South Africa Math Olympiad Question

[@ramitmazumder8633]

Substitution is an easier approach, get x in terms of y from eq 2, substitute in eq 1 and solve the quadratic equation

[@GereHadush]

Agreed 👍. The simpler is the better

[@СергейАньков-н2б]

Это было гораздо более очевидным и простым вариантом

[@kpopforever786]

I did same

[@AlanCanon2222]

Solvable by inspection. 35 is nearly prime, it only has 5 and 7 as prime factors. So x,y = 35, 1 or 7, 5. x+y = 12. ps. forgot -7,-5 as solns.

[@ollie4276]

Omg....... I feel so stupid. I went into a rabbit whole of using polar coordinates. I got it, but in the most ugly way ever

[@AlanCanon2222]

@@ollie4276 I love a convoluted proof!

[@michaelbell5984]

But isn't that assuming integer solutions?

[@ronrobinson3872]

i did same way and shes still going n and on and on ..lol

[@fahim8682]

Ho do you apply the 2i into the equation?

[@pourianozari1009]

Alternatively, multiply the first eq by x^2 => x^4 - x^2*y^2=24*x^2 => x^4 - 24x^2 - (xy)^2=0 now plug xy=35 into this => (x^2)^2 - 24x^2 - 35^2 = 0 Note that this could be factored as follows: (Try looking at the factors in 35^2= 5*5*7*7 to find two numbers from this the difference of which would be 24 => You will find 5*5 and 7*7) => (x^2 - 49) (x^2+25) = 0 => x^2 = 49 or x^2=-25 You can take it from here that x can be either +7 or -7 which makes x+y +12 or -12.

[@RenatoLousan44]

This only finds two solution of four needed, your method is good but only for real numbers. The other two solutions are imaginary numbers.

[@Haunter78]

@@RenatoLousan44you can apply [-b+-(b^2-4ac)^1/2]/2a to find the imaginary values

[@Cekcom]

@@RenatoLousan44 Not at all. The other two solutions follows from x^2=-25: x = +- i*5 => {xy=35} => y = -+ i*7 => x+y= +- i*2

[@hnnh4590]

It's very simple if xy=35 then y=35/× and we can use it in x^2-y^2=24

[@kebapgaming]

This is literally so easy, I just took the only 2 prime factors of 35 and substituted them and it works

[@qhawe_tricks]

We don't divide by variables

[@ronniedurham]

Awful example

[@UmeshB8811]

Yes! you are true but only when there is no possibility of variable becoming a zero. As you can see in the example xy= 35. It means that neither of the variables is zero. ​@@qhawe_tricks

[@saragiotis]

When squaring both sides of an equation, it is possible to introduce solutions that do not satisfy the initial system equations. Although for this particular system of equations, this will not change anything, imho it would be better practice to check the final result with the initial system of equations

[@ArnabSanyalakaDeathStroke]

Actually, if you keep solving for x and y, you will see that the solutions are x = 7, y = 5; x = -7, y = -5; x = 7i, y = -5i; x = -7i, y = 5i All 4 solutions satisfy xy = 35, but while the first two solutions satisfy the other equation, the last two do not. This is exactly because of the squaring process. @LKLogic please check.

[@kenkennio4452]

x+ y can be 4 different things: 12, -12, 2i and -2i

[@yamnitsky]

takes all of 3 seconds to solve by simple inspection: xy=35, the only factors of 35 that aren't trivial are 5 and 7. Checking 7^2-5^2 = 49-25=24. x=7, y=5. Similarly works with -7 and -5, so the answer is 12 or -12. No imaginary numbers necessary as far as I can tell.

[@jensraab2902]

Well, imaginary numbers *are* necessary if you need to find *all* solutions.

[@SerieusFrank]

it's just that 1x35 and 5x7 are the only quick integer opties to test out. I know x2-y2 is a symetric shape and xy=c a 1/x shape. thus 7,5 and negatives are properly the real solutions.

[@omu_omuomu]

The question doesn't say X and Y are integers.. Therefore it's wrong to get X=7 and Y=5 immediately.

[@saigonsong1954]

Ok

[@Generalist18]

That what I also thought

[@隋耀德-w2f]

(x,y)=(7,5)或(-7,-5)故x+y=12或-12

[@suryaentertainment5028]

​@@隋耀德-w2f 🎉

[@mathbigsmart]

Ur right

[@kuan-k4m]

Set Y ²= a, the equation is converted to 1225/a-a=24→ 1225-a ²= 24a, a=49, so Y=7, X=5

[@kpdywo848]

No Y=5, X=7 your solution give : X²-Y²= -24., so Solution is Y=5, X=7 and also Y=-5, X=-7, Two solutions X+Y=12, X+Y=-12.

[@MadChristoph]

Man, they made this simple math problem complex

[@yuki-dw1jr]

If you define x + y ≡ k, you can consider that x and y are solutions of the following equation: X^2 - (x+y) X + xy = X^2 - kX + 35 = 0. So you can get x - y = √(k^2 - 140) by solving this equation. You can obtain x^2 - y^2 = (x + y)(x - y) = k √(k^2 - 140) = 35, which means k = ±12, ±2i. This problem is basic level for Japanese high school students.

[@RTX2050izyygaming]

Can u explain it step by step, i want to learn ur way

[@sozkev9762]

how soln of X^2 - kX + 35 = 0 gives x-y = √(k^2 - 140) ??? Is this comment a scam?

[@ihave2enemies]

​@@sozkev9762No, x-y is difference of roots which is equal to root(D)/a where D is discriminant and a is coffeciant of x²

[@kanizfatema8958]

X^2 + Y^2= -70 is not valid, since the sum of the square of two number is always positive, It must be X^2 + Y^2 = 70.

[@TheLoveKusano]

what's the result of i² + i²?

[@otakurocklee]

We are including complex solutions.

[@carlowood9834]

I'd subst. k=x+y, m=x-y, because that is the obvious thing to do. Then km = 24 and k^2 - m^2 = 4xy = 140. Multiply with k^2: k^4 - 140 k^2 - 24^2 = 0 = (k^2 - 144)(k^2 + 4) and we're done.

[@egghead1969]

very interesting. should take less than 2 sec to solve xy=35 x^2-y^2 =24 Given that 35 is an odd number and a small number (a limited number sets to get 35), it is either 35*1 = 35 or 7*5 = 35. 7^2 - 5^2 = 24. So, x+y = 12. Am I missing something ???

[@rmtsapphire0]

Yes, the negative solution -12 and the two complex solutions; 2i and -2i

[@IamtheHolyMan]

The question does not say that x and y are positive or negative integers.

[@oenrn]

Aren't you introducing extraneous solutions when you square both sides at 0:55?

[@mustafasabir12]

I solved it in 2 seconds. First approach was to see 35 is prime so x and y could be 7 and 5. To make sure 7 sqaure - 5 square is also 24. So x + y is 12

[@jensraab2902]

Congratulations! You found one solution in 2 seconds. But what about the others?

[@lnmukund6152]

Correct my idea is same Mukund

[@EnriqueAThieleSolivan]

@@jensraab2902 Once you get one the other follow as negative numbers squared give positive numbers, the there are the imaginary ones whose difference is in notation, putting the i.

[@alfal4239]

Denote P = x^2, Q =- y^2 Then P and Q are the roots of t^2 - 24*t - 35^2 = 0

[@59lotus52]

This is correct and the most beatiful solution! Answer is 12 or -12

[@christopherdean1326]

I did this by little more than random guesswork in about two freaking seconds! The first two factors of 35 that came to mind were 7x5. 7 squared is 49, 5 squared is 25, 49 - 25 is 24.

[@JerryRigged]

Lol same

[@hellowill]

Ya not sure why easy questions are getting heaps of views. I guess school kids are studying using KZbin.

[@petermak3980]

It is good to say in which set to solve the equation. x, y are Integer , Real or complex;

[@ramunasstulga8264]

I didn't know you can plug in xy in equation as I'm used to isolate variable first, thanks for teaching a trick to find the solution

[@DuyHoang-mr1ob]

Có hiểu kết quả không? Phép tính cộng còn không biết thì học hành gì

[@d.m.7096]

Two more solutions. X = 5i, Y = -7i Or X = -5i, Y = 7i Where i = square root of -1.

[@Barghaest]

Both covered by x+y=2i or -2i.

[@today8084]

Why do you keep putting parentheses on numbers when marking 24 squares😫😫😫

[@xiaoxiahu5414]

Simply replace 24 with 24/35 xy, then solve the first equation, youbcan easily get x equals 7 ,and y is 5.

[@johnconnor555]

I don't know why but I found the result mentaly in about 10 seconds. x=7, y=5 and x+y=12...

[@tassiedevil2200]

I am a little surprised that one would be expected to just know 5476 is 74^2, although I suppose in a test problem one might guess that it would be a perfect square and it would not take long to find the square root, but, as several Commenters remark: one can see by eliminating y that there are four solutions expected since a quartic for x results. Since by inspection {x,y} ={7,5} and {-7,-5} are obviously solutions, it takes little more thought to realise that similarly {5i,-7i} and {-5i,7i} are the others and the results for x+y follow.

[@User-jr7vf]

She knew tht 5476 is 74^2 because she solved the problem before recording for KZbin, and she probably used a calculator to get to that result. Don't fool yourself, nobody is expected to know that from memory.

[@nando9723]

Here’s how I did it. You can see that 5476 is a factor of 4 and so split it up into 4*1369. Then it’s a matter of finding the square root of 1369. This is clearly between 900 & 1600 and a square number only ends in 9 if the square root ends in 3 or 7, therefore if 1369 is a square number then the square root is either 33 or 37. Since it’s closer to 1600 than 900 that suggests it could be 37^2. It doesn’t take long to try calculating 37^2 by hand and it turns out that it is indeed 1369! Therefore 5476 = 4*1369 = (2)^2 * (37)^2 = (2*37)^2 = 74^2 :)

[@tassiedevil2200]

@nando9723 Nice! Square root is easier than I first thought, although I still prefer the approach I offered.

[@yasarkaya5445]

Number pairs whose product result can be 35, 5x7 and 1x35

[@kekohokko7213]

The use of videos like this escapes me. If you already know maths, know the principles involved and know what to infer from the question, surely you could do this solution yourself. Numptys like me, who check out at the 30 seconds mark, would benefit from an explanation of the problem and the steps and tools required to solve it.

[@zeinders81]

한국어로 답변해서 미안하지만 답은 두개지만 한개로 5와7 입니다. 수식을 어찌저찌 계산하던 구구단 공식 35의 답은 5와7로 곱을했을경우입니다. 그래서 그결과의 x+y= 12 빠른 돌출된 결과 나옵니다.

[@RakeshSharma-wc7pm]

I'm not much interested in maths but the way she explained i liked it.❤❤

[@reddappaogeti5152]

Simplest solution. Given xy = 35 Factors for 35 are 5,7 and x, y values are 5,7 or 7,5 Substitute values x=7 and y=5 in equation x^2-y^2=24, 49-25=24, given equation is satisfied, the values x =7 and y=5 are correct. Now x+y=7+5=12

[@abhiroopdas6725]

Congratulations on reaching 1 Million Subscribers 🎉🎉🎉

[@orestesdd]

I wonder if there such property which allows you take power to both side of an equal equation. For instance, if a = b, can you say that a^2=b^2? is true that you can do this, a=b, then a.a = b.b? I don't think so. 🤔

[@service7755]

Dude. Your example is wrong. Think. Whay are you multiplying different number on each side?

[@dellagobaikal8205]

This can be easily resolved in complex digits, x and y determined.

[@amazingcalvin]

I did this by substituting y=35/x in the first equation and then multiplying the entire equation by x^2 to make it as x^4 -24x^2 - 1225 = 0. From there, it's just a quadratic which can be factored and the 4 values of x and y can be determined, leading to 4 different values for x+y

[@celtica5027]

I did it the same way as you.

[@mathmadeeasymme]

I did the same as you

[@jasonluu6711]

@@mathmadeeasymme How I tried and I cant figure out how to factor it after getting to x^4 -24x^2 - 1225 = 0, can anyone help please?

[@mathmadeeasymme]

@@jasonluu6711 from there,just make x^2 = m, so ur new equation will be m^2 -24m -1225=0 Solve quadratically, then don't forget you equated m = x^2😁 I have a maths channel too,u can check it out by clicking on my profile

[@PakShuMan]

Me three. I was actually taught to do it that way in school.

[@pg894]

Try to solve it via integrals and dif. functions. Solving is to be harder

[@davidshen5916]

X+Y=2M, X-Y=2N,X=M+N, Y=M-N, MN=6, M^2-N^2=35, M^2(-N^2)=-36, M^2=36, -N^2=-1, M=-+6, X+Y=-+12

[@askthewise]

What’s wrong with 7 and 5 as immediate answers?

[@AbasKial-fo5rp]

Yesssssd very mach goooood😊😊😊😊❤❤❤

[@androidgamerbinayak4050]

All of this is only needed in school exams,but to solve it faster we can just look at the factors of 35 which are 1,5,7 and 35,now after getting this we can put the value and just test,since 35²-1² will be way more than 24(just by looking we can know),we have to put the other value and voila we got the answer as x=7 and y=5

[@joepollard3228]

I simply went to equation 2, and got 7x5 =35. Then 7²=49 and 5²=25, giving me 24. Then 7+5=12 Take the simple way. It can save you a bunch of time.

[@bowlchamps37]

Yeah, save a bunch of time, but x+y=12 only gets you 2 out of 10 points for the question.

[@danielrybuk1905]

Wow what a beautiful question, i got 7 and 5, but missed the complex solutions. Again beautiful question.

[@CJBrunt]

Just factor 35 into 7 and 5, 7sq=49 minus 5sq=25 equals 24. So 7+5=12.

[@lawrencebartos]

All this math and it’s too complicated. Sometimes you just have to “see through the trash”. Not too many integers where the product of them is 35. In fact the choices are 1 and 35 or 5 and 7. Simply try that first and you’ll find that if x=7 and y=5, the x^2-y^2, 49-25=24. X+y=12

[@alejandroalmodovar7555]

Who said is an integer

[@alejandroalmodovar7555]

Actually there are complex solutions

[@bonecu6045]

It's fun. I want to study math again after university. 20years ago...

[@wildmouse6971]

I can only say it is a shame for International Olympic Committee to have a question like that.

[@Brandon-rc9vp]

This feels more like a middle school algebra question, if this is what passes for math Olympiad these days I fear over who will be designing bridges in our future.

[@alchemist9905]

lol. I guarantee you the engineers actually making a bridge would have just gone for the simple but obvious +,- 7 and 5 as solutions for x , y

[@kumarvaibhav8252]

Best solution: Y = 35/x Put this in the first equation you will get X ^4 -24 x^ 2 - 35 ^ 2 =0 Now assume X^2 = Z Equation will be Z^2 - 24Z -35^2 =0 This is quadratic Equation like a X^2 + b X + c= 0 a b c constant now as per formula X = ( -b +-✓b^2 -4ac)/2a You will get Z then X then Y

[@gsnijjar7931]

Simple 35 has only one combination - 7 & 5 Therefore x=7 and y=5

[@damsssoltani5126]

Step1: x=35/y Step2: substitute and you get a quadratic equation with y^4,y^2 and a number Step3: take y^2=a You'll get a 2nd degree equation, solve it in one minute and you get the final results easily as possible.

[@pholdway5801]

The squares of which numbers when the larger is taken from the smaller leaves 24.? what happens when these are added.......?. 12........,.,

[@jesseshaffer3951]

The difference between squares is twice the summation of all natural numbers between the two terms, less the sum of the two terms ( the sum from x to y, time two , minus x+y) - it makes finding the answer much easier

[@jesseshaffer3951]

5 , 6, 6, 7 = 24

[@musicsubicandcebu1774]

Interesting problem, thanks for posting

[@jibrank3]

Easiest math olympiad question ever

[@Vertical-Limit_BURST]

You said it. Even I was able to solve this in 5 minutes lol

[@CharlesChen-el4ot]

n = x+y n^2 - 70 = x^2+y^2 x ^2 - y^2 =24 2x^2 = 24+n^2-70 =n^2-46 2y^2 =24+70-n^2 4x^2*y^2=4*70*70 =(n^2-46)*(94-n^2) (n^2)^2-140n^2+4*49*123

[@AhmadHaggag-t5l]

Why you don’t substitute y=35/x in the other equation given not equal 0

[@slshusker]

Just eyeball it and see 7 * 5 = 35 and is the answer for both equations.

[@jeffd.9321]

Seriously, you could’ve just used substitution right from the start. It would’ve been easier.

[@gibbogle]

Try this: X^2 - Y^2 = (X+Y)(X-Y) = 24. Let Z = X+Y, then X-Y = 24/Z, (X-Y)^2 = X^2 + Y^2 - 2XY = 24^2/Z^2 = 576/U, where U = Z^2. Also U = Z^2 = (X+Y)^2 = X^2 + Y^2 + 2XY, subtracting the previous equation from this gives 4XY = U - 576/U, U^2 - 140U + 576 = 0. Observing that 576 = 144*4 we can factorise to get (U - 144)(U + 4) = 0, i.e. U = Z^2 = 144 or -4. Only 144 gives the real answer Z = X+Y = 12.

[@xreev0x]

Why are you just not removing Y as a variable by saying that Y = 35/X?

[@ashwanibeohar8172]

Simple xy=35 =(+/-7)*(+/-5) Or (x,y)= (7,5), (-7,-5) Why complecate it

[@KikeAmusan]

Or (x, y)= (7, 5),(-7, -5 )

[@KikeAmusan]

👋👆👆👍

[@proud333yt]

How u would find +/- 2i?

[@renatobustamante981]

In the later solution , It seems (x+y)² = (x²+y²) I can't understand why?

[@jim2376]

Hint: 49 - 25 = 24. The instructor has a lovely feminine voice. I could listen to her reading a telephone book.

[@sanklan1992]

answer is wrong. x^2 + y^2 = +74 or -74. But -74 will not be considered as x square and y square both are positive integer. How can they sum to negative.

[@emilioferrer9706]

The square root of a number is just a positive number not plus or minus a number....

[@jensraab2902]

Yes, the square root symbol is used for the principal root (the positive one when referring to real numbers); she didn't use the notation correctly. But her solutions are correct nonetheless because we're dealing with quadratic equations which in this case yield one positive and one negative solution.

[@spacechannelfiver]

35 is only divisible by 1, 5, 7 no need to overthink this; just use your times tables. Oh and obviously itself.

[@anthonyvalenti9093]

Well, I will leave out a few steps, (X^2-49)(x^2+25) = 0 does the trick. Hint: 35^2 = (49*25)

[@rembautimes8808]

Why the algo choose to place this video in my stream surprises me. As to why I actually tried and watched the solution baffles me even more. My days of spending my holidays solving calculus problems from Thomas & Finney were in the last century

[@Pandemonis]

Yeah, an equation of two polynomials from the second degree has 4 solutions in |C (for a perfect demonstration, you should have added a line x and y are both =|= 0, so you can resolve for the rest, but given the problem, it's obvious.)

[@DoKillaEnte]

There are infinite amounts of ways to solve this and you picked the nost complicated and confusing one.

[@kpdywo848]

x²-y²=24 et xy=35 --> x=35/y --> (35/y)²-y²=24 Multiplions par y² et regroupons on a : 1225-y^4-24y²=0, posons y²=Y, l'équation devient: -Y²-24Y+1225=0 --> deux solutions Y'=-49, Y''=25. Alors seul Y''=25 fonctionne puisque Y=y², -->i y=5. Puis : x²-25=24 --> x²=49 --> x=7. Alors x+y=12, et cela fonctionne avec x=-7, y=-5, x+y=-12... Inutile de travailler dans C.

[@Jay_D_Ashe]

Well we all know it's x=7 and y=5, but we needed that demo

[@lalalandblur8870]

the only number that i can think of xy =35 is 7x5. 7x7=49, 49-24= 25. 5x5=25 x=7 y=5 x+y=7+5= 12

[@ahhdimentia]

Quadratic simultaneous equation

[@tobefree8510]

35 = 7 X 5 So x times y = 7 times 5 x^2 - y ^2 = (x +y)(x-y) = (7+5)(7-5) = 12 times 2 = 24 so x + y = 7 + 5

[@cosimo7770]

Mental calculation in 5 secs. x,y = 7,5 or -7,-5. So x+y = +12 or -12. Maths teachers should teach insight, or not teach.

[@ph1shk4]

Jeez this is substitute problem You substitute to maid only one unknown - either x or y.

[@williamlemaire1984]

Are You sure its South african olympiad ? Because ive seen the same exercice with a title : Indian olympiad and Bihar Olympiad.

[@juancarlosgarcia9002]

Much easier if you simply solve the system of 2 first equations

[@ГрушаРыжая]

7 и 5 , методом подбора решается за 2 секунды

[@scuendo]

если не считать комплексные числа, то ты минимум пропустил -5,-7 и ответ на вопрос «-12».

[@rasolerv]

uhmm you can do more easy way, 35 is 35*1 or 7*5, you test in the cuadratic equation and is fast probe 7 7 and 5 works

[@nv8972]

Just by looking at them you can say it is 7 and 5.

[@balann8188]

1×35 or 5×7 that' all. square of 7 - square of 5 = 24. Therefore 7+5= 12.

[@johancoetzer2165]

As an engineer I find all the math amusing. There are only two integers that are factors of 35. Why do so much algebra when the answer is right there? 😮 Oh wait, this is an olympiad and the easy way would be wrong...

[@jensraab2902]

It's a Math Olympiad, not an Engineering Olympiad! 😛 That's why they most likely want to have *all* solutions.

[@edwincg5837]

Descomponiendo 35 en sus divisores X.Y= 35 => 7×5= 35 X² - Y²= 24 (7+5)(7-2)=24 X+Y= 12

[@drrjr1985]

An unnecessary complication. I'ts enough to do a simple variable substitution in order to get a bi-quadratic equation (solved transfoming it in a quadratic equation).

[@petekutheis3822]

this one was too simple.

[@oluwatosinakinpeloye5111]

True but tricky at first

[@ilmaio]

7 and 5, -7 and -5. If you can't do it in your mind in less than one minute you'll better go back to junior high school fast.

[@oenrn]

Once you graduate from junior high school you will learn that complex numbers exist.

[@gordontsai11]

7×5=35,x+y=12

[@SUNILJIGARG]

No need to do such long solution. You do not have so much time in examinations. Just do it mentally. observe xy=35 and possible prime factors are 7 and 5 that comes first to your mind. Test the same on previous equation and you get that it is right. It is hardly 5 second mental question. Such questions are deliberately given so that your ability to do such things quickly can be seen. Normally the options given in the question are put back in the option. If negative value option is given that can also be tested to be true. Imaginary value options can also be tried similarly.

[@LKLogic]

😊

[@lakshmikutty2255]

x²-y²=24 xy=35 To find x+y. (x+y) (x-y) =24 x+y =12 x-y=2 x+y=12 x=7 y=5

[@jose22805]

i would dare to bet that she did it the hardest way

[@syedmdabid7191]

Valde valde facile!!! Solutio realis: x=7& y= 5,responsi.

[@徐岩-k1l]

you are good at math。very nice person。

[@eduard.constantinescu]

WRONG! X + Y it is equal to 348/7 and -132/5

[@rvrse2485]

A typical soviet schoolboy solve this in 2 minutes

[@YoelIsaac-vb9xb]

Solución x=5 ...y=7....sin mucha complicación...por método de sustitución...Saludos

[@moniqueboyke5879]

Great video and great question

[@tamirerez2547]

X•Y= 35 It possible only if 1•35 or 7•5 Only 7^2 - 5^2 = 24 Therefore: X=7 Y=5 Q.E.D.

[@dback68]

(-7,-5) is also a solution to (x+y)(x-y)=24 and xy=35

[@Hyde199999999999]

You can’t wright sqrt of a negative number.

[@jensraab2902]

You can in the complex world.

[@Hyde199999999999]

@@jensraab2902 no. It is tolerated to write for instance « i is a square root of -1 ». But the function sqrt (when you use the symbol) is defined only on R+.

[@jensraab2902]

@@Hyde199999999999 In the complex world √x is defined as the principal square root of x. I've seen the square root symbol in combination with negative numbers (e.g. √-1, √-15, etc.) countless times. You'll find videos here on KZbin by distinguished mathematicians, including university professors of mathematics, using it that way. Not sure what else to say.