Russia| A Nice Math Olympiad algebraic problem| Can you solve this?

Hi, I saw this question and I was like, wow, this is the first time you're tuning in—you're highly Welcome! And if you're a returning viewer, I love you so much. Okay, today we are faced with another mathematics Olympiad problem, and we are asked to solve for the values of a and b, okay? Now, let's call So if we say equation one minus equation two, one minus two, okay, so it's going to be a squared. minus b squared, right? But before that, let me just note that a is not equal to b—that's a condition. Then we are looking for a and b. Let me wipe this so that I have enough space to work with. Okay. So I said one minus two, and when we do that, we have um a squared minus b squared, this minus this. will be equal to b plus one one one okay minus a plus one one one okay, so this is a squared minus b squared equals b plus one one one; now minus times a is minus a. Okay, and minus times one one one is minus one one one, so plus one one one minus one one one. will give us zero, so we have a squared minus b squared is equal to b minus a. All right, recall. the algebraic identity that a squared minus b squared, the difference of two squares, is the same thing as a plus b, brackets a minus b. All right, now let's bring this to the other side of the equation and equate this to be equal to zero; so if we do that, we are going to have minus a becomes plus a and my this becomes plus b, which already becomes minus b, all right, is equal to zero. Now notice that we have a minus b times a minus b, so let's factor that out; if we factor it out, we have a minus b, we open the bracket; now, if this is divided by this, you can see, this leaves you with only That's right, so you have a plus b. Now, if this is divided by this, you have what—one. So we have. plus one is equal to zero now in this situation we now have two values, right? The first one is that a minus b equals zero, okay, but when you solve this, when you solve this, you will notice that your a will be equal to minus b; we cross to become plus b, and recall where we are told that a is not equal to b, so this is not acceptable; we are not going to accept this, we are going to forget it. Then the second equation we are going to have here is that a plus b times a plus b plus one. is equal to zero, so this is the one we'll be working with. So if we do that, let's make a the subject of the formula here, so we are going to have that a is equal to plus b, we cross over to become minus b and plus one, we cross over to become minus one, so this becomes our a, all right Now that we have this as our a, let us call this a very good equation; let's call it. Equation three: So let's form this particular equation. We have that a squared is equal to b. plus one uh one one one going to write our minus b minus one squared right is equal to b plus one One one now this is what um an identity in algebra that states that a minus b squared okay is equal to a squared plus b squared minus 2 a b okay a squared plus b squared minus 2 a b so in this case our a is minus b and our b is one, so if we should solve that we are going to have remember we have a squared plus b squared minus 2ab, right? So it's going to be a squared. which is minus b squared, right, plus b squared, which is one squared, okay, minus two times our a Is minus b okay, and our b is one, all right? That is, it is going to be equal to b plus one, one one. minus b squared is b squared, one squared is one, then minus two minus b is what plus two b that is minus two times minus b will give us plus two b times one is still plus two b is equal to b plus one one one, all right, so we have b squared plus two b. Now let's collect like terms. so this b becomes minus b if we cross this equation, so we have minus b is equal to We have one, one, one here already; one, one, one. Now this one, we cross over to become minus one. Alright, so now we have that b squared plus two b minus b, which gives you b, right, is equal to 110 Now notice that we are forming a quadratic equation here, right? So we can now write that b squared plus b minus one ten equals zero, so this is a quadratic equation. Now we need Two factors of minus 110 that, when added, give one. The question for b here is one. so we need factors of 110 minus 110—that's when you add them you get one now 110 the factors are when 10 divides this, you get 11, so the factors are 11 and 10. Now, because this is negative and We need one when we add them; it's going to be 11 minus 10, giving us positive one, so it's to be 11, so we are going to replace b with what 11b and minus 10b, so that 11b minus 10b will Still give us this b in the middle here, so we are going to replace that, so we have b squared. plus 11b minus 10b minus 110 equals zero, okay. All right, now let's factorize by grouping. It makes me want to clean this now. Let's factor by grouping; these two have b in common, so we Bring that b. All right now, b squared divided by b. You have only one b remaining, so we write the b. now plus 11b divided by this b you are left with plus 11 plus 11, all right? Now this minus 10b and Minus 110, they have minus 10 in common, right? So we factor out the minus 10. Now it's minus 10b. divided by minus 10, you have what you have, only b remaining, so we have b then minus 110 divided by minus 10, you are left with only what plus 11 equals zero, okay? So we have b plus 11b plus 11 here, so we pick one; then we are going to pick this b minus 10 is equal to zero. So we now have two values for our b: they are that b plus 11 is equal to zero or b minus 10 is equal to zero, so if we solve this side, we have that b is equal to this plus we cross over become minus 11, then we have b is equal to this minus 10; we cross to become plus 10, so these are The two values of b are b equals minus 11 and b equals 10. So we need, um, if we call this b1 and we call this B2; we need A1 for this, and we need A2 for this. Now recall that we are told That b squared equals a plus one one one, so we have b squared equals a plus one. One one now for B1, if we need to get our A1, so it's going to be remember, let's substitute minus 11 So this is for a1 now, so this minus 11 squared, okay, is equal to a plus one one one. Now for this side is going to be what, recall that b squared is equal to a plus one one one. So we are going to replace our b with 10 for this side; or let's write this first to avoid confusion. Okay, so this side b is 10, so we have 10 squared is equal to a plus one one one. Alright, now let's solve minus 11 squared is, um, 11 times 11, which is 121, so we have one two one is equal to a plus 11, so to get the value of a here is going to be one two one minus One one one is equal to A. So, one two one minus one one one gives us 10. So we have that. Alright, this one is A1, right? That A1 is equal to 10. All right, now let's solve for A2. Let's solve for A2. So a² here, 10 squared is 100, is equal to a plus 111, then we have, um, to get the value of a is It will be 100 minus 111, which is equal to a. Okay, so 100 minus 111 will give you minus 11, which is equal to to a, so that means this is A2. Don't worry if this is the first time you're coming across it. Please subscribe to this channel and turn on your notification bell, yes, so that I won't be See my next video, okay? So, on your notification bell, so that you'll be notified if I upload. A new video now, I want this not to confuse us, so these are what we got, these are what we got. It means me to clarify because we are going to check the results; we are going to check whether we Are they correct or not? Okay, so let's wipe here and do our check. So we are going to pick our values. Let's place them in the middle so that a1 b1 equals 10 and minus 11, therefore a1 b1 equals 10. Alright, now a2 comma b2 is equal to what? Um, our a2 is what? Our a2 is minus 11, right? Minus 11. Comma B2 is 10, all right? So this is what we have for A2 B2, and this is what we have for A1 B1. Now let's check, let's check what we have. So let's substitute this into the first equation that says that a squared is equal to b plus 111, so let's use the first value. So we have our a here is what 10 times 10 squared is equal to our b here, which is minus 11, right, plus 111. Now, 10 squared is what? 100. is equal to now minus 11 plus what 1 1 1 is 4 1 1 minus 11 which is also 100 so the left hand side is equal to the right-hand side, so this is correct. Now let's substitute this again into the the second one to make sure we are 100 percent correct, so if we do that for the second one, for The second one, we have b squared is equal to a plus 111. Okay, so let's substitute this into So our b here is minus 11, so we have minus 11 squared is equal to our a here is 10, 10 plus 1 minus 11 squared is the same as 11 squared, which is 121. So we have 121. is equal to now 111 plus 10 is 121, so the left-hand side is also equal to the right-hand side. Okay, we have the second values for a and b. Please substitute those into the equation as well. To verify our result, but it is correct. Thank you so much for watching, and please watch another one. All right, bye.