# Thread: A minimum problem for the year 2010

1. ## A minimum problem for the year 2010

Let $a,b,c \in \mathbb{N}^+, \quad a+b+c = 2010$
Among all possible $a,b,c$, find the smallest $m$ such that $d = \frac{a!b!c!}{10^m} \in \mathbb{N}, \quad 10 \not| d$

2. the number of trailing zeros of n! is totally determined by the multiplicity of factor 5 in n!, which has a formula $\sum_{i=1}^\infty\lfloor\frac{n}{5^i}\rfloor$. Note that even written in an infinite sum form, the sum is actually finite since n is finite. Start from here.

3. Originally Posted by elim
Let $a,b,c \in \mathbb{N}^+, \quad a+b+c = 2010$
Among all possible $a,b,c$, find the smallest $m$ such that $d = \frac{a!b!c!}{10^m} \in \mathbb{N}, \quad 10 \not| d$
Procrastinating in the computer lab I came up with a brute force solution, I made a small Matlab script to solve this problem. Upon writing this script I got some insight into deriving the solution, but frankly I don't have the time right now. I will say however it does deal with minimizing $\lfloor\log_5(a)\rfloor+\lfloor\log_5(b)\rfloor+\l floor\log_5(c)\rfloor$ where $i,j,l$ are the largest possible integers to make each quotient greater than or equal to one.

Here's the code if you want to see it:
Spoiler:

Code:
function [a_s,b_s,c_s,m,t] = min_m(k)
c=clock; t_i=60*c(5)+c(6);
a_s=0; b_s=0; c_s=0; m=k;
for a=1:k-2
for b=1:k-a-1
c=k-a-b;
i=floor(log(a)/log(5));
j=floor(log(b)/log(5));
l=floor(log(c)/log(5));
i=i*(i+1)/2; j=j*(j+1)/2; l=l*(l+1)/2;
if i+j+l<m
m=i+j+l;
a_s=a;
b_s=b;
c_s=c;
end
end
end
c=clock; t_f=60*c(5)+c(6);
t=t_f-t_i;
I realize my code isn't the most efficient, but whatcha gonna do?

Here's the answer assuming my program works:
Spoiler:

$a=1,\;b=1,\;c=2008,\;m=10$ with a running time of $13.8357$ seconds

4. This cannot be true. 2008! get a lot more 0's than just 4

5. Originally Posted by elim
This cannot be true. 2008! get a lot more 0's than just 4
You're right, there's an error in my algorithm and I see what it is. I'll have a fix in a couple hours.

6. Code:
 def cc(n):
s=0
d=5
c=n/5
while c>0:
s+=c
d*=5
c=n/d
return s

def mm():
mm=3*cc(670)
ll=[670,670,670]
for a in range(1,671):
for b in range(a,(2010-a)/2+1):
c=2010-a-b
d=cc(a)+cc(b)+cc(c)
if mm>d:
mm=d
ll[0]=a
ll[1]=b
ll[2]=c
return mm,ll

mm()
(493, [137, 624, 1249])
when a = 137, b = 624, c = 1249, we get a smallest m = 493

7. There's a fundamental error in my program. I only take into account the powers of 5 less than 2010, not multiples of 5 less than 2010. My mistake! I believe your code is correct though.

8. When a,b,c are integer and 1 ≤ a ≤ b ≤ c，a + b + c = 1， there are 2712 ways to make m the smallest. where (a,b,c) = (624, 624, 762) makes a!b!c! the biggest among the other solutions.

2712.0/336675 = 0.0080552461572733353.. means the probability you randomly pick (a,b,c) that happen to be a solution.

Now the solution list:

[137, 624, 1249], [138, 623, 1249], [138, 624, 1248], [139, 622, 1249], [139, 623, 1248],
[139, 624, 1247], [142, 619, 1249], [142, 624, 1244], [143, 618, 1249], [143, 619, 1248],
[143, 623, 1244], [143, 624, 1243], [144, 617, 1249], [144, 618, 1248], [144, 619, 1247],
[144, 622, 1244], [144, 623, 1243], [144, 624, 1242], [147, 614, 1249], [147, 619, 1244],
[147, 624, 1239], [148, 613, 1249], [148, 614, 1248], [148, 618, 1244], [148, 619, 1243],
[148, 623, 1239], [148, 624, 1238], [149, 612, 1249], [149, 613, 1248], [149, 614, 1247],
[149, 617, 1244], [149, 618, 1243], [149, 619, 1242], [149, 622, 1239], [149, 623, 1238],
[149, 624, 1237], [162, 599, 1249], [162, 624, 1224], [163, 598, 1249], [163, 599, 1248],
[163, 623, 1224], [163, 624, 1223], [164, 597, 1249], [164, 598, 1248], [164, 599, 1247],
[164, 622, 1224], [164, 623, 1223], [164, 624, 1222], [167, 594, 1249], [167, 599, 1244],
[167, 619, 1224], [167, 624, 1219], [168, 593, 1249], [168, 594, 1248], [168, 598, 1244],
[168, 599, 1243], [168, 618, 1224], [168, 619, 1223], [168, 623, 1219], [168, 624, 1218],
[169, 592, 1249], [169, 593, 1248], [169, 594, 1247], [169, 597, 1244], [169, 598, 1243],
[169, 599, 1242], [169, 617, 1224], [169, 618, 1223], [169, 619, 1222], [169, 622, 1219],
[169, 623, 1218], [169, 624, 1217], [172, 589, 1249], [172, 594, 1244], [172, 599, 1239],
[172, 614, 1224], [172, 619, 1219], [172, 624, 1214], [173, 588, 1249], [173, 589, 1248],
[173, 593, 1244], [173, 594, 1243], [173, 598, 1239], [173, 599, 1238], [173, 613, 1224],
[173, 614, 1223], [173, 618, 1219], [173, 619, 1218], [173, 623, 1214], [173, 624, 1213],
[174, 587, 1249], [174, 588, 1248], [174, 589, 1247], [174, 592, 1244], [174, 593, 1243],
[174, 594, 1242], [174, 597, 1239], [174, 598, 1238], [174, 599, 1237], [174, 612, 1224],
[174, 613, 1223], [174, 614, 1222], [174, 617, 1219], [174, 618, 1218], [174, 619, 1217],
[174, 622, 1214], [174, 623, 1213], [174, 624, 1212], [187, 574, 1249], [187, 599, 1224],
[187, 624, 1199], [188, 573, 1249], [188, 574, 1248], [188, 598, 1224], [188, 599, 1223],
[188, 623, 1199], [188, 624, 1198], [189, 572, 1249], [189, 573, 1248], [189, 574, 1247],
[189, 597, 1224], [189, 598, 1223], [189, 599, 1222], [189, 622, 1199], [189, 623, 1198],
[189, 624, 1197], [192, 569, 1249], [192, 574, 1244], [192, 594, 1224], [192, 599, 1219],
[192, 619, 1199], [192, 624, 1194], [193, 568, 1249], [193, 569, 1248], [193, 573, 1244],
[193, 574, 1243], [193, 593, 1224], [193, 594, 1223], [193, 598, 1219], [193, 599, 1218],
[193, 618, 1199], [193, 619, 1198], [193, 623, 1194], [193, 624, 1193], [194, 567, 1249],
[194, 568, 1248], [194, 569, 1247], [194, 572, 1244], [194, 573, 1243], [194, 574, 1242],
[194, 592, 1224], [194, 593, 1223], [194, 594, 1222], [194, 597, 1219], [194, 598, 1218],
[194, 599, 1217], [194, 617, 1199], [194, 618, 1198], [194, 619, 1197], [194, 622, 1194],
[194, 623, 1193], [194, 624, 1192], [197, 564, 1249], [197, 569, 1244], [197, 574, 1239],
[197, 589, 1224], [197, 594, 1219], [197, 599, 1214], [197, 614, 1199], [197, 619, 1194],
[197, 624, 1189], [198, 563, 1249], [198, 564, 1248], [198, 568, 1244], [198, 569, 1243],
[198, 573, 1239], [198, 574, 1238], [198, 588, 1224], [198, 589, 1223], [198, 593, 1219],
[198, 594, 1218], [198, 598, 1214], [198, 599, 1213], [198, 613, 1199], [198, 614, 1198],
[198, 618, 1194], [198, 619, 1193], [198, 623, 1189], [198, 624, 1188], [199, 562, 1249],
[199, 563, 1248], [199, 564, 1247], [199, 567, 1244], [199, 568, 1243], [199, 569, 1242],
[199, 572, 1239], [199, 573, 1238], [199, 574, 1237], [199, 587, 1224], [199, 588, 1223],
[199, 589, 1222], [199, 592, 1219], [199, 593, 1218], [199, 594, 1217], [199, 597, 1214],
[199, 598, 1213], [199, 599, 1212], [199, 612, 1199], [199, 613, 1198], [199, 614, 1197],
[199, 617, 1194], [199, 618, 1193], [199, 619, 1192], [199, 622, 1189], [199, 623, 1188],
[199, 624, 1187], [212, 549, 1249], [212, 574, 1224], [212, 599, 1199], [212, 624, 1174],
[213, 548, 1249], [213, 549, 1248], [213, 573, 1224], [213, 574, 1223], [213, 598, 1199],
[213, 599, 1198], [213, 623, 1174], [213, 624, 1173], [214, 547, 1249], [214, 548, 1248],
[214, 549, 1247], [214, 572, 1224], [214, 573, 1223], [214, 574, 1222], [214, 597, 1199],
[214, 598, 1198], [214, 599, 1197], [214, 622, 1174], [214, 623, 1173], [214, 624, 1172],
[217, 544, 1249], [217, 549, 1244], [217, 569, 1224], [217, 574, 1219], [217, 594, 1199],
[217, 599, 1194], [217, 619, 1174], [217, 624, 1169], [218, 543, 1249], [218, 544, 1248],
[218, 548, 1244], [218, 549, 1243], [218, 568, 1224], [218, 569, 1223], [218, 573, 1219],
[218, 574, 1218], [218, 593, 1199], [218, 594, 1198], [218, 598, 1194], [218, 599, 1193],
[218, 618, 1174], [218, 619, 1173], [218, 623, 1169], [218, 624, 1168], [219, 542, 1249],
[219, 543, 1248], [219, 544, 1247], [219, 547, 1244], [219, 548, 1243], [219, 549, 1242],
[219, 567, 1224], [219, 568, 1223], [219, 569, 1222], [219, 572, 1219], [219, 573, 1218],
[219, 574, 1217], [219, 592, 1199], [219, 593, 1198], [219, 594, 1197], [219, 597, 1194],
[219, 598, 1193], [219, 599, 1192], [219, 617, 1174], [219, 618, 1173], [219, 619, 1172],
[219, 622, 1169], [219, 623, 1168], [219, 624, 1167], [222, 539, 1249], [222, 544, 1244],
[222, 549, 1239], [222, 564, 1224], [222, 569, 1219], [222, 574, 1214], [222, 589, 1199],
[222, 594, 1194], [222, 599, 1189], [222, 614, 1174], [222, 619, 1169], [222, 624, 1164],
[223, 538, 1249], [223, 539, 1248], [223, 543, 1244], [223, 544, 1243], [223, 548, 1239],
[223, 549, 1238], [223, 563, 1224], [223, 564, 1223], [223, 568, 1219], [223, 569, 1218],
[223, 573, 1214], [223, 574, 1213], [223, 588, 1199], [223, 589, 1198], [223, 593, 1194],
[223, 594, 1193], [223, 598, 1189], [223, 599, 1188], [223, 613, 1174], [223, 614, 1173],
[223, 618, 1169], [223, 619, 1168], [223, 623, 1164], [223, 624, 1163], [224, 537, 1249],
[224, 538, 1248], [224, 539, 1247], [224, 542, 1244], [224, 543, 1243], [224, 544, 1242],
[224, 547, 1239], [224, 548, 1238], [224, 549, 1237], [224, 562, 1224], [224, 563, 1223],
[224, 564, 1222], [224, 567, 1219], [224, 568, 1218], [224, 569, 1217], [224, 572, 1214],
[224, 573, 1213], [224, 574, 1212], [224, 587, 1199], [224, 588, 1198], [224, 589, 1197],
[224, 592, 1194], [224, 593, 1193], [224, 594, 1192], [224, 597, 1189], [224, 598, 1188],
[224, 599, 1187], [224, 612, 1174], [224, 613, 1173], [224, 614, 1172], [224, 617, 1169],
[224, 618, 1168], [224, 619, 1167], [224, 622, 1164], [224, 623, 1163], [224, 624, 1162],
[237, 524, 1249], [237, 549, 1224], [237, 574, 1199], [237, 599, 1174], [237, 624, 1149],
[238, 523, 1249], [238, 524, 1248], [238, 548, 1224], [238, 549, 1223], [238, 573, 1199],
[238, 574, 1198], [238, 598, 1174], [238, 599, 1173], [238, 623, 1149], [238, 624, 1148],
[239, 522, 1249], [239, 523, 1248], [239, 524, 1247], [239, 547, 1224], [239, 548, 1223],
[239, 549, 1222], [239, 572, 1199], [239, 573, 1198], [239, 574, 1197], [239, 597, 1174],
[239, 598, 1173], [239, 599, 1172], [239, 622, 1149], [239, 623, 1148], [239, 624, 1147],
[242, 519, 1249], [242, 524, 1244], [242, 544, 1224], [242, 549, 1219], [242, 569, 1199],
[242, 574, 1194], [242, 594, 1174], [242, 599, 1169], [242, 619, 1149], [242, 624, 1144],
[243, 518, 1249], [243, 519, 1248], [243, 523, 1244], [243, 524, 1243], [243, 543, 1224],
[243, 544, 1223], [243, 548, 1219], [243, 549, 1218], [243, 568, 1199], [243, 569, 1198],
[243, 573, 1194], [243, 574, 1193], [243, 593, 1174], [243, 594, 1173], [243, 598, 1169],
[243, 599, 1168], [243, 618, 1149], [243, 619, 1148], [243, 623, 1144], [243, 624, 1143],
[244, 517, 1249], [244, 518, 1248], [244, 519, 1247], [244, 522, 1244], [244, 523, 1243],
[244, 524, 1242], [244, 542, 1224], [244, 543, 1223], [244, 544, 1222], [244, 547, 1219],
[244, 548, 1218], [244, 549, 1217], [244, 567, 1199], [244, 568, 1198], [244, 569, 1197],
[244, 572, 1194], [244, 573, 1193], [244, 574, 1192], [244, 592, 1174], [244, 593, 1173],
[244, 594, 1172], [244, 597, 1169], [244, 598, 1168], [244, 599, 1167], [244, 617, 1149],
[244, 618, 1148], [244, 619, 1147], [244, 622, 1144], [244, 623, 1143], [244, 624, 1142],
[247, 514, 1249], [247, 519, 1244], [247, 524, 1239], [247, 539, 1224], [247, 544, 1219],
[247, 549, 1214], [247, 564, 1199], [247, 569, 1194], [247, 574, 1189], [247, 589, 1174],
[247, 594, 1169], [247, 599, 1164], [247, 614, 1149], [247, 619, 1144], [247, 624, 1139],
[248, 513, 1249], [248, 514, 1248], [248, 518, 1244], [248, 519, 1243], [248, 523, 1239],
[248, 524, 1238], [248, 538, 1224], [248, 539, 1223], [248, 543, 1219], [248, 544, 1218],
[248, 548, 1214], [248, 549, 1213], [248, 563, 1199], [248, 564, 1198], [248, 568, 1194],
[248, 569, 1193], [248, 573, 1189], [248, 574, 1188], [248, 588, 1174], [248, 589, 1173],
[248, 593, 1169], [248, 594, 1168], [248, 598, 1164], [248, 599, 1163], [248, 613, 1149],
[248, 614, 1148], [248, 618, 1144], [248, 619, 1143], [248, 623, 1139], [248, 624, 1138],
[249, 512, 1249], [249, 513, 1248], [249, 514, 1247], [249, 517, 1244], [249, 518, 1243],
[249, 519, 1242], [249, 522, 1239], [249, 523, 1238], [249, 524, 1237], [249, 537, 1224],
[249, 538, 1223], [249, 539, 1222], [249, 542, 1219], [249, 543, 1218], [249, 544, 1217],
[249, 547, 1214], [249, 548, 1213], [249, 549, 1212], [249, 562, 1199], [249, 563, 1198],
[249, 564, 1197], [249, 567, 1194], [249, 568, 1193], [249, 569, 1192], [249, 572, 1189],
[249, 573, 1188], [249, 574, 1187], [249, 587, 1174], [249, 588, 1173], [249, 589, 1172],
[249, 592, 1169], [249, 593, 1168], [249, 594, 1167], [249, 597, 1164], [249, 598, 1163],
[249, 599, 1162], [249, 612, 1149], [249, 613, 1148], [249, 614, 1147], [249, 617, 1144],
[249, 618, 1143], [249, 619, 1142], [249, 622, 1139], [249, 623, 1138], [249, 624, 1137],
[262, 499, 1249], [262, 624, 1124], [263, 498, 1249], [263, 499, 1248], [263, 623, 1124],
[263, 624, 1123], [264, 497, 1249], [264, 498, 1248], [264, 499, 1247], [264, 622, 1124],
[264, 623, 1123], [264, 624, 1122], [267, 494, 1249], [267, 499, 1244], [267, 619, 1124],
[267, 624, 1119], [268, 493, 1249], [268, 494, 1248], [268, 498, 1244], [268, 499, 1243],
[268, 618, 1124], [268, 619, 1123], [268, 623, 1119], [268, 624, 1118], [269, 492, 1249],
[269, 493, 1248], [269, 494, 1247], [269, 497, 1244], [269, 498, 1243], [269, 499, 1242],
[269, 617, 1124], [269, 618, 1123], [269, 619, 1122], [269, 622, 1119], [269, 623, 1118],
[269, 624, 1117], [272, 489, 1249], [272, 494, 1244], [272, 499, 1239], [272, 614, 1124],
[272, 619, 1119], [272, 624, 1114], [273, 488, 1249], [273, 489, 1248], [273, 493, 1244],
[273, 494, 1243], [273, 498, 1239], [273, 499, 1238], [273, 613, 1124], [273, 614, 1123],
[273, 618, 1119], [273, 619, 1118], [273, 623, 1114], [273, 624, 1113], [274, 487, 1249],
[274, 488, 1248], [274, 489, 1247], [274, 492, 1244], [274, 493, 1243], [274, 494, 1242],
[274, 497, 1239], [274, 498, 1238], [274, 499, 1237], [274, 612, 1124], [274, 613, 1123],
[274, 614, 1122], [274, 617, 1119], [274, 618, 1118], [274, 619, 1117], [274, 622, 1114],
[274, 623, 1113], [274, 624, 1112], [287, 474, 1249], [287, 499, 1224], [287, 599, 1124],
[287, 624, 1099], [288, 473, 1249], [288, 474, 1248], [288, 498, 1224], [288, 499, 1223],
[288, 598, 1124], [288, 599, 1123], [288, 623, 1099], [288, 624, 1098], [289, 472, 1249],
[289, 473, 1248], [289, 474, 1247], [289, 497, 1224], [289, 498, 1223], [289, 499, 1222],
[289, 597, 1124], [289, 598, 1123], [289, 599, 1122], [289, 622, 1099], [289, 623, 1098],
[289, 624, 1097], [292, 469, 1249], [292, 474, 1244], [292, 494, 1224], [292, 499, 1219],
[292, 594, 1124], [292, 599, 1119], [292, 619, 1099], [292, 624, 1094], [293, 468, 1249],
[293, 469, 1248], [293, 473, 1244], [293, 474, 1243], [293, 493, 1224], [293, 494, 1223],
[293, 498, 1219], [293, 499, 1218], [293, 593, 1124], [293, 594, 1123], [293, 598, 1119],
[293, 599, 1118], [293, 618, 1099], [293, 619, 1098], [293, 623, 1094], [293, 624, 1093],
[294, 467, 1249], [294, 468, 1248], [294, 469, 1247], [294, 472, 1244], [294, 473, 1243],
[294, 474, 1242], [294, 492, 1224], [294, 493, 1223], [294, 494, 1222], [294, 497, 1219],
[294, 498, 1218], [294, 499, 1217], [294, 592, 1124], [294, 593, 1123], [294, 594, 1122],
[294, 597, 1119], [294, 598, 1118], [294, 599, 1117], [294, 617, 1099], [294, 618, 1098],
[294, 619, 1097], [294, 622, 1094], [294, 623, 1093], [294, 624, 1092], [297, 464, 1249],
[297, 469, 1244], [297, 474, 1239], [297, 489, 1224], [297, 494, 1219], [297, 499, 1214],
[297, 589, 1124], [297, 594, 1119], [297, 599, 1114], [297, 614, 1099], [297, 619, 1094],
[297, 624, 1089], [298, 463, 1249], [298, 464, 1248], [298, 468, 1244], [298, 469, 1243],
[298, 473, 1239], [298, 474, 1238], [298, 488, 1224], [298, 489, 1223], [298, 493, 1219],
[298, 494, 1218], [298, 498, 1214], [298, 499, 1213], [298, 588, 1124], [298, 589, 1123],
[298, 593, 1119], [298, 594, 1118], [298, 598, 1114], [298, 599, 1113], [298, 613, 1099],
[298, 614, 1098], [298, 618, 1094], [298, 619, 1093], [298, 623, 1089], [298, 624, 1088],
[299, 462, 1249], [299, 463, 1248], [299, 464, 1247], [299, 467, 1244], [299, 468, 1243],
[299, 469, 1242], [299, 472, 1239], [299, 473, 1238], [299, 474, 1237], [299, 487, 1224],
[299, 488, 1223], [299, 489, 1222], [299, 492, 1219], [299, 493, 1218], [299, 494, 1217],
[299, 497, 1214], [299, 498, 1213], [299, 499, 1212], [299, 587, 1124], [299, 588, 1123],
[299, 589, 1122], [299, 592, 1119], [299, 593, 1118], [299, 594, 1117], [299, 597, 1114],
[299, 598, 1113], [299, 599, 1112], [299, 612, 1099], [299, 613, 1098], [299, 614, 1097],
[299, 617, 1094], [299, 618, 1093], [299, 619, 1092], [299, 622, 1089], [299, 623, 1088],
[299, 624, 1087], [312, 449, 1249], [312, 474, 1224], [312, 499, 1199], [312, 574, 1124],
[312, 599, 1099], [312, 624, 1074], [313, 448, 1249], [313, 449, 1248], [313, 473, 1224],
[313, 474, 1223], [313, 498, 1199], [313, 499, 1198], [313, 573, 1124], [313, 574, 1123],
[313, 598, 1099], [313, 599, 1098], [313, 623, 1074], [313, 624, 1073], [314, 447, 1249],
[314, 448, 1248], [314, 449, 1247], [314, 472, 1224], [314, 473, 1223], [314, 474, 1222],
[314, 497, 1199], [314, 498, 1198], [314, 499, 1197], [314, 572, 1124], [314, 573, 1123],
[314, 574, 1122], [314, 597, 1099], [314, 598, 1098], [314, 599, 1097], [314, 622, 1074],
[314, 623, 1073], [314, 624, 1072], [317, 444, 1249], [317, 449, 1244], [317, 469, 1224],
[317, 474, 1219], [317, 494, 1199], [317, 499, 1194], [317, 569, 1124], [317, 574, 1119],
[317, 594, 1099], [317, 599, 1094], [317, 619, 1074], [317, 624, 1069], [318, 443, 1249],
[318, 444, 1248], [318, 448, 1244], [318, 449, 1243], [318, 468, 1224], [318, 469, 1223],
[318, 473, 1219], [318, 474, 1218], [318, 493, 1199], [318, 494, 1198], [318, 498, 1194],
[318, 499, 1193], [318, 568, 1124], [318, 569, 1123], [318, 573, 1119], [318, 574, 1118],
[318, 593, 1099], [318, 594, 1098], [318, 598, 1094], [318, 599, 1093], [318, 618, 1074],
[318, 619, 1073], [318, 623, 1069], [318, 624, 1068], [319, 442, 1249], [319, 443, 1248],
[319, 444, 1247], [319, 447, 1244], [319, 448, 1243], [319, 449, 1242], [319, 467, 1224],
[319, 468, 1223], [319, 469, 1222], [319, 472, 1219], [319, 473, 1218], [319, 474, 1217],
[319, 492, 1199], [319, 493, 1198], [319, 494, 1197], [319, 497, 1194], [319, 498, 1193],
[319, 499, 1192], [319, 567, 1124], [319, 568, 1123], [319, 569, 1122], [319, 572, 1119],
[319, 573, 1118], [319, 574, 1117], [319, 592, 1099], [319, 593, 1098], [319, 594, 1097],
[319, 597, 1094], [319, 598, 1093], [319, 599, 1092], [319, 617, 1074], [319, 618, 1073],
[319, 619, 1072], [319, 622, 1069], [319, 623, 1068], [319, 624, 1067], [322, 439, 1249],
[322, 444, 1244], [322, 449, 1239], [322, 464, 1224], [322, 469, 1219], [322, 474, 1214],
[322, 489, 1199], [322, 494, 1194], [322, 499, 1189], [322, 564, 1124], [322, 569, 1119],
[322, 574, 1114], [322, 589, 1099], [322, 594, 1094], [322, 599, 1089], [322, 614, 1074],
[322, 619, 1069], [322, 624, 1064], [323, 438, 1249], [323, 439, 1248], [323, 443, 1244],
[323, 444, 1243], [323, 448, 1239], [323, 449, 1238], [323, 463, 1224], [323, 464, 1223],
[323, 468, 1219], [323, 469, 1218], [323, 473, 1214], [323, 474, 1213], [323, 488, 1199],
[323, 489, 1198], [323, 493, 1194], [323, 494, 1193], [323, 498, 1189], [323, 499, 1188],
[323, 563, 1124], [323, 564, 1123], [323, 568, 1119], [323, 569, 1118], [323, 573, 1114],
[323, 574, 1113], [323, 588, 1099], [323, 589, 1098], [323, 593, 1094], [323, 594, 1093],
[323, 598, 1089], [323, 599, 1088], [323, 613, 1074], [323, 614, 1073], [323, 618, 1069],
[323, 619, 1068], [323, 623, 1064], [323, 624, 1063], [324, 437, 1249], [324, 438, 1248],
[324, 439, 1247], [324, 442, 1244], [324, 443, 1243], [324, 444, 1242], [324, 447, 1239],
[324, 448, 1238], [324, 449, 1237], [324, 462, 1224], [324, 463, 1223], [324, 464, 1222],
[324, 467, 1219], [324, 468, 1218], [324, 469, 1217], [324, 472, 1214], [324, 473, 1213],
[324, 474, 1212], [324, 487, 1199], [324, 488, 1198], [324, 489, 1197], [324, 492, 1194],
[324, 493, 1193], [324, 494, 1192], [324, 497, 1189], [324, 498, 1188], [324, 499, 1187],
[324, 562, 1124], [324, 563, 1123], [324, 564, 1122], [324, 567, 1119], [324, 568, 1118],
[324, 569, 1117], [324, 572, 1114], [324, 573, 1113], [324, 574, 1112], [324, 587, 1099],
[324, 588, 1098], [324, 589, 1097], [324, 592, 1094], [324, 593, 1093], [324, 594, 1092],
[324, 597, 1089], [324, 598, 1088], [324, 599, 1087], [324, 612, 1074], [324, 613, 1073],
[324, 614, 1072], [324, 617, 1069], [324, 618, 1068], [324, 619, 1067], [324, 622, 1064],
[324, 623, 1063], [324, 624, 1062], [337, 424, 1249], [337, 449, 1224], [337, 474, 1199],
[337, 499, 1174], [337, 549, 1124], [337, 574, 1099], [337, 599, 1074], [337, 624, 1049],
[338, 423, 1249], [338, 424, 1248], [338, 448, 1224], [338, 449, 1223], [338, 473, 1199],
[338, 474, 1198], [338, 498, 1174], [338, 499, 1173], [338, 548, 1124], [338, 549, 1123],
[338, 573, 1099], [338, 574, 1098], [338, 598, 1074], [338, 599, 1073], [338, 623, 1049],
[338, 624, 1048], [339, 422, 1249], [339, 423, 1248], [339, 424, 1247], [339, 447, 1224],
[339, 448, 1223], [339, 449, 1222], [339, 472, 1199], [339, 473, 1198], [339, 474, 1197],
[339, 497, 1174], [339, 498, 1173], [339, 499, 1172], [339, 547, 1124], [339, 548, 1123],
.......

[574, 617, 819], [574, 618, 818], [574, 619, 817], [574, 622, 814], [574, 623, 813],
[574, 624, 812], [587, 599, 824], [587, 624, 799], [588, 598, 824], [588, 599, 823],
[588, 623, 799], [588, 624, 798], [589, 597, 824], [589, 598, 823], [589, 599, 822],
[589, 622, 799], [589, 623, 798], [589, 624, 797], [592, 594, 824], [592, 599, 819],
[592, 619, 799], [592, 624, 794], [593, 593, 824], [593, 594, 823], [593, 598, 819],
[593, 599, 818], [593, 618, 799], [593, 619, 798], [593, 623, 794], [593, 624, 793],
[594, 594, 822], [594, 597, 819], [594, 598, 818], [594, 599, 817], [594, 617, 799],
[594, 618, 798], [594, 619, 797], [594, 622, 794], [594, 623, 793], [594, 624, 792],
[597, 599, 814], [597, 614, 799], [597, 619, 794], [597, 624, 789], [598, 598, 814],
[598, 599, 813], [598, 613, 799], [598, 614, 798], [598, 618, 794], [598, 619, 793],
[598, 623, 789], [598, 624, 788], [599, 599, 812], [599, 612, 799], [599, 613, 798],
[599, 614, 797], [599, 617, 794], [599, 618, 793], [599, 619, 792], [599, 622, 789],
[599, 623, 788], [599, 624, 787], [612, 624, 774], [613, 623, 774], [613, 624, 773],
[614, 622, 774], [614, 623, 773], [614, 624, 772], [617, 619, 774], [617, 624, 769],
[618, 618, 774], [618, 619, 773], [618, 623, 769], [618, 624, 768], [619, 619, 772],
[619, 622, 769], [619, 623, 768], [619, 624, 767], [622, 624, 764], [623, 623, 764],
[623, 624, 763], [624, 624, 762]

9. That's a pretty long list.

May I ask what this problem's for?

10. I got the question from a foreign math forum. I'm interested how to do the optimization over discrete math problems. Actually I don't think this problem is solved since we haven't find the math method for the solutions. These are just a programming result. the probability of obtain a solution by randomly take a,b,c
(a<=b<=c, a+b+c=2010) is about 0.008. I hope to find the something real in solving this problem