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rr ss trtr sts ts t rt Prr r

  1. ●rö❜♥❡r ❇❛s❡s ♦❢ ❙tr✉❝t✉r❡❞ ❙②st❡♠s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s t♦ ❈r②♣t♦❧♦❣② P✐❡rr❡✲❏❡❛♥ ❙♣❛❡♥❧❡❤❛✉❡r ■◆❘■❆✴❈◆❘❙✴❯♥✐✈✳ ▲♦rr❛✐♥❡✱ ❈❛r❛♠❡❧ Pr♦❥❡❝t ❏♦✉r♥é❡s ❈✷✱ ▼❛r❝❤ ✷✻✱ ✷✵✶✹ /* CARAMEL */ C,A, /* */ R,a, /* */ M,E, L,i= 5,e, d[5],Q[999 ]={0};main(N ){for (;i--;e=scanf("%" "d",d+i));for(A =*d; ++i<A ;++Q[ i*i% A],R= i[Q]? R:i); for(;i --;) for(M =A;M --;N +=!M*Q [E%A ],e+= Q[(A +E*E- R*L* L%A) %A]) for( E=i,L=M,a=4;a;C= i*E+R*M*L,L=(M*E +i*L) %A,E=C%A+a --[d]);printf ("%d" "\n", (e+N* N)/2 /* cc caramel.c; echo f3 f2 f1 f0 p | ./a.out */ -A);} ✶

  2. ❢ ✶ ① ✶ ① ♥ ✵ ❧✐st t❤❡ s♦❧✉t✐♦♥s ✐♥ ✳ ♥ ❢ ✶ ❢ ♠ ① ✶ ① ♥ ✳ ✳ ♥ ✇❤❡r❡ ✐s ❛ ✜♥✐t❡ ✜❡❧❞ ❢ ♠ ① ✶ ① ♥ ✵ ❙❆❚ r❡❞✉❝❡s t♦ P♦❙❙♦ ✇❤❡♥ ✷ ◆P✲❤❛r❞ ♣r♦❜❧❡♠ ❞ ♥ s♦❧✉t✐♦♥s ✐♥ ♥ ✳ ❇é③♦✉t t❤❡♦r❡♠ ❇✉t✳✳✳ ❝r②♣t♦❣r❛♣❤✐❝ ♣r♦♣❡rt✐❡s ❙tr✉❝t✉r❡❞ ❛❧❣❡❜r❛✐❝ s②st❡♠s✳ ◗✉❡st✐♦♥✿ ✐♠♣❛❝t ♦❢ str✉❝t✉r❡s ♦♥ ●❇ ❝♦♠♣✉t❛t✐♦♥s ✳ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ●❡♥❡r❛❧ ❢r❛♠❡✇♦r❦ ▼♦❞❡❧✐♥❣ ♦❢ ❛ ❝r②♣t♦s②st❡♠ ❜② ❛♥ ❛❧❣❡❜r❛✐❝ ♣♦❧②♥♦♠✐❛❧ s②st❡♠ ❀ ❈♦❡✣❝✐❡♥ts ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞ ❀ ❙♦❧✈✐♥❣ Ñ r❡tr✐❡✈✐♥❣ s❡❝r❡t ✐♥❢♦r♠❛t✐♦♥❀ ❈♦♠♣❧❡①✐t② Ñ s❡❝✉r✐t②❀ ●rö❜♥❡r ❜❛s❡s ❛❧❣♦r✐t❤♠s � ✇❡❧❧✲s✉✐t❡❞ ✇❤❡♥ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞ ✳ ✷

  3. ❙❆❚ r❡❞✉❝❡s t♦ P♦❙❙♦ ✇❤❡♥ ✷ ◆P✲❤❛r❞ ♣r♦❜❧❡♠ ❞ ♥ s♦❧✉t✐♦♥s ✐♥ ♥ ✳ ❇é③♦✉t t❤❡♦r❡♠ ❇✉t✳✳✳ ❝r②♣t♦❣r❛♣❤✐❝ ♣r♦♣❡rt✐❡s ❙tr✉❝t✉r❡❞ ❛❧❣❡❜r❛✐❝ s②st❡♠s✳ ◗✉❡st✐♦♥✿ ✐♠♣❛❝t ♦❢ str✉❝t✉r❡s ♦♥ ●❇ ❝♦♠♣✉t❛t✐♦♥s ✳ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ●❡♥❡r❛❧ ❢r❛♠❡✇♦r❦ ▼♦❞❡❧✐♥❣ ♦❢ ❛ ❝r②♣t♦s②st❡♠ ❜② ❛♥ ❛❧❣❡❜r❛✐❝ ♣♦❧②♥♦♠✐❛❧ s②st❡♠ ❀ ❈♦❡✣❝✐❡♥ts ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞ ❀ ❙♦❧✈✐♥❣ Ñ r❡tr✐❡✈✐♥❣ s❡❝r❡t ✐♥❢♦r♠❛t✐♦♥❀ ❈♦♠♣❧❡①✐t② Ñ s❡❝✉r✐t②❀ ●rö❜♥❡r ❜❛s❡s ❛❧❣♦r✐t❤♠s � ✇❡❧❧✲s✉✐t❡❞ ✇❤❡♥ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞ ✳ $ ❢ ✶ p ① ✶ , . . . , ① ♥ q “ ✵ ❧✐st t❤❡ s♦❧✉t✐♦♥s ✐♥ ’ & ✳ K ♥ ❢ ✶ , . . . , ❢ ♠ P K r ① ✶ , . . . , ① ♥ s , ✳ ù ñ ✳ ♥ ✇❤❡r❡ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞ ’ K % ❢ ♠ p ① ✶ , . . . , ① ♥ q “ ✵ ✷

  4. ❇✉t✳✳✳ ❝r②♣t♦❣r❛♣❤✐❝ ♣r♦♣❡rt✐❡s ❙tr✉❝t✉r❡❞ ❛❧❣❡❜r❛✐❝ s②st❡♠s✳ ◗✉❡st✐♦♥✿ ✐♠♣❛❝t ♦❢ str✉❝t✉r❡s ♦♥ ●❇ ❝♦♠♣✉t❛t✐♦♥s ✳ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ●❡♥❡r❛❧ ❢r❛♠❡✇♦r❦ ▼♦❞❡❧✐♥❣ ♦❢ ❛ ❝r②♣t♦s②st❡♠ ❜② ❛♥ ❛❧❣❡❜r❛✐❝ ♣♦❧②♥♦♠✐❛❧ s②st❡♠ ❀ ❈♦❡✣❝✐❡♥ts ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞ ❀ ❙♦❧✈✐♥❣ Ñ r❡tr✐❡✈✐♥❣ s❡❝r❡t ✐♥❢♦r♠❛t✐♦♥❀ ❈♦♠♣❧❡①✐t② Ñ s❡❝✉r✐t②❀ ●rö❜♥❡r ❜❛s❡s ❛❧❣♦r✐t❤♠s � ✇❡❧❧✲s✉✐t❡❞ ✇❤❡♥ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞ ✳ $ ❢ ✶ p ① ✶ , . . . , ① ♥ q “ ✵ ❧✐st t❤❡ s♦❧✉t✐♦♥s ✐♥ ’ & ✳ K ♥ ❢ ✶ , . . . , ❢ ♠ P K r ① ✶ , . . . , ① ♥ s , ✳ ù ñ ✳ ♥ ✇❤❡r❡ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞ ’ K % ❢ ♠ p ① ✶ , . . . , ① ♥ q “ ✵ " ❙❆❚ r❡❞✉❝❡s t♦ P♦❙❙♦ ✇❤❡♥ K “ F ✷ ◆P✲❤❛r❞ ♣r♦❜❧❡♠ � ❇é③♦✉t t❤❡♦r❡♠ � ❞ ♥ s♦❧✉t✐♦♥s ✐♥ K ♥ ✳ ✷

  5. ◗✉❡st✐♦♥✿ ✐♠♣❛❝t ♦❢ str✉❝t✉r❡s ♦♥ ●❇ ❝♦♠♣✉t❛t✐♦♥s ✳ ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ●❡♥❡r❛❧ ❢r❛♠❡✇♦r❦ ▼♦❞❡❧✐♥❣ ♦❢ ❛ ❝r②♣t♦s②st❡♠ ❜② ❛♥ ❛❧❣❡❜r❛✐❝ ♣♦❧②♥♦♠✐❛❧ s②st❡♠ ❀ ❈♦❡✣❝✐❡♥ts ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞ ❀ ❙♦❧✈✐♥❣ Ñ r❡tr✐❡✈✐♥❣ s❡❝r❡t ✐♥❢♦r♠❛t✐♦♥❀ ❈♦♠♣❧❡①✐t② Ñ s❡❝✉r✐t②❀ ●rö❜♥❡r ❜❛s❡s ❛❧❣♦r✐t❤♠s � ✇❡❧❧✲s✉✐t❡❞ ✇❤❡♥ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞ ✳ $ ❢ ✶ p ① ✶ , . . . , ① ♥ q “ ✵ ❧✐st t❤❡ s♦❧✉t✐♦♥s ✐♥ ’ & ✳ K ♥ ❢ ✶ , . . . , ❢ ♠ P K r ① ✶ , . . . , ① ♥ s , ✳ ù ñ ✳ ♥ ✇❤❡r❡ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞ ’ K % ❢ ♠ p ① ✶ , . . . , ① ♥ q “ ✵ " ❙❆❚ r❡❞✉❝❡s t♦ P♦❙❙♦ ✇❤❡♥ K “ F ✷ ◆P✲❤❛r❞ ♣r♦❜❧❡♠ � ❇é③♦✉t t❤❡♦r❡♠ � ❞ ♥ s♦❧✉t✐♦♥s ✐♥ K ♥ ✳ ❇✉t✳✳✳ ❝r②♣t♦❣r❛♣❤✐❝ ♣r♦♣❡rt✐❡s ù ñ ❙tr✉❝t✉r❡❞ ❛❧❣❡❜r❛✐❝ s②st❡♠s✳ ✷

  6. ❆❧❣❡❜r❛✐❝ ❝r②♣t❛♥❛❧②s✐s ●❡♥❡r❛❧ ❢r❛♠❡✇♦r❦ ▼♦❞❡❧✐♥❣ ♦❢ ❛ ❝r②♣t♦s②st❡♠ ❜② ❛♥ ❛❧❣❡❜r❛✐❝ ♣♦❧②♥♦♠✐❛❧ s②st❡♠ ❀ ❈♦❡✣❝✐❡♥ts ✐♥ ❛ ✜♥✐t❡ ✜❡❧❞ ❀ ❙♦❧✈✐♥❣ Ñ r❡tr✐❡✈✐♥❣ s❡❝r❡t ✐♥❢♦r♠❛t✐♦♥❀ ❈♦♠♣❧❡①✐t② Ñ s❡❝✉r✐t②❀ ●rö❜♥❡r ❜❛s❡s ❛❧❣♦r✐t❤♠s � ✇❡❧❧✲s✉✐t❡❞ ✇❤❡♥ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞ ✳ $ ❢ ✶ p ① ✶ , . . . , ① ♥ q “ ✵ ❧✐st t❤❡ s♦❧✉t✐♦♥s ✐♥ ’ & ✳ K ♥ ❢ ✶ , . . . , ❢ ♠ P K r ① ✶ , . . . , ① ♥ s , ✳ ù ñ ✳ ♥ ✇❤❡r❡ K ✐s ❛ ✜♥✐t❡ ✜❡❧❞ ’ K % ❢ ♠ p ① ✶ , . . . , ① ♥ q “ ✵ " ❙❆❚ r❡❞✉❝❡s t♦ P♦❙❙♦ ✇❤❡♥ K “ F ✷ ◆P✲❤❛r❞ ♣r♦❜❧❡♠ � ❇é③♦✉t t❤❡♦r❡♠ � ❞ ♥ s♦❧✉t✐♦♥s ✐♥ K ♥ ✳ ❇✉t✳✳✳ ❝r②♣t♦❣r❛♣❤✐❝ ♣r♦♣❡rt✐❡s ù ñ ❙tr✉❝t✉r❡❞ ❛❧❣❡❜r❛✐❝ s②st❡♠s✳ ◗✉❡st✐♦♥✿ ✐♠♣❛❝t ♦❢ str✉❝t✉r❡s ♦♥ ●❇ ❝♦♠♣✉t❛t✐♦♥s ✳ ✷

  7. ▼❛❝❛✉❧❛② ♠❛tr✐① ✐♥ ❞❡❣r❡❡ ❞ ❉❡❣r❡❡ ♦❢ r❡❣✉❧❛r✐t② ✿ ❢ ✶ ❢ ♠ ✵ ❞❡❣ ❢ ✐ ❞ ✐ ♠❛①✐♠❛❧ ❞❡❣r❡❡ r❡❛❝❤❡❞ ❘♦✇s ✿ ❛❧❧ ♣r♦❞✉❝ts t❢ ✐ ✇❤❡r❡ t ▼♦♥♦♠✐❛❧s ❞ ❞ ✐ ✳ ❍✐❧❜❡rt s❡r✐❡s ✿ ❈♦❧✉♠♥s ✿ ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ❞ ✳ ❣❡♥❡r❛t✐♥❣ s❡r✐❡s ♦❢ t❤❡ r❛♥❦ ❞❡❢❡❝ts ♠ ✶ ♠ ❳ ❞ ■ ❞ t ❞ ❍❙ t ❞✐♠ t ✶ ❢ ✶ ❞ ✳ ✳ ✳ ❞ r❡❣ ❞❡❣ ❍❙ ✶ t ❦ ❢ ♠ ✵✲❞✐♠❡♥s✐♦♥❛❧ s♦❧✈✐♥❣ str❛t❡❣② ✇✐t❤ ●rö❜♥❡r ❜❛s❡s ❢ ✶ “ ¨ ¨ ¨ “ ❢ ♠ “ ✵ ❈♦♠♣❧❡①✐t② ❆❧❣♦r✐t❤♠s ó ˘ ω ¯ ❇✉❝❤❜❡r❣❡r ✭✶✾✻✺✮ ❘♦✇ ❊❝❤❡❧♦♥ ❢♦r♠s ♦❢ ▼❛❝❛✉❧❛② ´ ` ♥ ` ❞ r❡❣ ✏❣r❡✈❧❡①✑ ●❜ ❖ ♠ ❋ ✹ ✭❋❛✉❣èr❡ ✶✾✾✾✮ ♠❛tr✐❝❡s ✉♣ t♦ ❞❡❣r❡❡ ❞ r❡❣ ♥ ❋ ✺ ✭❋❛✉❣èr❡ ✷✵✵✷✮ ó ▲✐♥❡❛r ❛❧❣❡❜r❛ ✐♥ K r ❳ s ` ♥ ❉❊● p ■ q ✸ ˘ ✏❧❡①✑ ●❜ ❛s ❛ K ✲ ❖ ❋●▲▼ ■ ✈❡❝t✳ s♣❛❝❡ ♦❢ ❞✐♠✳ ❉❊● p ■ q ❋❛✉❣èr❡✱ ●✐❛♥♥✐✱ � ❣ p ✉ q “ ✵ , ① ✐ “ ❤ ✐ p ✉ q ▲❛③❛r❞✱ ▼♦r❛ ✭✶✾✾✸✮ ✸

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