1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429 | #include <iostream>
#include <set>
#include <map>
#include <limits>
#include <vector>
#include <iomanip> // for std::setw
#include <ios> // for std::noskipws, streamsize
#include <istream> // for std::istream
#include <ostream> // for std::ostream
#include <sstream> // for std::ostringstream
#include <cstddef> // for NULL
#include <stdexcept> // for std::domain_error
#include <string> // for std::string implicit constructor
#include <cstdlib> // for std::abs
#include <limits> // for std::numeric_limits
#include<type_traits>
using namespace std;
# define throw(x) throw(x)
namespace boost {
template<bool B, class T = void>
struct enable_if_c {
typedef T type;
};
template<class T>
struct enable_if_c<false, T> {
};
template<bool B, class T = void>
struct disable_if_c {
typedef T type;
};
template<class T>
struct disable_if_c<true, T> {
};
};
#include <cassert>
#include <climits>
#include <iterator>
#include <algorithm>
#include <limits>
#include <type_traits>
namespace boost {
template <class I>
class rational;
namespace integer {
namespace gcd_detail{
template <class T>
inline constexpr T constexpr_min(T const& a, T const& b) noexcept
{
return a < b ? a : b;
}
template <class T>
inline constexpr auto constexpr_swap(T&a, T& b) noexcept -> decltype(a.swap(b))
{
return a.swap(b);
}
template <class T, class U>
inline constexpr void constexpr_swap(T&a, U& b...) noexcept
{
T t(static_cast<T&&>(a));
a = static_cast<T&&>(b);
b = static_cast<T&&>(t);
}
template <class T, bool a =
std::is_unsigned<T>::value ||
(std::numeric_limits<T>::is_specialized && !std::numeric_limits<T>::is_signed)>
struct gcd_traits_abs_defaults
{
inline static constexpr const T& abs(const T& val) noexcept { return val; }
};
template <class T>
struct gcd_traits_abs_defaults<T, false>
{
inline static T constexpr abs(const T& val) noexcept
{
// This sucks, but std::abs is not constexpr :(
return val < T(0) ? -val : val;
}
};
enum method_type
{
method_euclid = 0,
method_binary = 1,
method_mixed = 2
};
struct any_convert
{
template <class T>
any_convert(const T&);
};
struct unlikely_size
{
char buf[9973];
};
unlikely_size operator <<= (any_convert, any_convert);
unlikely_size operator >>= (any_convert, any_convert);
template <class T>
struct gcd_traits_defaults : public gcd_traits_abs_defaults<T>
{
inline static constexpr unsigned make_odd(T& val) noexcept
{
unsigned r = 0;
while(0 == (val & 1u))
{
val >>= 1;
++r;
}
return r;
}
inline static constexpr bool less(const T& a, const T& b) noexcept
{
return a < b;
}
static T& get_value();
static const bool has_operator_left_shift_equal = sizeof(get_value() <<= 2) != sizeof(unlikely_size);
static const bool has_operator_right_shift_equal = sizeof(get_value() >>= 2) != sizeof(unlikely_size);
static const method_type method = std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer && has_operator_left_shift_equal && has_operator_right_shift_equal ? method_mixed : method_euclid;
};
//
// Default gcd_traits just inherits from defaults:
//
template <class T>
struct gcd_traits : public gcd_traits_defaults<T> {};
//
// The Mixed Binary Euclid Algorithm
// Sidi Mohamed Sedjelmaci
// Electronic Notes in Discrete Mathematics 35 (2009) 169-176
//
template <class T>
constexpr T mixed_binary_gcd(T u, T v) noexcept
{
if(gcd_traits<T>::less(u, v))
constexpr_swap(u, v);
unsigned shifts = 0;
if(u == T(0))
return v;
if(v == T(0))
return u;
shifts = constexpr_min(gcd_traits<T>::make_odd(u), gcd_traits<T>::make_odd(v));
while(gcd_traits<T>::less(1, v))
{
u %= v;
v -= u;
if(u == T(0))
return v << shifts;
if(v == T(0))
return u << shifts;
gcd_traits<T>::make_odd(u);
gcd_traits<T>::make_odd(v);
if(gcd_traits<T>::less(u, v))
constexpr_swap(u, v);
}
return (v == 1 ? v : u) << shifts;
}
/** Stein gcd (aka 'binary gcd')
*
* From Mathematics to Generic Programming, Alexander Stepanov, Daniel Rose
*/
template <typename SteinDomain>
constexpr SteinDomain Stein_gcd(SteinDomain m, SteinDomain n) noexcept
{
assert(m >= 0);
assert(n >= 0);
if (m == SteinDomain(0))
return n;
if (n == SteinDomain(0))
return m;
// m > 0 && n > 0
unsigned d_m = gcd_traits<SteinDomain>::make_odd(m);
unsigned d_n = gcd_traits<SteinDomain>::make_odd(n);
// odd(m) && odd(n)
while (m != n)
{
if (n > m)
constexpr_swap(n, m);
m -= n;
gcd_traits<SteinDomain>::make_odd(m);
}
// m == n
m <<= constexpr_min(d_m, d_n);
return m;
}
/** Euclidean algorithm
*
* From Mathematics to Generic Programming, Alexander Stepanov, Daniel Rose
*
*/
template <typename EuclideanDomain>
inline constexpr EuclideanDomain Euclid_gcd(EuclideanDomain a, EuclideanDomain b) noexcept
{
while (b != EuclideanDomain(0))
{
a %= b;
constexpr_swap(a, b);
}
return a;
}
template <typename T>
inline constexpr typename enable_if_c<gcd_traits<T>::method == method_mixed, T>::type
optimal_gcd_select(T const &a, T const &b) noexcept
{
return gcd_detail::mixed_binary_gcd(a, b);
}
template <typename T>
inline constexpr typename enable_if_c<gcd_traits<T>::method == method_binary, T>::type
optimal_gcd_select(T const &a, T const &b) noexcept
{
return gcd_detail::Stein_gcd(a, b);
}
template <typename T>
inline constexpr typename enable_if_c<gcd_traits<T>::method == method_euclid, T>::type
optimal_gcd_select(T const &a, T const &b) noexcept
{
return gcd_detail::Euclid_gcd(a, b);
}
template <class T>
inline constexpr T lcm_imp(const T& a, const T& b) noexcept
{
T temp = boost::integer::gcd_detail::optimal_gcd_select(a, b);
return (temp != T(0)) ? T(a / temp * b) : T(0);
}
} // namespace detail
template <typename Integer>
inline constexpr Integer gcd(Integer const &a, Integer const &b) noexcept
{
if(a == (std::numeric_limits<Integer>::min)())
return a == static_cast<Integer>(0) ? gcd_detail::gcd_traits<Integer>::abs(b) : boost::integer::gcd(static_cast<Integer>(a % b), b);
else if (b == (std::numeric_limits<Integer>::min)())
return b == static_cast<Integer>(0) ? gcd_detail::gcd_traits<Integer>::abs(a) : boost::integer::gcd(a, static_cast<Integer>(b % a));
return gcd_detail::optimal_gcd_select(static_cast<Integer>(gcd_detail::gcd_traits<Integer>::abs(a)), static_cast<Integer>(gcd_detail::gcd_traits<Integer>::abs(b)));
}
template <typename Integer>
inline constexpr Integer lcm(Integer const &a, Integer const &b) noexcept
{
return gcd_detail::lcm_imp(static_cast<Integer>(gcd_detail::gcd_traits<Integer>::abs(a)), static_cast<Integer>(gcd_detail::gcd_traits<Integer>::abs(b)));
}
} // namespace integer
} // namespace boost
namespace boost {
namespace rational_detail{
template <class FromInt, class ToInt>
struct is_compatible_integer
{
static const bool value = ((std::numeric_limits<FromInt>::is_specialized && std::numeric_limits<FromInt>::is_integer
&& (std::numeric_limits<FromInt>::digits <= std::numeric_limits<ToInt>::digits)
&& (std::numeric_limits<FromInt>::radix == std::numeric_limits<ToInt>::radix)
&& ((std::numeric_limits<FromInt>::is_signed == false) || (std::numeric_limits<ToInt>::is_signed == true))
&& is_convertible<FromInt, ToInt>::value)
|| is_same<FromInt, ToInt>::value)
|| (is_class<ToInt>::value && is_class<FromInt>::value && is_convertible<FromInt, ToInt>::value);
};
}
class bad_rational : public std::domain_error
{
public:
explicit bad_rational() : std::domain_error("bad rational: zero denominator") {}
explicit bad_rational( char const *what ) : std::domain_error( what ) {}
};
template <typename IntType>
class rational
{
// Class-wide pre-conditions
static_assert( ::std::numeric_limits<IntType>::is_specialized );
// Helper types
using param_type = IntType;
struct helper { IntType parts[2]; };
typedef IntType (helper::* bool_type)[2];
public:
// Component type
typedef IntType int_type;
constexpr
rational() : num(0), den(1) {}
template <class T>
constexpr rational(const T& n) : num(n), den(1) {}
template <class T, class U>
constexpr rational(const T& n, const U& d) : num(n), den(d) {
normalize();
}
template < typename NewType >
constexpr explicit
rational(rational<NewType> const &r, typename enable_if_c<rational_detail::is_compatible_integer<NewType, IntType>::value>::type const* = 0)
: num(r.numerator()), den(is_normalized(int_type(r.numerator()),
int_type(r.denominator())) ? r.denominator() :
(throw(bad_rational("bad rational: denormalized conversion")), 0)){}
template < typename NewType >
constexpr explicit
rational(rational<NewType> const &r, typename disable_if_c<rational_detail::is_compatible_integer<NewType, IntType>::value>::type const* = 0)
: num(r.numerator()), den(is_normalized(int_type(r.numerator()),
int_type(r.denominator())) && is_safe_narrowing_conversion(r.denominator()) && is_safe_narrowing_conversion(r.numerator()) ? r.denominator() :
(throw(bad_rational("bad rational: denormalized conversion")), 0)){}
// Default copy constructor and assignment are fine
// Add assignment from IntType
template <class T>
constexpr typename enable_if_c<
rational_detail::is_compatible_integer<T, IntType>::value, rational &
>::type operator=(const T& n) { return assign(static_cast<IntType>(n), static_cast<IntType>(1)); }
// Assign in place
template <class T, class U>
constexpr typename enable_if_c<
rational_detail::is_compatible_integer<T, IntType>::value && rational_detail::is_compatible_integer<U, IntType>::value, rational &
>::type assign(const T& n, const U& d)
{
return *this = rational<IntType>(static_cast<IntType>(n), static_cast<IntType>(d));
}
//
// The following overloads should probably *not* be provided -
// but are provided for backwards compatibity reasons only.
// These allow for construction/assignment from types that
// are wider than IntType only if there is an implicit
// conversion from T to IntType, they will throw a bad_rational
// if the conversion results in loss of precision or undefined behaviour.
//
template <class T>
constexpr rational(const T& n, typename enable_if_c<
std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
&& !rational_detail::is_compatible_integer<T, IntType>::value
&& (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
&& is_convertible<T, IntType>::value
>::type const* = 0)
{
assign(n, static_cast<T>(1));
}
template <class T, class U>
constexpr rational(const T& n, const U& d, typename enable_if_c<
(!rational_detail::is_compatible_integer<T, IntType>::value
|| !rational_detail::is_compatible_integer<U, IntType>::value)
&& std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
&& (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
&& is_convertible<T, IntType>::value &&
std::numeric_limits<U>::is_specialized && std::numeric_limits<U>::is_integer
&& (std::numeric_limits<U>::radix == std::numeric_limits<IntType>::radix)
&& is_convertible<U, IntType>::value
>::type const* = 0)
{
assign(n, d);
}
template <class T>
constexpr typename enable_if_c<
std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
&& !rational_detail::is_compatible_integer<T, IntType>::value
&& (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
&& is_convertible<T, IntType>::value,
rational &
>::type operator=(const T& n) { return assign(n, static_cast<T>(1)); }
template <class T, class U>
constexpr typename enable_if_c<
(!rational_detail::is_compatible_integer<T, IntType>::value
|| !rational_detail::is_compatible_integer<U, IntType>::value)
&& std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
&& (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
&& is_convertible<T, IntType>::value &&
std::numeric_limits<U>::is_specialized && std::numeric_limits<U>::is_integer
&& (std::numeric_limits<U>::radix == std::numeric_limits<IntType>::radix)
&& is_convertible<U, IntType>::value,
rational &
>::type assign(const T& n, const U& d)
{
if(!is_safe_narrowing_conversion(n) || !is_safe_narrowing_conversion(d))
throw(bad_rational());
return *this = rational<IntType>(static_cast<IntType>(n), static_cast<IntType>(d));
}
// Access to representation
constexpr
const IntType& numerator() const { return num; }
constexpr
const IntType& denominator() const { return den; }
// Arithmetic assignment operators
constexpr rational& operator+= (const rational& r);
constexpr rational& operator-= (const rational& r);
constexpr rational& operator*= (const rational& r);
constexpr rational& operator/= (const rational& r);
template <class T>
constexpr typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator+= (const T& i)
{
num += i * den;
return *this;
}
template <class T>
constexpr typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator-= (const T& i)
{
num -= i * den;
return *this;
}
template <class T>
constexpr typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator*= (const T& i)
{
// Avoid overflow and preserve normalization
IntType gcd = integer::gcd(static_cast<IntType>(i), den);
num *= i / gcd;
den /= gcd;
return *this;
}
template <class T>
constexpr typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator/= (const T& i)
{
// Avoid repeated construction
IntType const zero(0);
if(i == zero) throw(bad_rational());
if(num == zero) return *this;
// Avoid overflow and preserve normalization
IntType const gcd = integer::gcd(num, static_cast<IntType>(i));
num /= gcd;
den *= i / gcd;
if(den < zero) {
num = -num;
den = -den;
}
return *this;
}
// Increment and decrement
constexpr const rational& operator++() { num += den; return *this; }
constexpr const rational& operator--() { num -= den; return *this; }
constexpr rational operator++(int)
{
rational t(*this);
++(*this);
return t;
}
constexpr rational operator--(int)
{
rational t(*this);
--(*this);
return t;
}
// Operator not
constexpr
bool operator!() const { return !num; }
// Boolean conversion
constexpr
operator bool_type() const { return operator !() ? 0 : &helper::parts; }
// Comparison operators
constexpr bool operator< (const rational& r) const;
constexpr bool operator> (const rational& r) const { return r < *this; }
constexpr
bool operator== (const rational& r) const;
template <class T>
constexpr typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator< (const T& i) const
{
// Avoid repeated construction
int_type const zero(0);
// Break value into mixed-fraction form, w/ always-nonnegative remainder
assert(this->den > zero);
int_type q = this->num / this->den, r = this->num % this->den;
while(r < zero) { r += this->den; --q; }
// Compare with just the quotient, since the remainder always bumps the
// value up. [Since q = floor(n/d), and if n/d < i then q < i, if n/d == i
// then q == i, if n/d == i + r/d then q == i, and if n/d >= i + 1 then
// q >= i + 1 > i; therefore n/d < i iff q < i.]
return q < i;
}
template <class T>
constexpr typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator>(const T& i) const
{
return operator==(i) ? false : !operator<(i);
}
template <class T>
constexpr typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator== (const T& i) const
{
return ((den == IntType(1)) && (num == i));
}
private:
// Implementation - numerator and denominator (normalized).
// Other possibilities - separate whole-part, or sign, fields?
IntType num;
IntType den;
// Helper functions
static constexpr
int_type inner_gcd( param_type a, param_type b, int_type const &zero =
int_type(0) )
{ return b == zero ? a : inner_gcd(b, a % b, zero); }
static constexpr
int_type inner_abs( param_type x, int_type const &zero = int_type(0) )
{ return x < zero ? -x : +x; }
// Representation note: Fractions are kept in normalized form at all
// times. normalized form is defined as gcd(num,den) == 1 and den > 0.
// In particular, note that the implementation of abs() below relies
// on den always being positive.
constexpr bool test_invariant() const;
constexpr void normalize();
static constexpr
bool is_normalized( param_type n, param_type d, int_type const &zero =
int_type(0), int_type const &one = int_type(1) )
{
return d > zero && ( n != zero || d == one ) && inner_abs( inner_gcd(n,
d, zero), zero ) == one;
}
//
// Conversion checks:
//
// (1) From an unsigned type with more digits than IntType:
//
template <class T>
constexpr static typename boost::enable_if_c<(std::numeric_limits<T>::digits > std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T& val)
{
return val < (T(1) << std::numeric_limits<IntType>::digits);
}
//
// (2) From a signed type with more digits than IntType, and IntType also signed:
//
template <class T>
constexpr static typename boost::enable_if_c<(std::numeric_limits<T>::digits > std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == true), bool>::type is_safe_narrowing_conversion(const T& val)
{
// Note that this check assumes IntType has a 2's complement representation,
// we don't want to try to convert a std::numeric_limits<IntType>::min() to
// a T because that conversion may not be allowed (this happens when IntType
// is from Boost.Multiprecision).
return (val < (T(1) << std::numeric_limits<IntType>::digits)) && (val >= -(T(1) << std::numeric_limits<IntType>::digits));
}
//
// (3) From a signed type with more digits than IntType, and IntType unsigned:
//
template <class T>
constexpr static typename boost::enable_if_c<(std::numeric_limits<T>::digits > std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T& val)
{
return (val < (T(1) << std::numeric_limits<IntType>::digits)) && (val >= 0);
}
//
// (4) From a signed type with fewer digits than IntType, and IntType unsigned:
//
template <class T>
constexpr static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T& val)
{
return val >= 0;
}
//
// (5) From an unsigned type with fewer digits than IntType, and IntType signed:
//
template <class T>
constexpr static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == false) && (std::numeric_limits<IntType>::is_signed == true), bool>::type is_safe_narrowing_conversion(const T&)
{
return true;
}
//
// (6) From an unsigned type with fewer digits than IntType, and IntType unsigned:
//
template <class T>
constexpr static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == false) && (std::numeric_limits<IntType>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T&)
{
return true;
}
//
// (7) From an signed type with fewer digits than IntType, and IntType signed:
//
template <class T>
constexpr static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == true), bool>::type is_safe_narrowing_conversion(const T&)
{
return true;
}
};
// Unary plus and minus
template <typename IntType>
constexpr
inline rational<IntType> operator+ (const rational<IntType>& r)
{
return r;
}
template <typename IntType>
constexpr
inline rational<IntType> operator- (const rational<IntType>& r)
{
return rational<IntType>(static_cast<IntType>(-r.numerator()), r.denominator());
}
// Arithmetic assignment operators
template <typename IntType>
constexpr rational<IntType>& rational<IntType>::operator+= (const rational<IntType>& r)
{
// This calculation avoids overflow, and minimises the number of expensive
// calculations. Thanks to Nickolay Mladenov for this algorithm.
//
// Proof:
// We have to compute a/b + c/d, where gcd(a,b)=1 and gcd(b,c)=1.
// Let g = gcd(b,d), and b = b1*g, d=d1*g. Then gcd(b1,d1)=1
//
// The result is (a*d1 + c*b1) / (b1*d1*g).
// Now we have to normalize this ratio.
// Let's assume h | gcd((a*d1 + c*b1), (b1*d1*g)), and h > 1
// If h | b1 then gcd(h,d1)=1 and hence h|(a*d1+c*b1) => h|a.
// But since gcd(a,b1)=1 we have h=1.
// Similarly h|d1 leads to h=1.
// So we have that h | gcd((a*d1 + c*b1) , (b1*d1*g)) => h|g
// Finally we have gcd((a*d1 + c*b1), (b1*d1*g)) = gcd((a*d1 + c*b1), g)
// Which proves that instead of normalizing the result, it is better to
// divide num and den by gcd((a*d1 + c*b1), g)
// Protect against self-modification
IntType r_num = r.num;
IntType r_den = r.den;
IntType g = integer::gcd(den, r_den);
den /= g; // = b1 from the calculations above
num = num * (r_den / g) + r_num * den;
g = integer::gcd(num, g);
num /= g;
den *= r_den/g;
return *this;
}
template <typename IntType>
constexpr rational<IntType>& rational<IntType>::operator-= (const rational<IntType>& r)
{
// Protect against self-modification
IntType r_num = r.num;
IntType r_den = r.den;
// This calculation avoids overflow, and minimises the number of expensive
// calculations. It corresponds exactly to the += case above
IntType g = integer::gcd(den, r_den);
den /= g;
num = num * (r_den / g) - r_num * den;
g = integer::gcd(num, g);
num /= g;
den *= r_den/g;
return *this;
}
template <typename IntType>
constexpr rational<IntType>& rational<IntType>::operator*= (const rational<IntType>& r)
{
// Protect against self-modification
IntType r_num = r.num;
IntType r_den = r.den;
// Avoid overflow and preserve normalization
IntType gcd1 = integer::gcd(num, r_den);
IntType gcd2 = integer::gcd(r_num, den);
num = (num/gcd1) * (r_num/gcd2);
den = (den/gcd2) * (r_den/gcd1);
return *this;
}
template <typename IntType>
constexpr rational<IntType>& rational<IntType>::operator/= (const rational<IntType>& r)
{
// Protect against self-modification
IntType r_num = r.num;
IntType r_den = r.den;
// Avoid repeated construction
IntType zero(0);
// Trap division by zero
if (r_num == zero)
throw(bad_rational());
if (num == zero)
return *this;
// Avoid overflow and preserve normalization
IntType gcd1 = integer::gcd(num, r_num);
IntType gcd2 = integer::gcd(r_den, den);
num = (num/gcd1) * (r_den/gcd2);
den = (den/gcd2) * (r_num/gcd1);
if (den < zero) {
num = -num;
den = -den;
}
return *this;
}
//
// Non-member operators: previously these were provided by Boost.Operator, but these had a number of
// drawbacks, most notably, that in order to allow inter-operability with IntType code such as this:
//
// rational<int> r(3);
// assert(r == 3.5); // compiles and passes!!
//
// Happens to be allowed as well :-(
//
// There are three possible cases for each operator:
// 1) rational op rational.
// 2) rational op integer
// 3) integer op rational
// Cases (1) and (2) are folded into the one function.
//
template <class IntType, class Arg>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
operator + (const rational<IntType>& a, const Arg& b)
{
rational<IntType> t(a);
return t += b;
}
template <class Arg, class IntType>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
operator + (const Arg& b, const rational<IntType>& a)
{
rational<IntType> t(a);
return t += b;
}
template <class IntType, class Arg>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
operator - (const rational<IntType>& a, const Arg& b)
{
rational<IntType> t(a);
return t -= b;
}
template <class Arg, class IntType>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
operator - (const Arg& b, const rational<IntType>& a)
{
rational<IntType> t(a);
return -(t -= b);
}
template <class IntType, class Arg>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
operator * (const rational<IntType>& a, const Arg& b)
{
rational<IntType> t(a);
return t *= b;
}
template <class Arg, class IntType>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
operator * (const Arg& b, const rational<IntType>& a)
{
rational<IntType> t(a);
return t *= b;
}
template <class IntType, class Arg>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
operator / (const rational<IntType>& a, const Arg& b)
{
rational<IntType> t(a);
return t /= b;
}
template <class Arg, class IntType>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
operator / (const Arg& b, const rational<IntType>& a)
{
rational<IntType> t(b);
return t /= a;
}
template <class IntType, class Arg>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, bool>::type
operator <= (const rational<IntType>& a, const Arg& b)
{
return !(a > b);
}
template <class Arg, class IntType>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
operator <= (const Arg& b, const rational<IntType>& a)
{
return a >= b;
}
template <class IntType, class Arg>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, bool>::type
operator >= (const rational<IntType>& a, const Arg& b)
{
return !(a < b);
}
template <class Arg, class IntType>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
operator >= (const Arg& b, const rational<IntType>& a)
{
return a <= b;
}
template <class IntType, class Arg>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, bool>::type
operator != (const rational<IntType>& a, const Arg& b)
{
return !(a == b);
}
template <class Arg, class IntType>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
operator != (const Arg& b, const rational<IntType>& a)
{
return !(b == a);
}
template <class Arg, class IntType>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
operator < (const Arg& b, const rational<IntType>& a)
{
return a > b;
}
template <class Arg, class IntType>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
operator > (const Arg& b, const rational<IntType>& a)
{
return a < b;
}
template <class Arg, class IntType>
constexpr
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
operator == (const Arg& b, const rational<IntType>& a)
{
return a == b;
}
// Comparison operators
template <typename IntType>
constexpr
bool rational<IntType>::operator< (const rational<IntType>& r) const
{
// Avoid repeated construction
int_type const zero( 0 );
// This should really be a class-wide invariant. The reason for these
// checks is that for 2's complement systems, INT_MIN has no corresponding
// positive, so negating it during normalization keeps it INT_MIN, which
// is bad for later calculations that assume a positive denominator.
assert( this->den > zero );
assert( r.den > zero );
// Determine relative order by expanding each value to its simple continued
// fraction representation using the Euclidian GCD algorithm.
struct { int_type n, d, q, r; }
ts = { this->num, this->den, static_cast<int_type>(this->num / this->den),
static_cast<int_type>(this->num % this->den) },
rs = { r.num, r.den, static_cast<int_type>(r.num / r.den),
static_cast<int_type>(r.num % r.den) };
unsigned reverse = 0u;
// Normalize negative moduli by repeatedly adding the (positive) denominator
// and decrementing the quotient. Later cycles should have all positive
// values, so this only has to be done for the first cycle. (The rules of
// C++ require a nonnegative quotient & remainder for a nonnegative dividend
// & positive divisor.)
while ( ts.r < zero ) { ts.r += ts.d; --ts.q; }
while ( rs.r < zero ) { rs.r += rs.d; --rs.q; }
// Loop through and compare each variable's continued-fraction components
for ( ;; )
{
// The quotients of the current cycle are the continued-fraction
// components. Comparing two c.f. is comparing their sequences,
// stopping at the first difference.
if ( ts.q != rs.q )
{
// Since reciprocation changes the relative order of two variables,
// and c.f. use reciprocals, the less/greater-than test reverses
// after each index. (Start w/ non-reversed @ whole-number place.)
return reverse ? ts.q > rs.q : ts.q < rs.q;
}
// Prepare the next cycle
reverse ^= 1u;
if ( (ts.r == zero) || (rs.r == zero) )
{
// At least one variable's c.f. expansion has ended
break;
}
ts.n = ts.d; ts.d = ts.r;
ts.q = ts.n / ts.d; ts.r = ts.n % ts.d;
rs.n = rs.d; rs.d = rs.r;
rs.q = rs.n / rs.d; rs.r = rs.n % rs.d;
}
// Compare infinity-valued components for otherwise equal sequences
if ( ts.r == rs.r )
{
// Both remainders are zero, so the next (and subsequent) c.f.
// components for both sequences are infinity. Therefore, the sequences
// and their corresponding values are equal.
return false;
}
else
{
#ifdef BOOST_MSVC
#pragma warning(push)
#pragma warning(disable:4800)
#endif
// Exactly one of the remainders is zero, so all following c.f.
// components of that variable are infinity, while the other variable
// has a finite next c.f. component. So that other variable has the
// lesser value (modulo the reversal flag!).
return ( ts.r != zero ) != static_cast<bool>( reverse );
#ifdef BOOST_MSVC
#pragma warning(pop)
#endif
}
}
template <typename IntType>
constexpr
inline bool rational<IntType>::operator== (const rational<IntType>& r) const
{
return ((num == r.num) && (den == r.den));
}
// Invariant check
template <typename IntType>
constexpr
inline bool rational<IntType>::test_invariant() const
{
return ( this->den > int_type(0) ) && ( integer::gcd(this->num, this->den) ==
int_type(1) );
}
// Normalisation
template <typename IntType>
constexpr void rational<IntType>::normalize()
{
// Avoid repeated construction
IntType zero(0);
if (den == zero)
throw(bad_rational());
// Handle the case of zero separately, to avoid division by zero
if (num == zero) {
den = IntType(1);
return;
}
IntType g = integer::gcd(num, den);
num /= g;
den /= g;
if (den < -(std::numeric_limits<IntType>::max)()) {
throw(bad_rational("bad rational: non-zero singular denominator"));
}
// Ensure that the denominator is positive
if (den < zero) {
num = -num;
den = -den;
}
assert( this->test_invariant() );
}
#ifndef BOOST_NO_IOSTREAM
namespace detail {
// A utility class to reset the format flags for an istream at end
// of scope, even in case of exceptions
struct resetter {
resetter(std::istream& is) : is_(is), f_(is.flags()) {}
~resetter() { is_.flags(f_); }
std::istream& is_;
std::istream::fmtflags f_; // old GNU c++ lib has no ios_base
};
}
// Input and output
template <typename IntType>
std::istream& operator>> (std::istream& is, rational<IntType>& r)
{
using std::ios;
IntType n = IntType(0), d = IntType(1);
char c = 0;
detail::resetter sentry(is);
if ( is >> n )
{
if ( is.get(c) )
{
if ( c == '/' )
{
if ( is >> std::noskipws >> d )
try {
r.assign( n, d );
} catch ( bad_rational & ) { // normalization fail
try { is.setstate(ios::failbit); }
catch ( ... ) {} // don't throw ios_base::failure...
if ( is.exceptions() & ios::failbit )
throw; // ...but the original exception instead
// ELSE: suppress the exception, use just error flags
}
}
else
is.setstate( ios::failbit );
}
}
return is;
}
// Add manipulators for output format?
template <typename IntType>
std::ostream& operator<< (std::ostream& os, const rational<IntType>& r)
{
// The slash directly precedes the denominator, which has no prefixes.
std::ostringstream ss;
ss.copyfmt( os );
ss.tie( NULL );
ss.exceptions( std::ios::goodbit );
ss.width( 0 );
ss << std::noshowpos << std::noshowbase << '/' << r.denominator();
// The numerator holds the showpos, internal, and showbase flags.
std::string const tail = ss.str();
std::streamsize const w =
os.width() - static_cast<std::streamsize>( tail.size() );
ss.clear();
ss.str( "" );
ss.flags( os.flags() );
ss << std::setw( w < 0 || (os.flags() & std::ios::adjustfield) !=
std::ios::internal ? 0 : w ) << r.numerator();
return os << ss.str() + tail;
}
#endif // BOOST_NO_IOSTREAM
// Type conversion
template <typename T, typename IntType>
constexpr
inline T rational_cast(const rational<IntType>& src)
{
return static_cast<T>(src.numerator())/static_cast<T>(src.denominator());
}
// Do not use any abs() defined on IntType - it isn't worth it, given the
// difficulties involved (Koenig lookup required, there may not *be* an abs()
// defined, etc etc).
template <typename IntType>
constexpr
inline rational<IntType> abs(const rational<IntType>& r)
{
return r.numerator() >= IntType(0)? r: -r;
}
} // namespace boost
using rational = boost::rational<unsigned long long>;
struct Teacup {
unsigned long long temperature;
rational volume;
Teacup(int temperature, rational volume) : temperature(temperature), volume(volume) {}
bool operator==(const Teacup& second) const noexcept {
return this->temperature == second.temperature;
}
bool operator<(const Teacup& second) const noexcept {
return this->temperature < second.temperature;
}
};
ostream& operator<<(ostream& os, const Teacup& teacup)
{
os << "(t: " << teacup.temperature << ", v: " << teacup.volume << ")";
return os;
}
//ostream& operator<<(ostream& os, const pair<int,int>& teacup)
//{
// os << "(t: " << teacup.first << ", v: " << teacup.second << ")";
// return os;
//}
ostream& operator<<(ostream& os, const pair<rational, rational>& teacup)
{
os << "(t: " << teacup.first << ", v: " << teacup.second << ")";
return os;
}
//#define DEBUG
#ifdef DEBUG
#define dbg cerr
#else
#define dbg 0 && cerr
#endif
template<typename T>
void clear_with_zeros(T& mapa) {
std::vector<rational> keys_to_remove;
for(const auto& [key, value]: mapa) {
if (value == 0ull) {
keys_to_remove.push_back(key);
}
}
for (const auto& key: keys_to_remove) {
mapa.erase(key);
}
}
bool solve() {
int n;
cin >> n;
map<rational, rational> all_start, all_end;
for (int i = 0; i < n; ++i) {
int l, start, end;
cin >> l >> start >> end;
if (all_start.count(1ull * start) == 0) {
all_start[1ull * start] = 0;
}
if (all_end.count(1ull* end) == 0) {
all_end[1ull* end] = 0;
}
all_start[1ull* start] += 1ull* l;
all_end[1ull* end] += 1ull* l;
}
#ifdef DEBUG
dbg << "Start: ";
for(const auto& x: all_start) {
dbg << x << " ";
}
dbg << endl << endl << "End: ";
for(const auto& x: all_end) {
dbg << x << " ";
}
dbg << endl << endl;
#endif
// satisfy correct cups
for(auto& [temp, volume]: all_end) {
if (all_start.count(temp) > 0) {
auto min = std::min(volume, all_start[temp]);
volume -= min;
all_start[temp] -= min;
}
}
// clear with zero value
clear_with_zeros(all_start);
clear_with_zeros(all_end);
#ifdef DEBUG
dbg << "\nStart: ";
for(const auto& x: all_start) {
dbg << x << " ";
}
dbg << endl << endl << "End: ";
for(const auto& x: all_end) {
dbg << x << " ";
}
dbg << endl << endl;
#endif
if (all_start.size() == 0) {
if (all_end.size() == 0) {
return true;
}
return false;
}
if (all_end.size() == 0) {
return false;
}
if (all_start.begin()->first > all_end.begin()->first) {
return false;
}
if (all_start.rbegin()->first < all_end.rbegin()->first) {
return false;
}
{
rational power_total_start, power_total_end;
for (const auto&[temp, volume]: all_start) {
power_total_start += temp * volume;
}
for (const auto&[temp, volume]: all_end) {
power_total_end += temp * volume;
}
if (power_total_start != power_total_end) {
return false;
}
}
while (all_end.size() > 1 && all_start.size() > 1) {
dbg << endl;
auto now = all_end.begin();
#ifdef DEBUG
dbg << "Trying now " << *now << endl;
dbg << "Using: ";
for(const auto x: all_start) {
dbg << x << " ";
}
dbg << endl;
#endif
auto s1 = all_start.begin(), s2 = s1;
s2++;
#define TEMP(x) (x->first)
#define VOL(x) (x->second)
if (TEMP(now) < TEMP(s1)) {
return false;
}
if (TEMP(now) > TEMP(s2)) {
// merge s1 and s2
auto vol = VOL(s1) + VOL(s2);
auto temp = (TEMP(s1) * VOL(s1) + TEMP(s2) * VOL(s2)) / vol;
all_start[temp] = vol;
dbg << "Merge " << *s1 << " and " << *s2 << " into " << temp << ", " << vol << endl;
all_start.erase(s1);
all_start.erase(s2);
continue;
}
// here, TEMP(s1) < TEMP(now) < TEMP(s2)
// calculate how much to take from s1 and s2
auto Ax1 = VOL(now) * (TEMP(s2) - TEMP(now)) / (TEMP(s2) - TEMP(s1));
auto Ax2 = VOL(now) * (TEMP(now) - TEMP(s1)) / (TEMP(s2) - TEMP(s1));
dbg << "Best to take " << Ax1 << " from first, and " << Ax2 << " from second" << endl;
auto A1 = Ax1;
auto A2 = Ax2;
if (Ax1 > VOL(s1)) {
A1 = VOL(s1);
A2 = Ax2 * VOL(s1) / Ax1;
}
Ax1 = A1;
Ax2 = A2;
if (Ax2 > VOL(s2)) {
A1 = Ax1 * VOL(s2) / Ax2;
A2 = VOL(s2);
}
assert(Ax1 <= VOL(s1));
dbg << "After reduction: " << A1 << ", " << A2 << endl;
VOL(s1) -= A1;
VOL(s2) -= A2;
VOL(now) -= (A1 + A2);
if (VOL(s1) == 0ull) {
dbg << "dropping first!" << endl;
all_start.erase(s1);
}
if (VOL(s2) == 0ull) {
dbg << "dropping second!" << endl;
all_start.erase(s2);
}
if (VOL(now) == 0ull) {
dbg << "dropping requirement!" << endl;
all_end.erase(now);
}
}
#ifdef DEBUG
dbg << "\nAt the end:\n";
dbg << "Start: ";
for(const auto& x: all_start) {
dbg << x << " ";
}
dbg << endl << endl << "End: ";
for(const auto& x: all_end) {
dbg << x << " ";
}
dbg << endl << endl;
#endif
rational power_total_start, power_total_end;
rational volume_total_start, volume_total_end;
for(const auto& [temp, volume]: all_start) {
power_total_start += temp*volume;
volume_total_start += volume;
}
for(const auto& [temp, volume]: all_end) {
power_total_end += temp*volume;
volume_total_end += volume;
}
if (power_total_start == power_total_end && volume_total_start == volume_total_end) {
return true;
} else {
return false;
}
}
int main() {
// std::ios_base::sync_with_stdio(false);
// std::cin.tie(nullptr);
int Z;
cin >> Z;
while(Z--) {
if (solve()) {
cout << "TAK\n";
} else {
cout << "NIE\n";
}
}
}
|