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  1. ➽❲✐♥♥✐♥❣ ❜② ▲♦s✐♥❣➾✿ ■♥❝❡♥t✐✈❡ ■♥❝♦♠♣❛t✐❜✐❧✐t② ✐♥ ▼✉❧t✐♣❧❡ ◗✉❛❧✐✜❡rs ❉♠✐tr② ❉❛❣❛❡✈ 1 ❑♦♥st❛♥t✐♥ ❙♦♥✐♥ 2 1 ❍✐❣❤❡r ❙❝❤♦♦❧ ♦❢ ❊❝♦♥♦♠✐❝s 2 ◆❡✇ ❊❝♦♥♦♠✐❝ ❙❝❤♦♦❧ ▼♦s❝♦✇ ❋❡❜r✉❛r② ✷✻✱ ✷✵✶✸ ❉♠✐tr② ❉❛❣❛❡✈✱ ❑♦♥st❛♥t✐♥ ❙♦♥✐♥ ❲✐♥♥✐♥❣ ❜② ▲♦s✐♥❣

  2. P❧❛♥ ◮ Pr♦❜❧❡♠ ♦❢ ♠❛♥✐♣✉❧❛❜✐❧✐t② ♦❢ ❢♦♦t❜❛❧❧ ♠❛t❝❤❡s ♦✉t❝♦♠❡s ◮ ❘❡❛❧✲✇♦r❧❞ ❡①❛♠♣❧❡s ◮ ▲✐t❡r❛t✉r❡ r❡✈✐❡✇ ◮ ❋♦r♠❛❧ ♠♦❞❡❧ ◮ P♦ss✐❜❧❡ ✜①❡s ❉♠✐tr② ❉❛❣❛❡✈✱ ❑♦♥st❛♥t✐♥ ❙♦♥✐♥ ❲✐♥♥✐♥❣ ❜② ▲♦s✐♥❣

  3. Pr♦❜❧❡♠ ❉♦ t❤❡ ❝♦♠♣❡t✐♥❣ t❡❛♠s ❤❛✈❡ ♣❡r✈❡rs❡ ✐♥❝❡♥t✐✈❡s t♦ ❧♦s❡ ❛ ❣❛♠❡ ❞❡❧✐❜❡r❛t❡❧②❄ ➽❨❡s➾✱ ✐❢ ◮ t❡❛♠ ✐s ❜r✐❜❡❞ ◮ r❛♥❦✐♥❣ r✉❧❡ ♦r ❞✐str✐❜✉t✐♦♥ ♦❢ ♣r✐③❡s ✐s ♥♦t ♠♦♥♦t♦♥♦✉s ◮ t❤❡ ❣❛♠❡ ✐s ❝♦❛❧✐t✐♦♥❛❧ ❲❤❛t ✐❢ ❛ t♦✉r♥❛♠❡♥t ✐s ❢❛✐r❄ ■♥ ❛ s✐♥❣❧❡ t♦✉r♥❛♠❡♥t ✖ ➽◆♦➾✳ ❇✉t ✐♥ ♠✉❧t✐♣❧❡ t♦✉r♥❛♠❡♥ts✳✳✳ ❉♠✐tr② ❉❛❣❛❡✈✱ ❑♦♥st❛♥t✐♥ ❙♦♥✐♥ ❲✐♥♥✐♥❣ ❜② ▲♦s✐♥❣

  4. ❘❡❛❧✲✇♦r❧❞ ❡①❛♠♣❧❡ ✶ ✿ ❘✉ss✐❛✲✷✵✶✶✴✷✵✶✷ ❙t❛♥❞✐♥❣s ❛s ♦❢ ▼❛② ✽✱ ✷✵✶✷✿ P❧❛❝❡ ❚❡❛♠ Pts ✶ ❩❡♥✐t ❙t✳P❡t❡rs❜✉r❣ ✽✺ ✷ ❈❙❑❆ ▼♦s❝♦✇ ✼✸ ✸ ❙♣❛rt❛❦ ▼♦s❝♦✇ ✼✷ ✹ ❉②♥❛♠♦ ▼♦s❝♦✇ ✼✶ ✺ ❆♥③❤✐ ▼❛❦❤❛❝❤❦❛❧❛ ✼✵ ✻ ▲♦❦♦♠♦t✐✈ ▼♦s❝♦✇ ✻✻ ✼ ❘✉❜✐♥ ❑❛③❛♥✬ ✻✺ ✽ ❑✉❜❛♥✬ ❑r❛s♥♦❞❛r ✻✵ ❘❡♠❛✐♥✐♥❣ ♠❛t❝❤❡s ✭▼❛② ✶✸✮✿ ❑✉❜❛♥✬ ✕ ❉②♥❛♠♦✱ ❘✉❜✐♥ ✕ ❈❙❑❆✱ ▲♦❦♦♠♦t✐✈ ✕ ❙♣❛rt❛❦✱ ❆♥③❤✐ ✕ ❩❡♥✐t✳ ❈✉♣ ✜♥❛❧ ✭▼❛② ✾✮✿ ❘✉❜✐♥ ✕ ❉②♥❛♠♦✳ ✶ ❈r❡❞✐ts t♦ ❉r✳ ❆♥❞r❡✐ ❇r✐❝❤❦✐♥ ✇❤♦ ✐♥✐t✐❛❧❧② ♥♦t✐❝❡❞ t❤✐s ❞r❛✇❜❛❝❦ ❉♠✐tr② ❉❛❣❛❡✈✱ ❑♦♥st❛♥t✐♥ ❙♦♥✐♥ ❲✐♥♥✐♥❣ ❜② ▲♦s✐♥❣

  5. ❘❡❛❧✲✇♦r❧❞ ❡①❛♠♣❧❡✿ ❘✉ss✐❛✲✷✵✶✶✴✷✵✶✷ ❚✇♦ ♠❛❥♦r ❊✉r♦♣❡❛♥ ✐♥t❡r♥❛t✐♦♥❛❧ t♦✉r♥❛♠❡♥ts✿ ❯❊❋❆ ❈❤❛♠♣✐♦♥s ▲❡❛❣✉❡ ❛♥❞ ❯❊❋❆ ❊✉r♦♣❛ ▲❡❛❣✉❡ ❇❡rt❤s ❛r❡ ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ r✉❧❡s✿ ✶✳ ❚❡❛♠s t❤❛t ❛r❡ r❛♥❦❡❞ ✶st ❛♥❞ ✷♥❞ ✐♥ t❤❡ ❘✉ss✐❛♥ ♥❛t✐♦♥❛❧ ❝❤❛♠♣✐♦♥s❤✐♣ q✉❛❧✐❢② ❢♦r t❤❡ ❈❤❛♠♣✐♦♥s ▲❡❛❣✉❡✳ ✷✳ ❚❡❛♠s t❤❛t ❛r❡ r❛♥❦❡❞ ✸r❞ t♦ ✺t❤ ✐♥ t❤❡ ♥❛t✐♦♥❛❧ ❝❤❛♠♣✐♦♥s❤✐♣ q✉❛❧✐❢② ❢♦r t❤❡ ❊✉r♦♣❛ ▲❡❛❣✉❡✳ ✸✳ ❘✉ss✐❛♥ ❝✉♣ ✇✐♥♥❡r q✉❛❧✐✜❡s ❢♦r t❤❡ ❊✉r♦♣❛ ▲❡❛❣✉❡✳ ✹✳ ■❢ ❝✉♣ ✇✐♥♥❡r ✐s r❛♥❦❡❞ ✶st ♦r ✷♥❞ ✐♥ t❤❡ ♥❛t✐♦♥❛❧ ❝❤❛♠♣✐♦♥s❤✐♣✱ t❤❡♥ t❤✐s t❡❛♠ q✉❛❧✐✜❡s ❢♦r t❤❡ ❈❤❛♠♣✐♦♥s ▲❡❛❣✉❡✱ ❛♥❞ t❤❡ ❝✉♣ r✉♥♥❡r✲✉♣ q✉❛❧✐✜❡s ❢♦r t❤❡ ❊✉r♦♣❛ ▲❡❛❣✉❡✳ ✺✳ ■❢ ❈✉♣ ✇✐♥♥❡r ✐s r❛♥❦❡❞ ✸r❞ t♦ ✺t❤ ✐♥ t❤❡ ♥❛t✐♦♥❛❧ ❝❤❛♠♣✐♦♥s❤✐♣✱ t❤❡♥ t❤❡ t❡❛♠ r❛♥❦❡❞ ✻t❤ ✐♥ t❤❡ ♥❛t✐♦♥❛❧ ❝❤❛♠♣✐♦♥s❤✐♣ q✉❛❧✐✜❡s ❢♦r t❤❡ ❊✉r♦♣❛ ▲❡❛❣✉❡✳ ❉♠✐tr② ❉❛❣❛❡✈✱ ❑♦♥st❛♥t✐♥ ❙♦♥✐♥ ❲✐♥♥✐♥❣ ❜② ▲♦s✐♥❣

  6. ❘❡❛❧✲✇♦r❧❞ ❡①❛♠♣❧❡✿ ❘✉ss✐❛✲✷✵✶✶✴✷✵✶✷ ❙✉♣♣♦s❡ t❤❛t ❉②♥❛♠♦ ✇✐♥s t❤❡ ❘✉ss✐❛♥ ❈✉♣ ❛♥❞ ❜❡❛ts ❑✉❜❛♥✬✱ ❛♥❞ ❘✉❜✐♥ ✕ ❈❙❑❆ ✐s ❛ ❞r❛✇✳ ❆♥③❤✐ ✕ ❩❡♥✐t ❣❛♠❡ ✐s ✐rr❡❧❡✈❛♥t✳ ▲♦❦♦♠♦t✐✈✬s ✇✐♥ ❉r❛✇ ▲♦❦♦♠♦t✐✈✬s ❧♦ss ➑ ❚❡❛♠ Pts ➑ ❚❡❛♠ Pts ➑ ❚❡❛♠ Pts ✶ ❩❡♥✐t ✽✺ ✶ ❩❡♥✐t ✽✺ ✶ ❩❡♥✐t ✽✺ ✷ ❉②♥❛♠♦ ✼✹ ✷ ❉②♥❛♠♦ ✼✹ ✷ ❙♣❛rt❛❦ ✼✺ ✸ ❈❙❑❆ ✼✹ ✸ ❈❙❑❆ ✼✹ ✸ ❉②♥❛♠♦ ✼✹ ✹ ❙♣❛rt❛❦ ✼✷ ✹ ❙♣❛rt❛❦ ✼✸ ✹ ❈❙❑❆ ✼✹ ✺ ❆♥③❤✐ ✼✵ ✺ ❆♥③❤✐ ✼✵ ✺ ❆♥③❤✐ ✼✵ ✻ ▲♦❦♦♠♦t✐✈ ✻✾ ✻ ▲♦❦♦♠♦t✐✈ ✻✼ ✻ ▲♦❦♦♠♦t✐✈ ✻✻ ✼ ❘✉❜✐♥ ✻✻ ✼ ❘✉❜✐♥ ✻✻ ✼ ❘✉❜✐♥ ✻✻ ✽ ❑✉❜❛♥ ✻✵ ✽ ❑✉❜❛♥ ✻✵ ✽ ❑✉❜❛♥ ✻✵ ▲♦❦♦♠♦t✐✈ ❤❛s ❛❧❧ t❤❡ ✐♥❝❡♥t✐✈❡s t♦ ❧♦s❡✦ ❉♠✐tr② ❉❛❣❛❡✈✱ ❑♦♥st❛♥t✐♥ ❙♦♥✐♥ ❲✐♥♥✐♥❣ ❜② ▲♦s✐♥❣

  7. ▲✐t❡r❛t✉r❡ r❡✈✐❡✇ ◮ ✭❍❛r❛r②✱ ▼♦s❡r✱ ✶✾✻✻✮ ❘❛♥❦✐♥❣ t❡❛♠s ❂ ❆❣❣r❡❣❛t✐♦♥ ♦❢ ✈♦t❡rs✬ ♣r❡❢❡r❡♥❝❡s ◮ ✭❆rr♦✇✱ ✶✾✻✸✮ ❙❡✈❡r❛❧ ❤✐❣❤❧② ❞❡s✐r❡❞ ♣r♦♣❡rt✐❡s ♦❢ ❛❣❣r❡❣❛t✐♦♥ ♦❢ ✈♦t❡rs✬ ♣r❡❢❡r❡♥❝❡s r✉❧❡s✳ ❆rr♦✇✬s ✐♠♣♦ss✐❜✐❧✐t② t❤❡♦r❡♠✳ ◮ ✭❘✉❜✐♥st❡✐♥✱ ✶✾✽✺✮ ❙✐♠✐❧❛r ❛♣♣r♦❛❝❤ ❢♦r t❤❡ ♣r♦❜❧❡♠ ♦❢ r❛♥❦✐♥❣ ♣❛rt✐❝✐♣❛♥ts ✐♥ t❤❡ r♦✉♥❞✲r♦❜✐♥ t♦✉r♥❛♠❡♥t ◮ ✭●✐❜❜❛r❞✱ ✶✾✼✸❀ ❙❛tt❡rt❤✇❛✐t❡✱ ✶✾✼✺❀ ❉✉❣❣❛♥✱ ❙❝❤✇❛rt③✱ ✷✵✵✵✮ ❯♥❞❡r ➽❣♦♦❞ ❡♥♦✉❣❤➾ ❛❣❣r❡❣❛t✐♦♥ r✉❧❡s t❤❡r❡ ❛❧✇❛②s ❡①✐sts ❛ ✈♦t❡r ✇❤♦ ❝❛♥ ♣r♦✜t❛❜❧② ❞❡✈✐❛t❡ ❢r♦♠ ❤✐s tr✉❧② ♣r❡❢❡r❡♥❝❡s ◮ ✭❘✉ss❡❧❧✱ ❲❛❧s❤✱ ✷✵✵✾✮ ❈♦❛❧✐t✐♦♥❛❧ ♠❛♥✐♣✉❧❛t✐♥❣ ✐♥ ❝✉♣s ❛♥❞ r♦✉♥❞✲r♦❜✐♥ ❝♦♠♣❡t✐t✐♦♥s ❉♠✐tr② ❉❛❣❛❡✈✱ ❑♦♥st❛♥t✐♥ ❙♦♥✐♥ ❲✐♥♥✐♥❣ ❜② ▲♦s✐♥❣

  8. ❚❤❡♦r② ✕ ❞❡✜♥✐t✐♦♥s ❚♦✉r♥❛♠❡♥t ✐s ❛ ♣❛✐r ( X , v ( x , y )) ✱ ✇❤❡r❡ X ✐s ❛ ♥♦♥✲❡♠♣t② ✜♥✐t❡ s❡t ♦❢ t❤❡ t❡❛♠s ❛♥❞ v ( x , y ) ✐s ❛ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ❝♦♥❞✐t✐♦♥s✿ ✶✮ v ( x , y ) ✐s ❞❡✜♥❡❞ ♦♥ t❤❡ s❡t ( X × X ) \ { ( x , y ) | x = y } ❀ ✷✮ ✐♠❛❣❡ ♦❢ v ( x , y ) ✐s ❛ s✉❜s❡t ♦❢ t❤❡ s❡t {− 1 , 0 , 1 } ❀ ✸✮ ❢♦r ❡❛❝❤ x 0 , y 0 ∈ X ✱ x 0 � = y 0 ✱ t❤❡ ❡q✉❛❧✐t② v ( x 0 , y 0 ) = − v ( y 0 , x 0 ) ❤♦❧❞s✳ ❉♠✐tr② ❉❛❣❛❡✈✱ ❑♦♥st❛♥t✐♥ ❙♦♥✐♥ ❲✐♥♥✐♥❣ ❜② ▲♦s✐♥❣

  9. ❊①❛♠♣❧❡ ❊①❛♠♣❧❡ ✳ ❈♦♥s✐❞❡r ❛ t♦✉r♥❛♠❡♥t T = ( X , v 0 ) ✱ ✇❤❡r❡ X = { A , B , C , D } ❛♥❞ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ v 0 ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ t❛❜❧❡✿ A B C D A ✲ ✶ ✲✶ ✲✶ B ✲✶ ✲ ✶ ✵ C ✶ ✲✶ ✲ ✲✶ D ✶ ✵ ✶ ✲ ❉♠✐tr② ❉❛❣❛❡✈✱ ❑♦♥st❛♥t✐♥ ❙♦♥✐♥ ❲✐♥♥✐♥❣ ❜② ▲♦s✐♥❣

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