Olympiad Mathematics | Russian | Can You Solve This?

Added: 2026-05-10

[00:00:00]If you're ready, let's provide the solution to this one very quickly.

[00:00:09]We have x + y = 10.

[00:00:16]And this is our equation one.

[00:00:19]We also have xy = 10.

[00:00:24]And it's our equation two.

[00:00:27]We're going to solve these two equations simultaneously.

[00:00:32]So, what do we do first?

[00:00:35]From equation one, from equation one, we're going to make x the subject.

[00:00:44]So, that our x from equation one will be 10 - y.

[00:00:50]Right?

[00:00:51]And this will turn to our equation three.

[00:00:54]We're going to need this equation three to get the value of x.

[00:00:58]Now, put this equation two, I mean equation three into equation two.

[00:01:04]Okay, we're going to substitute this into equation two.

[00:01:12]And our equation two is xy = 10, right?

[00:01:19]But now, our x is 10 - y. So, here we have 10 - y to multiply y.

[00:01:28]And it's equal to 10.

[00:01:31]So, the next point is to open the bracket. 10 * y, that is 10y.

[00:01:37]- y * y, that is - y squared.

[00:01:41]And all of this is equal to 10.

[00:01:45]So, the next point is for us to make - y the subject. So, we have - y squared.

[00:01:54]Then we have + 10.

[00:01:57]This will become - 10.

[00:01:59]by way this is um 10 y this one here then this one becomes minus 10 and there is nothing more on the right hand side.

[00:02:12]The next point is to multiply everything by negative one so that y squared will have a positive coefficient.

[00:02:23]So this would give us y squared it will make this negative 10 y it will make this plus 10 and zero is there.

[00:02:34]Now we're going to use quadratic formula to solve this.

[00:02:38]Yes and the formula is y equals minus b plus or minus we have b squared minus 4 ac all over 2 times a.

[00:02:52]So what we have to do is to know our abc.

[00:02:56]Our a is the coefficient of y squared which is 1 b is the coefficient of y which is minus 10 and c is a constant which is 10 right?

[00:03:09]So from here we will put in the >> [clears throat] >> the values of abc and our y will be negative this negative is here and b itself is negative 10.

[00:03:23]Then we have plus or minus b squared is going to be negative 10 squared right?

[00:03:31]Negative 10 squared then we have minus 4 times 1 times 10.

[00:03:38]A [snorts] is 1 and c is 10.

[00:03:44]So all of this will be over 2 times 1 right? 2 times 1. So let's continue with this.

[00:03:56]Okay, so we have our Y now to be negative negative is positive.

[00:04:04]Then we have minus 10 squared is 100.

[00:04:09]Then 4 * 10 is 40.

[00:04:13]This is over two.

[00:04:16]Now, our Y is 10 plus minus 40 minus um 100 minus 40 is 60.

[00:04:27]And this is over two, right?

[00:04:31]Okay, so we continue with this.

[00:04:37]Okay, so our Y now will be equal to 10 plus or minus we have square root of um four multiplied by square root of 15.

[00:04:52]And everything is over two. Remember, root four times root 15 is root 60.

[00:04:59]So that our Y will be 10 plus or minus square root of four is two then multiplied by root 15.

[00:05:10]And everything is over two.

[00:05:13]And two will divide the two numerators.

[00:05:17]So that Y will now be two into 10 is five plus minus two into this it will cancel this two from there and we have just root 15.

[00:05:29]But this is a two-in-one kind of solution because Y is five plus root 15 or five minus root 15.

[00:05:45]But then, we need to have our corresponding values of X.

[00:05:50]And from my equation equation three, we say that let X be equal to 10 - Y.

[00:05:59]So, we're going to put in the first value of Y, which is 5 + √15.

[00:06:04]So, our X will now be 10 - open bracket 5 + the square root of 15.

[00:06:12]So, we open the brackets, and X will be equal to 10 - 5 - √15.

[00:06:21]Because the negative will open this, and it will make this to be negative, too.

[00:06:26]So, the value of X is equal to The value of X is um 10 - 5 is 5.

[00:06:35]Then, we have minus √15.

[00:06:39]So, we are saying that when the value of X is 5 - √15, the value of Y is equal to 5 + √15.

[00:06:52]But, we'll not stop here because we still have another value of Y, which is 5 - √15. So, let's go back there.

[00:07:03]>> [snorts] >> Okay, so we're going to pick this value of Y now.

[00:07:07]And um our X will still be 10 - Y.

[00:07:12]So, we now have X to be 10 - open bracket, put this, 5 - √15.

[00:07:22]We'll open the brackets like we did before.

[00:07:27]Yes, we open the bracket like we did before, and we have X to be equal to 10 - 5. Negative negative will turn to positive.

[00:07:38]And um we have that.

[00:07:41]At the end of the day, we have X to be equal to 10 minus 5 is 5 then we have plus root 15.

[00:07:51]So, we are also saying that at the at this point, when X is 5 when X [clears throat] when X is 5 plus root 15 our value of Y is 5 minus root 15.

[00:08:06]So, we have solved the equation completely.

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