1991 OIM Problems/Problem 4

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Problem

Find a number $N$ of five different and non-zero digits, which is equal to the sum of all the numbers of three different digits that can be formed with five digits of $N$.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

Let $N=d_1d_2d_3d_4d_5$ or in a better format: $N=10000d_1+1000d_2+100d_3+10d_4+d_5$

The total number of combinations is given the following way:

For the first digit of any three-digit number we have 5 numbers to chose from.

For the second digit we have 4 numbers to chose from.

For the third digit we have 3 numbers to chose from.

Total numbers of three digit numbers is (5)(4)(3)=60.

Now we need to find their sum.

From all 60 ways, in the first digit we will have each digit of N showing with (4)(3)=12 configurations.

From all 60 ways, in the second digit we will have each digit of N showing with (4)(3)=12 configurations.

From all 60 ways, in the last digit we will have each digit of N showing with (4)(3)=12 configurations.

Therefore the sum, since each digit of $N$ is shown in each position 12 times, then

$S=\left( 12\sum_{i=1}^{5}d_i \right)100+\left( 12\sum_{i=1}^{5}d_i \right)10+\left( 12\sum_{i=1}^{5}d_i \right)$

$S=1332 \sum_{i=1}^{5}d_i =N$

Since $12345\le N \le 98765$, then $15 \le \sum_{i=1}^{5}d_i \le 35$

${(15)(1332)=19980;1+9+9+8+0=27;2715;NO(15)(1332)=19980;1+9+9+8+0=27;2715;NO(16)(1332)=21312;2+1+3+1+2=9;916;NO(17)(1332)=22644;2+2+6+4+4=18;1817;NO(18)(1332)=23976;2+3+9+7+6=27;2718;NO(19)(1332)=25308;2+5+3+0+8=18;1819;NO(20)(1332)=26640;2+6+6+4+0=18;1820;NO(21)(1332)=27972;2+7+9+7+2=27;2721;NO(22)(1332)=29304;2+9+3+0+4=18;1822;NO(23)(1332)=30636;3+0+6+3+6=18;1823;NO(24)(1332)=31968;3+1+9+6+8=27;2724;NO(25)(1332)=33300;3+3+3+0+0=9;925;NO(26)(1332)=34632;3+4+6+3+2=18;1826;NO(27)(1332)=35964;3+5+9+6+4=27;27=27;YES(28)(1332)=37296;3+7+2+9+6=27;2728;NO(29)(1332)=38628;3+8+6+2+8=27;2729;NO(30)(1332)=39960;3+9+9+6+0=27;2730;NO(31)(1332)=41292;4+1+2+9+2=18;1831;NO(32)(1332)=42624;4+2+6+2+4=18;1832;NO(33)(1332)=43956;4+3+9+5+6=27;2733;NO(34)(1332)=45288;4+5+2+8+8=27;2734;NO(35)(1332)=46620;4+6+6+2+0=18;1835;NO$ (Error compiling LaTeX. Unknown error_msg)


Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

See also

https://www.oma.org.ar/enunciados/ibe6.htm