Olympiad Maths Prep

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Problem 893

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Combinatorics Difficulty 5.7 Find the answer

14. (15 points) In the following addition problem, different Chinese characters can represent the same digit. How many different equations satisfy the requirement?

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Official solution

【Analysis】According to the calculation method of integer addition.
【Solution】From the vertical form, we can get: “Hua” =1=1;
Since in the addition vertical form, different Chinese characters can represent the same number;
Therefore, the sum of “Yue” + “Ri” + “Sai” in the units place is 2121, 1111 or 1;
The sum of “Yue” + “Ri” + “Sai” in the units place is 21, carry 2 to the tens place;
In the tens place, 4+6+4+6+ “Jue” +2 ends with 1, by 4+6+9+2=214+6+9+2=21, we get “Jue” =9=9, carry 2 to the hundreds place; in the hundreds place, 1+1+ “Bei” +2 ends with 0, by 1+7+2=101+7+2=10, we get “Bei” =7=7, carry 1 to the thousands place;

In the thousands place, 1+11+1 is exactly 2;
From the above, as long as the sum in the units place is 21, “Hua”, “Bei”, and “Jue” are fixed numbers;
Similarly, if the sum of “Yue” + “Ri” + “Sai” in the units place is 11, we get “Hua” =1=1, “Bei” =9=9, “Jue” =0=0, which are also fixed numbers;

If the sum of “Yue” + “Ri” + “Sai” in the units place is 1, we get “Hua” =1=1, “Bei” =9=9, “Jue” =1=1, which are also fixed numbers;

Therefore, “Yue”, “Ri”, and “Sai” determine different equations;
(1) “Yue” + “Ri” + “Sai” =21=21;
7+7+7=217+7+7=21, we get 1 type;
6+7+8=216+7+8=21, we get 6 types;
6+6+9=216+6+9=21, we get 3 types;
5+8+8=215+8+8=21, we get 3 types;
5+7+9=215+7+9=21, we get 6 types;
4+8+9=214+8+9=21, we get 6 types;
3+9+9=213+9+9=21, we get 3 types;
Thus, the sum of “Yue” + “Ri” + “Sai” being 21 can result in 1+6+3+3+6+6+3=281+6+3+3+6+6+3=28 different equations;
(2) “Yue” + “Ri” + “Sai” =11=11;
2+0+9=112+0+9=11, we get 6 types;
3+0+8=113+0+8=11, we get 6 types;
4+0+7=114+0+7=11, we get 6 types;
5+0+6=215+0+6=21, we get 6 types;
1+1+9=111+1+9=11, we get 3 types;
2+1+8=112+1+8=11, we get 6 types;
3+1+7=113+1+7=11, we get 6 types;
4+1+6=114+1+6=11, we get 6 types;
5+1+5=115+1+5=11, we get 3 types;
2+2+7=112+2+7=11, we get 3 types;
3+2+6=113+2+6=11, we get 6 types;
4+2+5=114+2+5=11, we get 6 types;
3+3+5=113+3+5=11, we get 3 types;
4+3+4=114+3+4=11, we get 3 types;
Thus, the sum of “Yue” + “Ri” + “Sai” being 11 can result in 6+6+6+6+3+6+6+6+3+3+6+6+3+3=696+6+6+6+3+6+6+6+3+3+6+6+3+3=69 different equations;
(3) “Yue” + “Ri” + “Sai” =1=1;
0+0+1=10+0+1=1, we get 3 types;
Thus, the sum of “Yue” + “Ri” + “Sai” being 1 can result in 3 different equations;
In summary, there are a total of 28+69+3=10028+69+3=100 different equations.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.