘❲ ♦♥ Z s✉❝❤ t❤❛t ❳ ❳ ❡ t① ♣ ① ❁ ✶ ❢♦r s♠❛❧❧ t ♣ ① ❂ ✵ ❀ ① ① ▲❡t ✛ ✷ ❂ P ① ① ✷ ♣ ① ❛♥❞✱ ❢♦r X ❂ ✭ ❳ � ◆ ❀ ✿ ✿ ✿ ❀ ❳ ◆ ✮ ✱ ◆ � ✶ ❨ p ❘❲ ✭ X ✮ ❂ ♣ ❳ ✐ ✰✶ � ❳ ✐ ✐ ❂ � ◆

  10. ▲❡t ❍ ✕ ❜❡ t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ t♦ ❍ ✷ ❱ ✕ ✭ ❍ ✮ ❂ ✶ ✭♠❡❛s✉r❡s t❤❡ t❤✐❝❦♥❡ss ♦❢ t❤❡ ❧❛②❡r✮ ❆❞❞✐t✐♦♥❛❧ ❛ss✉♠♣t✐♦♥s ✭♦♥ ✭ ❱ ✕ ✮ ✕❃ ✵ ✮✿ ✕ ★ ✵ ❍ ✷ ✕ ★ ✵ ❍ ✕ ❂ ✰ ✶ ❀ ❧✐♠ ❧✐♠ ✕ ❱ ✕ ✭ r❍ ✕ ✮ ❂ q ✭ r ✮ ❀ ✷ ✭ ✇✐t❤ q ✷ ✰ ✮ s✉❝❤ t❤❛t ❧✐♠ r ✦✶ q ✭ r ✮ ❂ ✰ ✶ ✳ ❊①❛♠♣❧❡✿ ❱ ✕ ✭ ① ✮ ❂ ✕① ☛ ✱ ☛ ❃ ✵ ✱ ❍ ✕ ❂ ✕ � ✶ ❂ ✭✷✰ ☛ ✮ ✱ q ✭ r ✮ ❂ r ☛ ●❡♥❡r❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧ ✷✳ ❚❤❡ ♣♦t❡♥t✐❛❧s ✭ ❱ ✕ ✮ ✕❃ ✵ ▲❡t ❱ ✕ ✿ N ✦ R ✰ ❜❡ s✉❝❤ t❤❛t ❱ ✕ ✭✵✮ ❂ ✵ ❀ ❱ ✕ ✐♥❝r❡❛s✐♥❣✱ ① ✦✶ ❱ ✕ ✭ ① ✮ ❂ ✰ ✶ ❧✐♠

  11. ❆❞❞✐t✐♦♥❛❧ ❛ss✉♠♣t✐♦♥s ✭♦♥ ✭ ❱ ✕ ✮ ✕❃ ✵ ✮✿ ✕ ★ ✵ ❍ ✷ ❧✐♠ ✕ ★ ✵ ❍ ✕ ❂ ✰ ✶ ❀ ❧✐♠ ✕ ❱ ✕ ✭ r❍ ✕ ✮ ❂ q ✭ r ✮ ❀ ✷ ✭ ✇✐t❤ q ✷ ✰ ✮ s✉❝❤ t❤❛t ❧✐♠ r ✦✶ q ✭ r ✮ ❂ ✰ ✶ ✳ ❊①❛♠♣❧❡✿ ❱ ✕ ✭ ① ✮ ❂ ✕① ☛ ✱ ☛ ❃ ✵ ✱ ❍ ✕ ❂ ✕ � ✶ ❂ ✭✷✰ ☛ ✮ ✱ q ✭ r ✮ ❂ r ☛ ●❡♥❡r❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧ ✷✳ ❚❤❡ ♣♦t❡♥t✐❛❧s ✭ ❱ ✕ ✮ ✕❃ ✵ ▲❡t ❱ ✕ ✿ N ✦ R ✰ ❜❡ s✉❝❤ t❤❛t ❱ ✕ ✭✵✮ ❂ ✵ ❀ ❱ ✕ ✐♥❝r❡❛s✐♥❣✱ ① ✦✶ ❱ ✕ ✭ ① ✮ ❂ ✰ ✶ ❧✐♠ ▲❡t ❍ ✕ ❜❡ t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ t♦ ❍ ✷ ❱ ✕ ✭ ❍ ✮ ❂ ✶ ✭♠❡❛s✉r❡s t❤❡ t❤✐❝❦♥❡ss ♦❢ t❤❡ ❧❛②❡r✮

  12. ❊①❛♠♣❧❡✿ ❱ ✕ ✭ ① ✮ ❂ ✕① ☛ ✱ ☛ ❃ ✵ ✱ ❍ ✕ ❂ ✕ � ✶ ❂ ✭✷✰ ☛ ✮ ✱ q ✭ r ✮ ❂ r ☛ ●❡♥❡r❛❧ ❡✛❡❝t✐✈❡ ♠♦❞❡❧ ✷✳ ❚❤❡ ♣♦t❡♥t✐❛❧s ✭ ❱ ✕ ✮ ✕❃ ✵ ▲❡t ❱ ✕ ✿ N ✦ R ✰ ❜❡ s✉❝❤ t❤❛t ❱ ✕ ✭✵✮ ❂ ✵ ❀ ❱ ✕ ✐♥❝r❡❛s✐♥❣✱ ① ✦✶ ❱ ✕ ✭ ① ✮ ❂ ✰ ✶ ❧✐♠ ▲❡t ❍ ✕ ❜❡ t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ t♦ ❍ ✷ ❱ ✕ ✭ ❍ ✮ ❂ ✶ ✭♠❡❛s✉r❡s t❤❡ t❤✐❝❦♥❡ss ♦❢ t❤❡ ❧❛②❡r✮ ❆❞❞✐t✐♦♥❛❧ ❛ss✉♠♣t✐♦♥s ✭♦♥ ✭ ❱ ✕ ✮ ✕❃ ✵ ✮✿ ✕ ★ ✵ ❍ ✷ ❧✐♠ ✕ ★ ✵ ❍ ✕ ❂ ✰ ✶ ❀ ❧✐♠ ✕ ❱ ✕ ✭ r❍ ✕ ✮ ❂ q ✭ r ✮ ❀ ✇✐t❤ q ✷ C ✷ ✭ R ✰ ✮ s✉❝❤ t❤❛t ❧✐♠ r ✦✶ q ✭ r ✮ ❂ ✰ ✶ ✳

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