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[求助] 可以有多少个算式?

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发表于 2021-5-5 07:50:43 | 显示全部楼层 |阅读模式

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若正整数 a < x < y < b < 90 ,\(\D\frac{\sin(a^\circ)\sin(b^\circ)}{\sin(x^\circ)\sin(y^\circ)}=1\)  可以有多少个算式?
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2021-5-6 09:08:39 | 显示全部楼层
本帖最后由 王守恩 于 2021-5-6 09:14 编辑

   相近的话题。

    P 是三角形 ABC 内的点,我们恒有

\(\D \ \ \ \ \ \ \ \frac{\ \sin(∠PAC)\sin(∠PBA)\sin(∠PCB)\ \ \ \ \ \ \ \ }{\ \sin(∠PAB)\sin(∠PBC)\sin(∠PCA)\ \ \ \ \ \ \ \ }\equiv1\ \ \ \ \ (1)\)

  若∠PAC, ∠PAB, ∠PBA, ∠PBC, ∠PCB, ∠PCA 都是整数角度数,则(1)可以有 123 个算式。

  ∠PAC < ∠PAB,   ∠PAC < ∠PBA < ∠PCB,   ∠PAB < ∠PBC < ∠PCA

∠PAC, ∠PBA, ∠PCB, ∠PAB, ∠PBC, ∠PCA
001:01, 28, 091, 02, 28, 30,
002:01, 29, 061, 01, 30, 58,
003:01, 29, 089, 02, 29, 30,
004:01, 30, 087, 02, 29, 31,
005:01, 56, 061, 03, 28, 31,
006:01, 58, 059, 03, 29, 30,
007:02, 14, 104, 02, 28, 30,
008:02, 26, 092, 04, 26, 30,
009:02, 28, 062, 02, 30, 56,
010:02, 28, 088, 04, 28, 30,
011:02, 30, 084, 04, 28, 32,
012:02, 34, 094, 10, 16, 24,
013:02, 42, 044, 04, 14, 74,
014:02, 46, 062, 08, 12, 50,
015:02, 52, 062, 06, 26, 32,
016:02, 56, 058, 06, 28, 30,
017:03, 18, 093, 06, 12, 48,
018:03, 24, 081, 06, 15, 51,
019:03, 24, 093, 06, 24, 30,
020:03, 27, 063, 03, 30, 54,
021:03, 27, 087, 06, 27, 30,
022:03, 30, 075, 09, 12, 51,
023:03, 30, 081, 06, 27, 33,
024:03, 30, 093, 12, 18, 24,
025:03, 39, 066, 09, 15, 48,
026:03, 39, 075, 09, 24, 30,
027:03, 39, 081, 15, 18, 24,
028:03, 42, 075, 12, 21, 27,
029:03, 48, 054, 09, 15, 51,
030:03, 48, 063, 09, 24, 33,
031:03, 48, 069, 15, 18, 27,
032:03, 54, 057, 09, 27, 30,
033:03, 54, 063, 15, 21, 24,
034:04, 13, 103, 04, 26, 30,
035:04, 22, 094, 08, 22, 30,
036:04, 26, 064, 04, 30, 52,
037:04, 26, 086, 08, 26, 30,
038:04, 30, 078, 08, 26, 34,
039:04, 38, 064, 10, 18, 46,
040:04, 39, 043, 08, 13, 73,
041:04, 44, 064, 12, 22, 34,
042:04, 52, 056, 12, 26, 30,
043:05, 20, 095, 10, 20, 30,
044:05, 25, 065, 05, 30, 50,
045:05, 25, 085, 10, 25, 30,
046:05, 30, 075, 10, 25, 35,
047:05, 40, 065, 15, 20, 35,
048:05, 50, 055, 15, 25, 30,
049:06, 06, 126, 06, 12, 24,
050:06, 09, 099, 06, 12, 48,
051:06, 12, 096, 06, 18, 42,
052:06, 12, 102, 06, 24, 30,
053:06, 12, 114, 12, 18, 18,
054:06, 15, 105, 12, 18, 24,
055:06, 18, 078, 06, 24, 48,
056:06, 18, 084, 12, 12, 48,
057:06, 18, 096, 12, 18, 30,
058:06, 21, 057, 09, 12, 75,
059:06, 24, 066, 06, 30, 48,
060:06, 24, 084, 12, 24, 30,
061:06, 30, 066, 12, 18, 48,
062:06, 30, 072, 12, 24, 36,
063:06, 30, 078, 18, 24, 24,
064:06, 36, 042, 12, 12, 72,
065:06, 36, 066, 18, 18, 36,
066:06, 42, 048, 12, 18, 54,
067:06, 42, 054, 12, 24, 42,
068:06, 48, 054, 18, 24, 30,
069:07, 16, 097, 14, 16, 30,
070:07, 23, 067, 07, 30, 46,
071:07, 23, 083, 14, 23, 30,
072:07, 30, 069, 14, 23, 37,
073:07, 32, 067, 16, 21, 37,
074:07, 46, 053, 21, 23, 30,
075:08, 11, 101, 08, 22, 30,
076:08, 14, 098, 14, 16, 30,
077:08, 22, 068, 08, 30, 44,
078:08, 22, 082, 16, 22, 30,
079:08, 28, 068, 14, 24, 38,
080:08, 30, 066, 16, 22, 38,
081:08, 33, 041, 11, 16, 71,
082:08, 44, 052, 22, 24, 30,
083:09, 12, 099, 12, 18, 30,
084:09, 15, 075, 09, 18, 54,
085:09, 15, 081, 12, 15, 48,
086:09, 15, 087, 12, 18, 39,
087:09, 18, 069, 12, 15, 57,
088:09, 21, 069, 09, 30, 42,
089:09, 21, 081, 18, 21, 30,
090:09, 24, 069, 12, 27, 39,
091:09, 30, 063, 18, 21, 39,
092:09, 33, 054, 15, 21, 48,
093:09, 42, 051, 21, 27, 30,
094:10, 10, 100, 10, 20, 30,
095:10, 20, 070, 10, 30, 40,
096:10, 20, 080, 20, 20, 30,
097:10, 26, 070, 22, 24, 28,
098:10, 30, 060, 20, 20, 40,
099:10, 40, 050, 20, 30, 30,
100:11, 19, 071, 11, 30, 38,
101:11, 19, 079, 19, 22, 30,
102:11, 30, 057, 19, 22, 41,
103:11, 38, 049, 19, 30, 33,
104:12, 12, 084, 12, 18, 42,
105:12, 18, 072, 12, 30, 36,
106:12, 18, 078, 18, 24, 30,
107:12, 24, 066, 18, 30, 30,
108:12, 30, 048, 18, 18, 54,
109:12, 30, 054, 18, 24, 42,
110:12, 33, 039, 15, 18, 63,
111:12, 36, 048, 18, 30, 36,
112:13, 17, 073, 13, 30, 34,
113:13, 17, 077, 17, 26, 30,
114:13, 30, 051, 17, 26, 43,
115:13, 34, 047, 17, 30, 39,
116:14, 16, 074, 14, 30, 32,
117:14, 16, 076, 16, 28, 30,
118:14, 30, 048, 16, 28, 44,
119:14, 32, 046, 16, 30, 42,
120:15, 15, 075, 15, 30, 30,
121:15, 24, 057, 18, 27, 39,
122:15, 30, 051, 24, 27, 33,
123:18, 24, 054, 24, 30, 30,
毋因群疑而阻独见  毋任己意而废人言
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 楼主| 发表于 2021-5-7 08:26:19 | 显示全部楼层
wayne 发表于 2021-5-6 16:22
我穷举了下C(89,4)=2441626种情况,发现只有51个解,不包括$x=y$:

小结。
题目:正整数 a < x ≤ y < b ≤ 90 ,满足\(\D\frac{\sin(a^\circ)\sin(b^\circ)}{\sin(x^\circ)\sin(y^\circ)}=1\) 的不同算式有几个?
感谢 mathematica!感谢 mathe!感谢 sheng_jianguo!感谢 wayne!
可以有 56 个算式。
        a,   x,   y,   b,
01:01, 02, 30, 89,
02:02, 04, 30, 88,
03:03, 06, 30, 87,
04:04, 08, 30, 86,
05:05, 10, 30, 85,
06:06, 12, 30, 84,
07:07, 14, 30, 83,
08:08, 16, 30, 82,
09:09, 18, 30, 81,
10:10, 20, 30, 80,
11:11, 22, 30, 79,
12:12, 24, 30, 78,
13:13, 26, 30, 77,
14:14, 28, 30, 76,
15:15, 30, 30, 75,
16:16, 30, 32, 74,
17:17, 30, 34, 73,
18:18, 30, 36, 72,
19:19, 30, 38, 71,
20:20, 30, 40, 70,
21:21, 30, 42, 69,
22:22, 30, 44, 68,
23:23, 30, 46, 67,
24:24, 30, 48, 66,
25:25, 30, 50, 65,
26:26, 30, 52, 64,
27:27, 30, 54, 63,
28:28, 30, 56, 62,
29:29, 30, 58, 61,
   :30, 30, 60, 60,
30:30, 31, 59, 62,
31:30, 32, 58, 64,
32:30, 33, 57, 66,
33:30, 34, 56, 68,
34:30, 35, 55, 70,
35:30, 36, 54, 72,
36:30, 37, 53, 74,
37:30, 38, 52, 76,
38:30, 39, 51, 78,
39:30, 40, 50, 80,
40:30, 41, 49, 82,
41:30, 42, 48, 84,
42:30, 43, 47, 86,
43:30, 44, 46, 88,
44:30, 45, 45, 90,

45:06, 12, 24, 54,
46:06, 18, 18, 66,
47:09, 12, 48, 81,
48:12, 18, 30, 48,
49:12, 18, 42, 84,
50:15, 18, 54, 75,
   :15, 30, 30, 75,
51:18, 24, 48, 78,
52:18, 30, 30, 54,
53:24, 27, 63, 84,
54:24, 30, 54, 84,
   :30, 45, 45, 90,
55:42, 54, 54, 78,
56:48, 54, 66, 84,

前面44个有规律:\(\D\frac{\sin(x)\sin(\pi/2-x)}{\sin(2x)\sin(\pi/2)}=1\)
后面的好像没有规律?

  如何利用这 56 个算式,譬如:从这 56 个算式到 8 楼,
01:01, 02, 30, 89,\(\Rightarrow\)001:01, 28, 091, 02, 28, 30,
29:29, 30, 58, 61,\(\Rightarrow\)002:01, 29, 061, 01, 30, 58,
01:01, 02, 30, 89,\(\Rightarrow\)003:01, 29, 089, 02, 29, 30,
52:18, 30, 30, 54,\(\Rightarrow\)123:18, 24, 054, 24, 30, 30,
  如何利用这 56 个算式到 4 楼,...
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2021-5-7 14:52:35 | 显示全部楼层
如果我们查看R5:3R3, 为了保持单位向量的共轭性,3个R3中一个必须抵消R5中的1(自共轭),另外两个只能抵消R5中两个共轭的单位根。
如果$exp(\pm\frac{2\pi i}5)$被抵消,那么需要留下$exp(\pm\frac{4\pi i}5)$,并且产生$-exp(\pm\frac{2\pi i}5\pm\frac{2\pi i}3)$,而抵消1又产生$-exp(\pm\frac{2\pi i}3)$
所以得到8个共轭单位根$exp(\pm\frac{4\pi i}5), -exp(\pm\frac{2\pi i}5\pm\frac{2\pi i}3),-exp(\pm\frac{2\pi i}3) $,这个应该可以得出一组特殊解。另外类似抵消$exp(\pm\frac{4\pi i}5)$而留下$exp(\pm\frac{2\pi i}5)$也会产生一组可能的候选解

而对于(R5:R3)+R2,为了保持共轭性,其中R2只能代表$\pm i$, (R5:R3)同样只能抵消实数1,所以只能最多得到一组解。
而R3+R5对应使用两个1的情况(或者旋转180度对应两个-1?)也就是少数解。

但是R3+R3+R2中R2只能$\pm i$,但是两个R3可以相互共轭,而它们各自的起始角度就可以任意选择,所以如果可以对应无限组有理角度解(当然如果限定整数角度,会有限组)。
最后R2+R2+R2+R2也可以相互共轭,有无数组选择
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2021-5-5 07:59:41 | 显示全部楼层
穷举法!90^4=65610000,不超过这么多种类型!
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2021-5-5 08:31:38 | 显示全部楼层
的确计算很简单。结果比较有意思,大部分结果可以通过$\sin(2x)=2\sin(x)\cos(x)$得出
于是$\frac{\sin(x)\sin(\frac{\pi}2-x)}{\sin(2x)\sin(\frac{pi}6)}=1$

点评

正整数,穷举  发表于 2021-5-5 12:39
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2021-5-5 12:07:12 | 显示全部楼层

可以有多少个算式?

    P 是四边形 ABCD 内的点,我们恒有

\(\ \ \ \ \ \ \ \D\frac{\ \sin(∠PAD)\sin(∠PBA)\sin(∠PCB)\sin(∠PDC)\ \ \ \ \ \ \ \ }{\ \sin(∠PAB)\sin(∠PBC)\sin(∠PCD)\sin(∠PDA)\ \ \ \ \ \ \ \ }\equiv1\ \ \ \ \ (1)\)

  若∠PAD, ∠PAB, ∠PBA, ∠PBC, ∠PCB, ∠PCD, ∠PDC, ∠PDA 都是整数角度数,则(1)可以有多少个算式?

  ∠PAD < ∠PAB,   ∠PAD < ∠PBA < ∠PCB < ∠PDC,   ∠PAB < ∠PBC < ∠PCD < ∠PDA

■ 刨根究底范畴:对某处数学概念或推导过程,虽久经琢磨,仍存疑或参悟不透,方可请高人指点,以期拨云见日。


补充内容 (2021-5-6 10:43):
可以合并。谢谢!

点评

两帖子标题一样,想合并么?  发表于 2021-5-6 10:25
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2021-5-5 14:40:14 | 显示全部楼层
本帖最后由 王守恩 于 2021-5-5 14:52 编辑
mathe 发表于 2021-5-5 08:31
的确计算很简单。结果比较有意思,大部分结果可以通过$\sin(2x)=2\sin(x)\cos(x)$得出
于是$\frac{\sin(x) ...


可以有 34 个算式?还有吗?

01:\(\frac{\sin(01^\circ)\sin(89^\circ)}{\sin(02^\circ)\sin(30^\circ)}=1\)
02:\(\frac{\sin(02^\circ)\sin(88^\circ)}{\sin(04^\circ)\sin(30^\circ)}=1\)
03:\(\frac{\sin(03^\circ)\sin(87^\circ)}{\sin(06^\circ)\sin(30^\circ)}=1\)
04:\(\frac{\sin(04^\circ)\sin(86^\circ)}{\sin(08^\circ)\sin(30^\circ)}=1\)
05:\(\frac{\sin(05^\circ)\sin(85^\circ)}{\sin(10^\circ)\sin(30^\circ)}=1\)
06:\(\frac{\sin(06^\circ)\sin(54^\circ)}{\sin(12^\circ)\sin(24^\circ)}=1\)
07:\(\frac{\sin(06^\circ)\sin(84^\circ)}{\sin(12^\circ)\sin(30^\circ)}=1\)
08:\(\frac{\sin(07^\circ)\sin(83^\circ)}{\sin(14^\circ)\sin(30^\circ)}=1\)
09:\(\frac{\sin(08^\circ)\sin(82^\circ)}{\sin(16^\circ)\sin(30^\circ)}=1\)
10:\(\frac{\sin(09^\circ)\sin(81^\circ)}{\sin(12^\circ)\sin(48^\circ)}=1\)
11:\(\frac{\sin(09^\circ)\sin(81^\circ)}{\sin(18^\circ)\sin(30^\circ)}=1\)
12:\(\frac{\sin(10^\circ)\sin(80^\circ)}{\sin(20^\circ)\sin(30^\circ)}=1\)
13:\(\frac{\sin(11^\circ)\sin(79^\circ)}{\sin(22^\circ)\sin(30^\circ)}=1\)
14:\(\frac{\sin(12^\circ)\sin(84^\circ)}{\sin(18^\circ)\sin(42^\circ)}=1\)
15:\(\frac{\sin(12^\circ)\sin(78^\circ)}{\sin(24^\circ)\sin(30^\circ)}=1\)
16:\(\frac{\sin(13^\circ)\sin(77^\circ)}{\sin(26^\circ)\sin(30^\circ)}=1\)
17:\(\frac{\sin(14^\circ)\sin(76^\circ)}{\sin(28^\circ)\sin(30^\circ)}=1\)
18:\(\frac{\sin(15^\circ)\sin(75^\circ)}{\sin(18^\circ)\sin(54^\circ)}=1\)
19:\(\frac{\sin(15^\circ)\sin(75^\circ)}{\sin(30^\circ)\sin(30^\circ)}=1\)
20:\(\frac{\sin(16^\circ)\sin(74^\circ)}{\sin(30^\circ)\sin(32^\circ)}=1\)
21:\(\frac{\sin(17^\circ)\sin(73^\circ)}{\sin(30^\circ)\sin(34^\circ)}=1\)
22:\(\frac{\sin(18^\circ)\sin(78^\circ)}{\sin(24^\circ)\sin(48^\circ)}=1\)
23:\(\frac{\sin(18^\circ)\sin(72^\circ)}{\sin(30^\circ)\sin(36^\circ)}=1\)
24:\(\frac{\sin(19^\circ)\sin(71^\circ)}{\sin(30^\circ)\sin(38^\circ)}=1\)
25:\(\frac{\sin(20^\circ)\sin(70^\circ)}{\sin(30^\circ)\sin(40^\circ)}=1\)
26:\(\frac{\sin(21^\circ)\sin(69^\circ)}{\sin(30^\circ)\sin(42^\circ)}=1\)
27:\(\frac{\sin(22^\circ)\sin(68^\circ)}{\sin(30^\circ)\sin(44^\circ)}=1\)
28:\(\frac{\sin(23^\circ)\sin(67^\circ)}{\sin(30^\circ)\sin(46^\circ)}=1\)
29:\(\frac{\sin(24^\circ)\sin(66^\circ)}{\sin(30^\circ)\sin(48^\circ)}=1\)
30:\(\frac{\sin(25^\circ)\sin(65^\circ)}{\sin(30^\circ)\sin(50^\circ)}=1\)
31:\(\frac{\sin(26^\circ)\sin(64^\circ)}{\sin(30^\circ)\sin(52^\circ)}=1\)
32:\(\frac{\sin(27^\circ)\sin(63^\circ)}{\sin(30^\circ)\sin(54^\circ)}=1\)
33:\(\frac{\sin(28^\circ)\sin(62^\circ)}{\sin(30^\circ)\sin(56^\circ)}=1\)
34:\(\frac{\sin(29^\circ)\sin(61^\circ)}{\sin(30^\circ)\sin(58^\circ)}=1\)


毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2021-5-5 14:52:38 | 显示全部楼层
也可以大于30,比如
sin(31°)sin(59°)=sin(62°)sin(30°)
于是可以选择a=30,b=62,x=31,y=59.
此外还有
6 12 24 54
6 18 18 66
9 12 48 81
12 18 30 48
12 18 42 84
15 18 54 75
18 30 30 54
18 24 48 78
24 27 63 84
24 30 54 84
48 54 66 84
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2021-5-5 16:13:43 | 显示全部楼层
本帖最后由 王守恩 于 2021-5-5 16:36 编辑
mathe 发表于 2021-5-5 14:52
也可以大于30,比如
sin(31°)sin(59°)=sin(62°)sin(30°)
于是可以选择a=30,b=62,x=31,y=59.


可以有 55 个算式?还有吗?

01:01, 02, 30, 89,
02:02, 04, 30, 88,
03:03, 06, 30, 87,
04:04, 08, 30, 86,
05:05, 10, 30, 85,
06:06, 12, 24, 54,
07:06, 12, 30, 84,
08:06, 18, 18, 66,
09:07, 14, 30, 83,
10:08, 16, 30, 82,
11:09, 12, 48, 81,
12:09, 18, 30, 81,
13:10, 20, 30, 80,
14:11, 22, 30, 79,
15:12, 18, 30, 48,
16:12, 18, 42, 84,
17:12, 24, 30, 78,
18:13, 26, 30, 77,
19:14, 28, 30, 76,
20:15, 18, 54, 75,
21:15, 30, 30, 75,
22:16, 30, 32, 74,
23:17, 30, 34, 73,
24:18, 24, 48, 78,
25:18, 30, 30, 54,
26:18, 30, 36, 72,
27:19, 30, 38, 71,
28:20, 30, 40, 70,
29:21, 30, 42, 69,
30:22, 30, 44, 68,
31:23, 30, 46, 67,
32:24, 27, 63, 84,
33:24, 30, 48, 66,
34:24, 30, 54, 84,
35:25, 30, 50, 65,
36:26, 30, 52, 64,
37:27, 30, 54, 63,
38:28, 30, 56, 62,
39:29, 30, 58, 61,
40:30, 31, 59, 62,
41:30, 32, 58, 64,
42:30, 33, 57, 66,
43:30, 34, 56, 68,
44:30, 35, 55, 70,
45:30, 36, 54, 72,
46:30, 37, 53, 74,
47:30, 38, 52, 76,
48:30, 39, 51, 78,
49:30, 40, 50, 80,
50:30, 41, 49, 82,
51:30, 42, 48, 84,
52:30, 43, 47, 86,
53:30, 44, 46, 88,
54:30, 45, 45, 90,
55:48, 54, 66, 84,

毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2021-5-6 14:25:08 | 显示全部楼层
王守恩 发表于 2021-5-5 16:13
可以有 55 个算式?还有吗?

01:01, 02, 30, 89,

你的55个算式中,有多个不满足题目要求(如第54个等等):四个变量中没有两个相等,且小于90°。
我的计算结果为56个:
    1:   1,  2, 30, 89,
    2:   2,  4, 30, 88,
    3:   3,  6, 30, 87,
    4:   4,  8, 30, 86,
    5:   5, 10, 30, 85,
    6:   6, 12, 24, 54,
    7:   6, 12, 30, 84,
    8:   7,  9, 51, 86,
    9:   7, 14, 30, 83,
   10:   8, 16, 30, 82,
   11:   9, 12, 48, 81,
   12:   9, 18, 30, 81,
   13:  10, 20, 30, 80,
   14:  11, 22, 30, 79,
   15:  12, 18, 30, 48,
   16:  12, 18, 42, 84,
   17:  12, 24, 30, 78,
   18:  13, 26, 30, 77,
   19:  14, 28, 30, 76,
   20:  15, 18, 54, 75,
   21:  16, 30, 32, 74,
   22:  17, 30, 34, 73,
   23:  18, 24, 48, 78,
   24:  18, 30, 36, 72,
   25:  19, 30, 38, 71,
   26:  20, 30, 40, 70,
   27:  21, 30, 42, 69,
   28:  22, 30, 44, 68,
   29:  23, 30, 46, 67,
   30:  24, 27, 63, 84,
   31:  24, 30, 48, 66,
   32:  24, 30, 54, 84,
   33:  25, 30, 50, 65,
   34:  26, 30, 52, 64,
   35:  27, 30, 54, 63,
   36:  28, 30, 56, 62,
   37:  29, 30, 58, 61,
   38:  30, 31, 59, 62,
   39:  30, 32, 58, 64,
   40:  30, 33, 57, 66,
   41:  30, 34, 56, 68,
   42:  30, 35, 55, 70,
   43:  30, 36, 54, 72,
   44:  30, 37, 53, 74,
   45:  30, 38, 52, 76,
   46:  30, 39, 51, 78,
   47:  30, 40, 50, 80,
   48:  30, 41, 49, 82,
   49:  30, 42, 48, 84,
   50:  30, 43, 47, 86,
   51:  30, 44, 46, 88,
   52:  32, 38, 57, 77,
   53:  38, 39, 78, 89,
   54:  48, 54, 66, 84,
   55:  56, 58, 76, 83,
   56:  60, 66, 69, 80,
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2021-5-6 16:04:14 | 显示全部楼层
本帖最后由 王守恩 于 2021-5-6 16:09 编辑
sheng_jianguo 发表于 2021-5-6 14:25
你的55个算式中,有多个不满足题目要求(如第54个等等):四个变量中没有两个相等,且小于90°。
我的计 ...


谢谢 sheng_jianguo 分享。

你的55个算式中,有多个不满足题目要求(如第54个等等):四个变量中没有两个相等,且小于90°。
1,我的计算结果为56个:56(7#)=55(6#)+ 5(多了5个) - 4(少了4个)
多了5个   
     8:   7,  9, 51, 86,
   52:  32, 38, 57, 77,
   53:  38, 39, 78, 89,
   55:  56, 58, 76, 83,
   56:  60, 66, 69, 80,
少了4个
08:06, 18, 18, 66,
21:15, 30, 30, 75,
25:18, 30, 30, 54,
54:30, 45, 45, 90,

2,这4个还是挺不容易找的,丢了挺可惜,
是否题意改一下:若正整数 a < x ≤ y < b ≤ 90

3,算式应该还有(心里没底),譬如:60 就没有算式吗?!
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
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