Difference between revisions of "1991 OIM Problems/Problem 4"
Line 33: | Line 33: | ||
Since <math>12345\le N \le 98765</math>, then <math>15 \le \sum_{i=1}^{5}d_i \le 35</math> | Since <math>12345\le N \le 98765</math>, then <math>15 \le \sum_{i=1}^{5}d_i \le 35</math> | ||
+ | |||
+ | <math>x_{i}=\begin{cases} \frac{x_{0}}{2^{i}} & 1 \le i \le 1994 \\ \frac{1}{x_{i-1}} & i=1995\end{cases}</math> | ||
+ | |||
<math>\begin{cases} (15)(1332)=19980; & 1+9+9+8+0=27; & 27\ne 15; & \text{NO}\\(35)(1332)=46620; & 4+6+6+2+0=18; & 18\ne 35; & \text{NO}\end{cases}</math> | <math>\begin{cases} (15)(1332)=19980; & 1+9+9+8+0=27; & 27\ne 15; & \text{NO}\\(35)(1332)=46620; & 4+6+6+2+0=18; & 18\ne 35; & \text{NO}\end{cases}</math> |
Revision as of 23:25, 13 December 2023
Problem
Find a number 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
.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
Let or in a better format:
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 is shown in each position 12 times, then
Since , then
$\begin{cases} (15)(1332)=19980; & 1+9+9+8+0=27; & 27\ne 15; & \text{NO}\\(35)(1332)=46620; & 4+6+6+2+0=18; & 18\ne 35; & \text{NO}\end{cases}$ (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.