Grades 11-12 Video Solutions 2025-2025_11-12

Video Transcription

This is problem 21 for the grade 11 and 12, 2025 Math Kangaroo. A student starts with two numbers written on the board. He then deletes them and writes the sum of the numbers  and the positive difference of the two numbers. He then continues the same process with the new numbers. So he starts with the numbers 3 and 5 and repeats this process 50 times.  What are the two numbers that he will end up with? Our answers are 3 to the 25 and 5 to the 25, 3 to the 50 and 5 to the 50, 2 times 3 to the 25 and 2 times 5 to the 25,  D, 3 times 2 to the 25 and 5 times 2 to the 25, and E, none of the above. So yeah, take your time. Pause the video. All right. So let's just like simulate this for a little bit  because I think that can help us gain a lot of intuition. So we start with board 1, or I guess turn 0. 3 and 5 are written on the board. We play the game. Now we're on turn 1.  And we're going to end this on like turn 50. So we take the sum, which is 8, and the positive difference, 2.  So maybe there's some credence for exponents of 2 somewhere. Write it one more time. This is turn 2.  Sum and positive difference, which is 10 and 6. So something that I immediately notice is that as we go from turn 0 to turn 2,  we've doubled the 3 as 6, and we've doubled the 5 as 10. And we can sort of like prove this. So starting with A and B, we like turn 1. We get A plus B and A minus B.  And then when we add A plus B and A minus B, we end up with 2A because the Bs cancel. Similarly, if we do A plus B minus A minus B and like take the positive version of that,  we get that this ends up being 2B because we sort of assume that B started as a positive number. So starting with A and B, doing two rounds, like getting to A plus B and A minus B,  and then after another round, we get to 2A and 2B. Well, that means that every two turns, so turn N to N plus 2, we go from A comma B to 2A comma 2B.  Then we pretty much have the trick down. So if we do this 50 times, then that means there are 25 repetitions of this like two-step process.  And thus, we just repeat this like multiplication by 2 25 times for each thing. So 3 after 25 of these two-step processes goes to 3 times 2 to the 25.  And 5 after 25 two steps goes to 5 times 2 to the 25, which corresponds exactly to answer D. So that's our answer.

Video Summary

In this Math Kangaroo problem, a student repeatedly replaces two numbers on a board with their sum and positive difference. Starting with numbers 3 and 5, the process is repeated 50 times. Through each two-step cycle, the numbers double: after two turns, (3, 5) becomes (6, 10), and so on. This means every two turns, each number is multiplied by 2. After 50 turns, the transformation is applied 25 times, leading to the final numbers being \(3 \times 2^{25}\) and \(5 \times 2^{25}\). Therefore, the correct answer is option D.

Keywords

Math Kangaroo

number transformation

sum and difference

doubling process

final numbers