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, - ( , - ). . ( a b a0, b0.) a := a + b; {a = a0 + b0, b = b0} b := a - b; {a = a0 + b0, b = a0} a := a - b; {a = b0, b = a0} 1.1.3. ( - ) n. n. , - , n , - (, b) n. ( .) . k, 0 n, : b = (a k). k := 0; b := 1; {b = a k} while k <> n do begin | k := k + 1; | b := b * a; end; : k := n; b := 1; {a n = b * (a k)} while k <> 0 do begin | k := k - 1; | b := b * a; end; 1.1.4. , , - ( ) log n ( C*log n - C; log n - , 2, - n). . - : k := n; b := 1; c:=a; {a n = b * (c k)} while k <> 0 do begin | if k mod 2 = 0 then begin | | k:= k div 2; | | c:= c*c; | end else begin | | k := k - 1; | | b := b * c; | end; end; ( ) ( k , ), k . 1.1.5. , b. *b, +, -, =, <>. . var a, b, c, k : integer; k := 0; c := 0; {: c = a * k} while k <> b do begin | k := k + 1; | c := c + a; end; {c = a * k k = b, , c = a * b} 1.1.6. b. +b. <1> := <2>, <> := <>, <1> := <2> + 1. . ... {: c = a + k} ... 1.1.7. ( ) d. q r d, div mod. . , a = q * d + r, 0 <= r < d. {a >= 0; d > 0} r := a; q := 0; {: a = q * d + r, 0 <= r} while not (r < d) do begin | {r >= d} | r := r - d; {r >= 0} | q := q + 1; end; 1.1.8. n, n! (0!=1, n! = n * (n-1)!). 1.1.9. : a(0)= 1, a(1) = 1, a(k) = a(k-1) + a(k-2) k >= 2. n, a(n). 1.1.10. , , log n. ( - .) . - |1 1| |1 0| n. C*log n , - . 1.1.11. n, 1/0!+1/1!+...+1/n!. 1.1.12. , , ( ) C*n - . . : sum = 1/1! +...+ 1/k!, last = 1/k! ( k!). 1.1.13. a b, . (a,b) - b. (1 ). if a > b then begin | k := a; end else begin | k := b; end; {k = max (a,b)} {: , k, - } while not (((a mod k)=0) and ((b mod k)=0)) do begin | k := k - 1; end; {k - , - } (2 - ). , (0,0) = 0. (a,b) = (a-b,b) = (a,b-a); (a,0) = (0,a) = a a,b>=0. m := a; n := b; {: (a,b) = (m,n); m,n >= 0 } while not ((m=0) or (n=0)) do begin | if m >= n then begin | | m := m - n; | end else begin | | n := n - m; | end; end; if m = 0 then begin | k := n; end else begin | k := m; end; 1.1.14. - , (a, b) = (a mod b, b) a >= b, (a, b) = (a, b mod a) b >= a. 1.1.15. b, 0 . d = (a,b) x y, d = a*x + b*y. . p, q, r, s m = p*a + q*b; n = r*a + s*b. m:=a; n:=b; p := 1; q := 0; r := 0; s := 1; {: (a,b) = (m,n); m,n >= 0 m = p*a + q*b; n = r*a + s*b.} while not ((m=0) or (n=0)) do begin | if m >= n then begin | | m := m - n; p := p - r; q := q - s; | end else begin | | n := n - m; r := r - p; s := s - q; | end; end; if m = 0 then begin | k :=n; x := r; y := s; end else begin | k := m; x := p; y := q; end; 1.1.16. , . 1.1.17. (.). - u, v, z: m := a; n := b; u := b; v := a; {: (a,b) = (m,n); m,n >= 0 } while not ((m=0) or (n=0)) do begin | if m >= n then begin | | m := m - n; v := v + u; | end else begin | | n := n - m; u := u + v; | end; end; if m = 0 then begin | z:= v; end else begin {n=0} | z:= u; end; , z - a, b: z = 2 * (a,b). . , m*u + n*v . , 2*a*b (a, b) * (a, b) = a*b. 1.1.18. , (2*a, 2*b) = 2*(a,b) (2*a, b) = (a,b) b, , 2 . ( log k , k.) . m:= a; n:=b; d:=1; {(a,b) = d * (m,n)} while not ((m=0) or (n=0)) do begin | if (m mod 2 = 0) and (n mod 2 = 0) then begin | | d:= d*2; m:= m div 2; n:= n div 2; | end else if (m mod 2 = 0) and (n mod 2 = 1) then begin | | m:= m div 2; | end else if (m mod 2 = 1) and (n mod 2 = 0) then begin | | n:= n div 2; | end else if (m mod 2=1) and (n mod 2=1) and (m>=n)then begin | | m:= m-n; | end else if (m mod 2=1) and (n mod 2=1) and (m<=n)then begin | | n:= n-m; | end; end; {m=0 => =d*n; n=0 => =d*m} : m n . 1.1.19. x y, ax+by=(a,b). . ( .) , a b x y. , a b . ( .) , , p,q,r,s, m = ap + bq n = ar + bs , , , m 2 p q 2, ( -- ). - - (, ). d a b -- . , (2 i)* d = ax + by x,y i. , i > 1? x y , 2 . , x := x + b y := y - a ( ax+by). . , - , a b . a. y , x ( ax+by - ). y a y . 1.1.20. , - 0 n. . k:=0; writeln (k*k); {: k<=n, k } while not (k=n) do begin | k:=k+1; | writeln (k*k); end; 1.1.21. , - , - n. . k_square (square - ), k k_square = k*k: k := 0; k_square := 0; writeln (k_square); while not (k = n) do begin | k := k + 1; | {k_square = (k-1) * (k-1) = k*k - 2*k + 1} | k_square := k_square + k + k - 1; | writeln (k_square); end; 1.1.22. , - n > 0 ( - , - n; n = 1, ). (1 ). k := n; {: k n, } while not (k = 1) do begin | l := 2; | {: k (1,l)} | while k mod l <> 0 do begin | | l := l + 1; | end; | {l - k, 1, , | } | writeln (l); | k:=k div l; end; (2 ). k := n; l := 2; { k n; - ; k , l} while not (k = 1) do begin | if k mod l = 0 then begin | | {k l , | | l, , l } | | k := k div l; | | writeln (l); | end else begin | | { k l } | | l := l + 1; | end; end; 1.1.23. , - , , . . l:=l+1 - if l*l > k then begin | l:=k; end else begin | l:=l+1; end; 1.1.24. , n > 1 . 1.1.25. ( ). - n + mi ( Z[i]). (a) , - ( Z[i]); () ( Z[i]) . 1.1.26. write (i) i = 0,1,2,...,9. , - n > 0. ( n = 0 , , n = 0 - .) . base:=1; {base - 10, n} while 10 * base <= n do begin | base:= base * 10; end; {base - 10, n} k:=n; {: k , base; base = 100..00} while base <> 1 do begin | write(k div base); | k:= k mod base; | base:= base div 10; end; {base=1; k} write(k); ( : - . - , k < base; k .) 1.1.27. , . ( n = 173 371.) . k:= n; {: k } while k <> 0 do begin | write (k mod 10); | k:= k div 10; end; 1.1.28. n. x*x + y*y < n ( ) , . . k := 0; s := 0; {: s = x*x + y*y < n c x < k} while k*k < n do begin | ... | {t = k*k + y*y < n | ( k) } | k := k + 1; | s := s + t; end; {k*k >= n, s = } ... - , : l := 0; t := 0; {: t = k*k + y*y < n c y < l } while k*k + l*l < n do begin | l := l + 1; | t := t + 1; end; {k*k + l*l >= n, t = k*k + y*y < n} 1.1.29. , (n 1/2). ( , , n .) . ( - * ) , * * * (n 1/2). * * * * ( X) - * * * * . * * * * * , , , . . s - : k- ; s - . : l - l >= 0, - X; s - x, y, x < k - X. (). k := 0; l := 0; while <0,l> X do begin | l := l + 1; end; {k = 0, l - l >= 0, X } s := 0; {: } while not (l = 0) do begin | s := s + l; | {s - k- } | k := k + 1; | { X, , , | , } | while (l <> 0) and ( X) do begin | | l := l - 1; | end; end; {, l = 0, k- , s } : - (n 1/2) , - (n 1/2) . 1.1.30. n k, n > 1. k 1/n. ( - , - .) . . 1/n k , (10 k)/n. - , . . (10 k) n - . - , - . ( - ) r: l := 0; r := 1; {.: l 1/n, k - l r/n} while l <> k do begin | write ( (10 * r) div n); | r := (10 * r) mod n; | l := l + 1; end; 1.1.31. n > 1. - 1/n. . ( ; , , ). , , - n. (n+1)- - k, (n+1+k)- (n+1)-. l := 0; r := 1; {: r/n = l 1/n} while l <> n+1 do begin | r := (10 * r) mod n; | l := l + 1; end; c := r; {c = (n+1)- } r := (10 * r) mod n; k := 0; {r = (n+k+1)- } while r <> c do begin | r := (10 * r) mod n; | k := k + 1; end; 1.1.32 (. ). f : f(0) = 0, f(1) = 1, f (2n) = f(n), f (2n+1) = f (n) + f (n+1). f (n) n, log n . . k := n; a := 1; b := 0; {: 0 <= k, f (n) = a * f(k) + b * f (k+1)} while k <> 0 do begin | if k mod 2 = 0 then begin | | l := k div 2; | | {k = 2l, f(k) = f(l), f (k+1) = f (2l+1) = f(l) + f(l+1), | | f (n) = a*f(k) + b*f(k+1) = (a+b)*f(l) + b*f(l+1)} | | a := a + b; k := l; | end else begin | | l := k div 2; | | {k = 2l + 1, f(k) = f(l) + f(l+1), | | f(k+1) = f(2l+2) = f(l+1), | | f(n) = a*f(k) + b*f(k+1) = a*f(l) + (a+b)*f(l+1)} | | b := a + b; k := l; | end; end; {k = 0, f(n) = a * f(0) + b * f(1) = b, } 1.1.33. , f(0) = 13, f(1) = 17, f(2n) = 43 f(n) + 57 f(n+1), f(2n+1) = 91 f(n) + 179 f(n+1) n>=1. . f(n) . 1.1.34. b, b > 0. b, div mod, - 2 ; C1*log(a/b) + C2 C1, C2. . b1 := b; while b1 <= a do begin | b1 := b1 * 2; end; {b1 > a, b1 = b * ( 2)} q:=0; r:=a; {: q, r - a b1, b1 = b * ( 2)} while b1 <> b do begin | b1 := b1 div 2 ; q := q * 2; | { a = b1 * q + r, 0 <= r, r < 2 * b1} | if r >= b1 then begin | | r := r - b1; | | q := q + 1; | end; end; {q, r - a b} 1.2. . x, y, z - array [1..n] of integer (n - , 0), . 1.2.1. x . ( , , - x[1]..x[n] , x.) . i := 0; {: i x[1]..x[i] 0} while i <> n do begin | i := i + 1; | {x[1]..x[i-1] = 0} | x[i] := 0; end; 1.2.2. x. ( , x, - k x.) . ... {: k= x[1]...x[i] } ... 1.2.3. , , x:=y. . i := 0; {: y , x[l] = y[l] l <= i} while i <> n do begin | i := i + 1; | x[i] := y[i]; end; 1.2.4. x[1]..x[n]. . i := 1; max := x[1]; {: max = x[1]..x[i]} while i <> n do begin | i := i + 1; | {max = x[1]..x[i-1]} | if x[i] > max then begin | | max := x[i]; | end; end; 1.2.5. x: array [1..n] of integer, x[1] <= x[2] <= ... <= x[n]. . . (1 ) i := 1; k := 1; {: k - x[1]..x[i]} while i <> n do begin | i := i + 1; | if x[i] <> x[i-1] then begin | | k := k + 1; | end; end; (2 ) 1 i 1..n-1, x[i] <> x[i+1]. k := 1; for i := 1 to n-1 do begin | if x[i]<> x[i+1] then begin | | k := k + 1; | end; end; 1.2.6. ( ...) m n - mn . - . , - . , array [1..m] of array [1..n] of boolean; , . m*n. . . , , , . ( - , , .) 1.2.7. x: array [1..n] of integer. - . ( n*n.) 1.2.8. , , n* log n. (. - .) 1.2.9. , , - - 1 k n+k. 1.2.10. x [1]..x[n] . , - . . x [i] x [n+1-i] i, i < n + 1 - i, .. 2*i < n + 1 <=> 2*i <= n <=> i <= n div 2: for i := 1 to n div 2 do begin | ... x [i] x [n+1-i]; end; 1.2.11. ( .) x[1]..x[m+n], : x[1]..x[m] m x[m+1]..x[m+n] n. - , . ( m+n.) . (1 ). ( - ) , . (2 , ..). , , - . - , . (3 ). - x[p+1]..x[q] x[q+1]..x[s]. , ( A) ( B). B , A, - B1, B2. ( B = B1 + B2, .) A + B1 + B2 B1 + B2 + A. A B1 - - , ,- B1 + A + B2, A B2. . , : p := 0; q := m; r := m + n; {: x[p+1]..x[q], x[q+1]..x[s]} while (p <> q) and (q <> s) do begin | { } | if (q - p) <= (s - q) then begin | | .. x[p+1]..x[q] x[q+1]..x[q+(q-p)] | | pnew := q; qnew := q + (q - p); | | p := pnew; q := qnew; | end else begin | | .. x[q-(r-q)+1]..x[q] x[q+1]..x[r] | | qnew := q - (r - q); rnew := q; | | q := qnew; r := rnew; | end; end; : - A; A. 1.2.12. a: array [0..n] of integer (n - , ). x (. . a[n]*(x n)+...+a[1]*x+a[0]). . ( .) k := 0; y := a[n]; {: 0 <= k <= n, y= a[n]*(x k)+...+a[n-1]*(x k-1)+...+ + a[n-k]*(x 0)} while k<>n do begin | k := k + 1; | y := y * x + a [n - k]; end; 1.2.13. ( . ..- .) - . . - P(x) P(x)*x + c. x P(x)*x + P(x). ( : . - , , , .) 1.2.14. a:array [0..k] of integer b: array [0..l] of integer k l. - c: array [0..m] of integer - . ( k, l, m - , m = k + l; - i x i.) . for i:=0 to m do begin | c[i]:=0; end; for i:=0 to k do begin | for j:=0 to l do begin | | c[i+j] := c[i+j] + a[i]*b[j]; | end; end; 1.2.15. n*n n. ( n) , (n (log 4)/(log 3)) . . , - 2k. A(x)*x^k + B(x) C(x)*x^k + D(x) ( x^k x k). A(x)C(x)*x^{2k} + (A(x)D(x)+B(x)C(x))*x^k + B(x)D(x) AC, AD+BC, BD - k, - : AC, BD (A+B)(C+D), , AD+BC=(A+B)(C+D)-AC-BD. 1.2.16. x: array [1..k] of integer y: array [1..l] of integer. (. . t, - t = x[i] = y[j] i j). ( - k+l.) . k1:=0; l1:=0; n:=0; {: 0<=k1<=k; 0<=l1<=l; = n + x[k1+1]...x[k] y[l1+1]..y[l]} while (k1 <> k) and (l1 <> l) do begin | if x[k1+1] < y[l1+1] then begin | | k1 := k1 + 1; | end else if x[k1+1] > y[l1+1] then begin | | l1 := l1 + 1; | end else begin {x[k1+1] = y[l1+1]} | | k1 := k1 + 1; | | l1 := l1 + 1; | | n := n + 1; | end; end; {k1 = k l1 = l, , , , n } . k1, l1; . 1.2.17. , , x[1] <= ... <= x[k] y[1] <= ... <= y[l] ( ). . : k1 l1 1, - 1 x[k1+1]...x[k] x[l1+1]...x[l]. . ... end else begin {x[k1+1] = y[l1+1]} | t := x [k1+1]; | while (k1 k) or (l1 <> l) do begin | if k1 = k then begin | | {l1 < l} | | l1 := l1 + 1; | | z[k1+l1] := y[l1]; | end else if l1 = l then begin | | {k1 < k} | | k1 := k1 + 1; | | z[k1+l1] := x[k1]; | end else if x[k1+1] <= y[l1+1] then begin | | k1 := k1 + 1; | | z[k1+l1] := x[k1]; | end else if x[k1+1] >= y[l1+1] then begin | | l1 := l1 + 1; | | z[k1+l1] := y[l1]; | end else begin | | { } | end; end; {k1 = k, l1 = l, } . , . , , . 1.2.20. x[1] <= ... <= x[k] y[1] <= ... <= y[l]. , .. z[1] <= ... <= z[m], , z - x y. k+l. 1.2.21. x[1]<=...<=x[k] y[1]<=...<=y[l] q. x[i]+y[j], q. ( k+l, - - , - .) . - x[1]<=...<=x[k] q-y[l]<=..<=q-y[1], () . 1.2.22. ( .) x[1] <= ... <= x[p], y[1] <= ... <= y[q], z[1] <= ... <= z[r]. . p + q + r. . p1:=1; q1=1; r1:=1; {: x[p1]..x[p], y[q1]..y[q], z[r1]..z[r] } while not ((x[p1]=y[q1]) and (y[q1]=z[r1])) do begin | if x[p1] } while not eq do begin | s := 1; k := 1; | {a[s][b[s]] - a[1][b[1]]..a[k][b[k]]} | while k <> n do begin | | k := k + 1; | | if a[k][b[k]] < a[s][b[s]] then begin | | | s := k; | | end; | end; | {a[s][b[s]] - a[1][b[1]]..a[n][b[n]]} | b [s] := b [s] + 1; | for k := 2 to n do begin | | eq := eq and (a[1][b[1]] = a[k][b[k]]); | end; end; writeln (a[1][b[1]]); 1.2.25. - m*n*n . m*n. . . , : - , . 1.2.26. ( ) x[1] <= ... <= x[n] a. , a , . . i 1..n, - x[i]=a. ( log n.) . (, n > 0.) l := 1; r := n+1; { a , x[l]..x[r-1], r > l} while r - l <> 1 do begin | m := l + (r-l) div 2 ; | {l < m < r } | if x[m] <= a then begin | | l := m; | end else begin {x[m] > a} | | r := m; | end; end; ( , x[m] = a - .) r-l , - . . l + (r-l) div 2 = (2l + (r-l)) div 2 = (r+l) div 2. 1.2.27. ( .) x: array [1..n] of array [1..m] of integer, : x[i][j] <= x[i+1][j], x[i][j] <= x[i][j+1] a. , a x[i][j]. . a (- , ), , - a, . - x[i][j] 1<=i<=l k<=j<=m. 1 k m - 1| |***********| | |***********| | |***********| l| |***********| || | | n| | - ( l = 0 k = m+1). l:=n; k:=1; {l>=0, k<=m+1, a , } while (l > 0) and (k < m+1) and (x[l][k] <> a) do begin | if x[l][k] < a then begin | | k := k + 1; { a, } | end else begin {x[l][k] > a} | | l := l - 1; { a, } | end; end; {x[l][k] = a } answer:= (l > 0) and (k < m+1) ; . : x[l][k] - . (Ÿ .) 1.2.28. ( ) - a[1] <= a[2] <=...<= a[n]. , - ( - ). n. . , , a[1],...,a[k], 1 - N. a[k+1] > N+1, N+1 , a[1]..a[n]. a[k+1] <= N+1, , a[1]..a[k+1], 1 N+a[k+1]. k := 0; N := 0; {: , a[1]..a[k], 1..N} while (k <> n) and (a[k+1] <= N+1) do begin | N := N + a[k+1]; | k := k + 1; end; {(k = n) (a[k+1] > N+1); N+1} writeln (N+1); ( : .) 1.2.29. ( ) a[1]..a[n] 1..n ( ). () . ( (), () - n.) () , ( a[i]=j, a[j]=i). . () - . , - , , . () . 1.2.30. a[1..n] b. , , b, - b. . l:=0; r:=n; {: a[1]..a[l]<=b; a[r+1]..a[n]>=b} while l <> r do begin | if a[l+1] <= b then begin | | l:=l+1; | end else if a[r] >=b then begin | | r:=r-1; | end else begin {a[l+1]>b; a[r]b} while m <> r do begin | if a[m+1]=b then begin | | m:=m+1; | end else if a[m+1]>b then begin | | a[m+1] a[r] | | r:=r-1; | end else begin {a[m+1], n - N, m - M) N, f() = F (f (), x[n]). : k := 0; f := f0; {: f - } while k<> n do begin | k := k + 1; | f := F (f, x[k]); end; f0 - ( 0). f , k := 1; f := f ();. . f , - g, f ( , t, f () = t (g ()) ). , F ( , g F g). 1.3.1. : ) ; ) , ; ) (, , ); ) ; ) ( - ) - ; ) , ( - ). . ) < ; >; ) < , ; - >; ) < ; >; ) < ; - - ; >; ) < ; ; - - ; >; ) < , >. 1.3.2. ( ..) x[1]..x[n] y[1]..y[k] . , - , . . - , . n+k. . (1 ) . n1:=n; k1:=k; {: <=> x[1]..x[n1] - y[1]..y[k1] } while (n1 > 0) and (k1 > 0) do begin | if x[n1] = y[k1] then begin | | n1 := n1 - 1; | | k1 := k1 - 1; | end else begin | | n1 := n1 - 1; | end; end; {n1 = 0 k1 = 0; k1 = 0, - , k1 <> 0 ( n1 = 0), - } answer := (k1 = 0); , x[n1] = y[k1] y[1]..y[k1] - x[1]..x[n1], y[1]..y[k1-1] - - x[1]..x[n1-1]. (2 ) x[1]..x[n1] |-> ( k1, y[1]..y[k1] x[1]..x[n1]) - . 1.3.3. x[1]..x[n] y[1]..y[k] . , - . - n*k. ( .., ..). - f(n1,k1) - x[1]..x[n1] y[1]..y[k1]. x[n1] <> y[k1] => f(n1,k1) = max (f(n1,k1-1), f(n1-1,k1)); x[n1] = y[k1] => f(n1,k1) = max (f(n1,k1-1), f(n1-1,k1), f(n1-1,k1-1)+1 ); ( f(n1-1,k1-1)+1 >= f(n1,k1-1), f(n1-1,k1), .) f, n*k. k ( n), - ( n1) ( n1 ). 1.3.4 ( .) - x[1],..., x[n]. ( n*log(n)). . , : ( k) - u[1],...,u[k], u[i] = ( i). , u[1] <= ... <= u[k]. x u k - . n1 := 1; k := 1; u[1] := x[1]; {: k u } while n1 <> n do begin | n1 := n1 + 1; | ... | {i - 1..k, - | u[i] < x[n1]; , i=0 } | if i = k then begin | | k := k + 1; | | u[k+1] := x[n1]; | end else begin {i < k, u[i] < x[n1] <= u[i+1] } | | u[i+1] := x[n1]; | end; end; ... ; - u[0] , u[k+1] - ; : u[i] < x[n1] <= u[i+1]. i:=0; j:=k+1; {u[i] < x[n1] <= u[j], j > i} while (j - i) <> 1 do begin | s := i + (j-i) div 2; {i < s < j} | if u[s] >= x[n1] then begin | | j := s; | end else begin {u[s] < x[n1]} | | i := s; | end; end; {u[i] < x[n1] <= u[j], j-i = 1} . 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