2025 Mathcounts state sprint #22 problem and solution

 

2025 Mathcounts state sprint

#22: Let n be a positive integer less than or equal to 1000. If the last two digits of n are reversed, the resulting integer is exactly 85 percent of n. What is the sum of the possible values of n?

Try this question first. Then scroll down for solution. 

Let n be a positive integer less than or equal to 1000. If the last two digits of n are reversed, the resulting integer is exactly 85 percent of n. What is the sum of the possible values of n?

Let the original number be:

$$n=100h+10t+u$$

The number formed by swapping the tens and units digits is:

$$n^{\prime }=100h+10u+t$$

According to the problem:

$$n^{\prime }=\frac{17}{20}n$$

So $n$ has to be divisible by 20 (make sure you know why). This implies:

$$u=0,t\text{ is even}$$

Let:

$$t=2k,0\le k\le 4$$

Then:

$$n=100h+10t=100h+20k$$ $$n^{\prime }=100h+t=100h+2k$$

Now compute the difference:

$$n-n^{\prime }=18k$$

Also, from the given:

$$n-n^{\prime }=n-\frac{17}{20}n=\frac{3}{20}n$$

Equating both expressions:

$$18k=\frac{3}{20}n\Rightarrow n=120k$$

Since $k\ne 0$, we get:

$$n=120k$$

Valid values for $k\in \{1,2,3,4\}$, so the numbers are:

$$120,240,360,480$$

Their sum is:

$$120+240+360+480=120(1+2+3+4)=120\times 10=\boxed{{\displaystyle 1200}}$$

V's record

 V's record 

weekly homework about 30 to 45 minutes 

extra videos, links optional 

First lesson:

2010 chapter sprint: 

Hi, I got a score of 22 and got questions 16, 22, 23, 24,25,27,29, and 30 wrong. 

I just guessed these questions because I didn't really find a way to do them.

Second meet: 

2011-12 Mathcounts handbook  (40 questions total) 

Warm Up 1: 8
Warm Up 2: 19,20

Warm Up 3: 32, 39 

Workout 1: 23, 24, 30

Third meet: 

2010 Mathcounts school test : 

Hi, I finished trying the Sprint and Target questions:

 these are the problems that I got wrong 

Sprint: I got 7/15 correct.

     15,16: Attempted but answer was wrong

     19,20,26,27,29: Didn't attempt

       30: Couldn't find a good method to do, but was able to solve it by listing out all the possibilities.

Target: I got 6/8 correct.

    The first 6 were relatively easy and I could find a clear way to do them

    The last 2, I couldn't find a way to approach the problem.

Fourth meet : 

2011 Mathcounts school test : last 15 sprint and last 4 target 

Sprint: Out of the last 15, 

I got 7 correct.

The questions that I got wrong were 20,21,22,24,27,28,29,30. 

I tried 21,22, and 27 but the answers were incorrect. I wasn't able to attempt the rest. 

Target: I got every question other than #5. 

However, questions 1 and 2, 

I got wrong at first, but when I retried them, 

I was able to get them. I read the problems wrong and didn't fully understand them the first time.

Fifth meet : review 

Sixth meet : 

2011 chapter 

Hi Mrs Lin! I was able to try both the target and the sprint round questions and here are my results:

Target: I missed 3 & 8, and I didn't attempt them.

Sprint: I got 17,21,23,24,25, 27,29,30.

I didn't attempt 27,29, and 30,

2012 Mathcounts school 

Hi Mrs Lin! I was able to finish both the chapter and target, and here are my results:

Target: I got 5,6,and 8 wrong. I didn't try any of them because I didn't know how to do them.

Sprint: I got 18,23,25,26, and 30

Apr. 30, 2023 

2012 Mathcounts chapter 

Hi Mrs. Lin! I hope your having a very good day and week! I tried both the target and the sprint. 

Target: I got only number 8 wrong, but I didn't know how to do it.

Sprint: I got number 22, 24, 26,28,29, and 30 wrong. I tried to do 22, but I got it wrong. The rest I didn't know how to do them..

May 7th, review 

May 13th, 2023 

2013 Mathcounts School 

Hi Mrs.Lin, I was able to try both the Target and the Sprint and here are my results!

Target (5/8) I got 6,7, and 8 wrong, however, I was able to figure out the answer to problem 6 when I reviewed it. 

Sprint: I got 30,29,28, and 24 wrong. 

May 20th, 2023

Hi Mrs Lin!

2013 Mathcounts Chapter 

I was able to try both the target and sprint round tests and here are my results:

Target: 6/8. I attempted the first 6 problems and got all of them right. I understood the problems fairly well and was able to do all of them on the first try.

sprint: I got 10/15 right. I got all problems I attempted right, and didn't attempt 30,28,25,24,21, and 19.

2014 Mathcounts school 

Hi Mrs Lin, I was able to finish both the target and the sprint, and here were my results!

Sprint: 22/30, I didn't get 18,19,24,25,26,28,29, and 30. I attempted number 18, but didn't get it.

Target: I got 5 out of 8 on the target, but after reviewing my answers, I was able to figure out number 5.

2015 Mathcounts school 

Hi Mrs Lin, I was able to try both the sprint and target. For the sprint, I got problems 20 and 27 wrong, and I didn't attempt any problems past 24 other than 27.

For my  Target I got numbers 6,7, and 8 wrong, and I didn't know how to approach any of them.