South Africa Math Olympiad Question
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- Опубліковано 21 чер 2023
- Maths Olympiads are held all around the world to recognise students who excel in maths. The test is offered at many grade levels and provides them with numerous possibilities to win certifications, awards, and even scholarships for higher studies
It's very simple
if xy=35
then y=35/×
and we can use it in
x^2-y^2=24
This is literally so easy, I just took the only 2 prime factors of 35 and substituted them and it works
We don't divide by variables
Awful example
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.
Well, imaginary numbers *are* necessary if you need to find *all* solutions.
Substitution is an easier approach, get x in terms of y from eq 2, substitute in eq 1 and solve the quadratic equation
Agreed 👍. The simpler is the better
Это было гораздо более очевидным и простым вариантом
I did same
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.
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
@@ollie4276 I love a convoluted proof!
But isn't that assuming integer solutions?
i did same way and shes still going n and on and on ..lol
Ho do you apply the 2i into the equation?
x+ y can be 4 different things: 12, -12, 2i and -2i
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
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.
Man, they made this simple math problem complex
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.
This only finds two solution of four needed, your method is good but only for real numbers. The other two solutions are imaginary numbers.
@@RenatoLousan44you can apply [-b+-(b^2-4ac)^1/2]/2a to find the imaginary values
@@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
Wow what a beautiful question, i got 7 and 5, but missed the complex solutions. Again beautiful question.
Interesting problem, thanks for posting
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.
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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.
Lol same
Ya not sure why easy questions are getting heaps of views.
I guess school kids are studying using UA-cam.
It is good to say in which set to solve the equation. x, y are Integer , Real or complex;
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.
Parabéns. Exelente solução.
Set Y ²= a, the equation is converted to 1225/a-a=24→ 1225-a ²= 24a, a=49, so Y=7, X=5
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.
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
Có hiểu kết quả không? Phép tính cộng còn không biết thì học hành gì
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
Congratulations! You found one solution in 2 seconds. But what about the others?
Correct my idea is same
Mukund
@@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.
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.
what's the result of i² + i²?
We are including complex solutions.
Great video and great question
It's fun. I want to study math again after university. 20years ago...
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.
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.
Can u explain it step by step, i want to learn ur way
how soln of X^2 - kX + 35 = 0 gives
x-y = √(k^2 - 140) ???
Is this comment a scam?
@@sozkev9762No, x-y is difference of roots which is equal to root(D)/a where D is discriminant and a is coffeciant of x²
Denote P = x^2, Q =- y^2
Then P and Q are the roots of t^2 - 24*t - 35^2 = 0
This is correct and the most beatiful solution! Answer is 12 or -12
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.
Two more solutions.
X = 5i, Y = -7i
Or X = -5i, Y = 7i
Where i = square root of -1.
Both covered by x+y=2i or -2i.
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.
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
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 ???
Yes, the negative solution -12 and the two complex solutions; 2i and -2i
The question does not say that x and y are positive or negative integers.
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
I can only say it is a shame for International Olympic Committee to have a question like that.
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
I don't know why but I found the result mentaly in about 10 seconds. x=7, y=5 and x+y=12...
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...
It's a Math Olympiad, not an Engineering Olympiad! 😛
That's why they most likely want to have *all* solutions.
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.
Aren't you introducing extraneous solutions when you square both sides at 0:55?
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
5 , 6, 6, 7 = 24
Simply replace 24 with 24/35 xy, then solve the first equation, youbcan easily get x equals 7 ,and y is 5.
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.
Yeah, save a bunch of time, but x+y=12 only gets you 2 out of 10 points for the question.
Easiest math olympiad question ever
You said it.
Even I was able to solve this in 5 minutes lol
Descomponiendo 35 en sus divisores
X.Y= 35 => 7×5= 35
X² - Y²= 24
(7+5)(7-2)=24
X+Y= 12
Why do you keep putting parentheses on numbers when marking 24 squares😫😫😫
Great ,thanks very much
What’s wrong with 7 and 5 as immediate answers?
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
Just factor 35 into 7 and 5, 7sq=49 minus 5sq=25 equals 24. So 7+5=12.
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.)
Number pairs whose product result can be 35, 5x7 and 1x35
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.
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
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
This can be easily resolved in complex digits, x and y determined.
Well we all know it's x=7 and y=5, but we needed that demo
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.
She knew tht 5476 is 74^2 because she solved the problem before recording for UA-cam, and she probably used a calculator to get to that result. Don't fool yourself, nobody is expected to know that from memory.
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 :)
@nando9723 Nice! Square root is easier than I first thought, although I still prefer the approach I offered.
this one was too simple.
True but tricky at first
you are good at math。very nice person。
Чуть не умер, за эти не полные 7минут. Именно ЭТОТ пример решается без 100500К возведений в квадрат. Х =+/-7, У=+/-5 Х+У= +/- 12 . Если знать таблицу умножения на 5 и 7, а также таблицу квадратов, хотя-бы до 10, то решить эту задачу реально в течении 10-20 секунд
teşekkürler elinize sağlık
7 и 5 , методом подбора решается за 2 секунды
если не считать комплексные числа, то ты минимум пропустил -5,-7 и ответ на вопрос «-12».
Are You sure its South african olympiad ? Because ive seen the same exercice with a title : Indian olympiad and Bihar Olympiad.
Try to solve it via integrals and dif. functions.
Solving is to be harder
Hint: 49 - 25 = 24.
The instructor has a lovely feminine voice. I could listen to her reading a telephone book.
The squares of which numbers when the larger is taken from the smaller leaves 24.? what happens when these are added.......?. 12........,.,
Why you don’t substitute y=35/x in the other equation given not equal 0
Simple
35 has only one combination - 7 & 5
Therefore x=7 and y=5
X^2+y^2 is always positive dear
iam more simple, there is only one solution for: xy=35, easy as 7x5=35, it just match for x²-y²= 24, so x+y must be 12
За пол-минуты в уме решил.
There are infinite amounts of ways to solve this and you picked the nost complicated and confusing one.
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.
Once you graduate from junior high school you will learn that complex numbers exist.
Much easier if you simply solve the system of 2 first equations
great 👏🤘
In the later solution ,
It seems (x+y)² = (x²+y²)
I can't understand why?
x=7 y=5 x + y=12 or x=-7 y=-5 x + y= -12 I solved the equation in my mind
This was so satisfying...
Все такие умные 😁, а кто нибудь нашёл корни этих квадратных уравнений, потом подставил их в искомую сумму 😂 листа бумаги хватило😂
Quadratic simultaneous equation
South Africa Olympiad is easier than Maths exam of 10th standard in India 😂😂😂😂
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
Thanks you
Sees title
Me: Brain.exe has stopped working
Just by looking at them you can say it is 7 and 5.
This one is so easy. I don’t want to spend time on solving it
Why are you just not removing Y as a variable by saying that Y = 35/X?
Seriously, you could’ve just used substitution right from the start. It would’ve been easier.
Just eyeball it and see 7 * 5 = 35 and is the answer for both equations.
this is too simple, every Chinese grade 8 student knows how to do it
I think my brain getting rusty already
I had x = 7; y = 5 within 10 seconds. x + y = 12. 😂
I saw the same qustion in other videos
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
35 is only divisible by 1, 5, 7 no need to overthink this; just use your times tables. Oh and obviously itself.
Valde valde facile!!! Solutio realis: x=7& y= 5,responsi.
1×35 or 5×7 that' all. square of 7 - square of 5 = 24.
Therefore 7+5= 12.
7 and 5
Took me 3 seconds to figure it out
7×5=35,x+y=12
Congratulations
🫶🏻
vvery very impressive
Jeez this is substitute problem
You substitute to maid only one unknown - either x or y.
The question doesn't say X and Y are integers.. Therefore it's wrong to get X=7 and Y=5 immediately.
Ok
That what I also thought
(x,y)=(7,5)或(-7,-5)故x+y=12或-12
@@user-up4sj5yv6d 🎉
Ur right