Olympiad Challenge | Solve for (x^x)(y^y)-(x^y)(y^x)

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Olympiad Challenge | Solve for (x^x)(y^y)-(x^y)(y^x) | Math Olympiad Preparation

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Can you find the value of (x^x)(y^y)-(x^y)(y^x) in the given system of equations. Step-by-step tutorial by PreMath.com
Today I will teach you tips and tricks to solve the given olympiad math question in a simple and easy way. Learn how to prepare for Math Olympiad fast!
Step-by-step tutorial by PreMath.com
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Пікірлер: 32

@tanaysingh7913 Жыл бұрын

Nice question! ❤

@PreMath Жыл бұрын

Glad you liked it! Thank you! Cheers! 😀

@MultiYUUHI Жыл бұрын

The substitution of variables in mathematics makes the equation more eloquent. X^2+q^2=26① Y^2+p^2=5② Xp+Yq=7③ ①×②=④ ③^2=⑤ ④-⑤=81 ④-⑤=(XY-pq)^2=81 XY-pq=±9

@rabotaakk-nw9nm Жыл бұрын

±9

@davidfromstow Жыл бұрын

Brilliant - and you always make it look so easy to solve!

@PreMath Жыл бұрын

Glad you think so! Thank you, David! Cheers! 😀

@LNC4P Жыл бұрын

Wouldn't the answer be +/- 9?

@Hopebehindthecloud Жыл бұрын

It could be a question answer: Is it -9

@Okkk517 Жыл бұрын

If you consider x^x and y^x as the components of a vector “A” such that A(x^x,y^x,0) and similarly another vector “B” with components B(x^y,y^y,0). We can identify eq. 1 with |A|^2 and eq. 2 with |B|^2 where |A| and |B| are the magnitudes of A and B respectively. Moreover, eq. 3 represents the dot product of A and B, and the required value represents the cross product of the two vectors. Finally, by applying the vector identity (A.B)^2+(AxB)^2=|A|^2*|B|^2 we can easily arrive to the problem solution of 9.

@pietergeerkens6324 Жыл бұрын

Nice! An under rated comment. One might even suspect that the variables are chosen as (x,y) to hint at this solution.

@KAvi_YA666 Жыл бұрын

Thanks for video.Good luck sir!!!!!!!!

@JSSTyger Жыл бұрын

You are one of the best on KZfaq. Thanks.

@PreMath Жыл бұрын

Wow, thanks!

@rishudubey1533 Жыл бұрын

thankyou so much dear professor ❤

@PreMath Жыл бұрын

You are very welcome

@yalchingedikgedik8007 11 ай бұрын

Thanks Sir Difficult but useful. ❤❤❤❤

@kazusakurai7183 Жыл бұрын

I solved it using the inner product of mathematics. If you notice two sides that are orthogonal, it becomes a trigonometric problem.

@anahimaceda7070 Жыл бұрын

😮 oooh, wow!!!

@misterenter-iz7rz Жыл бұрын

Two puzzles today😮. But the second one is much more difficult. 😢. A foolish attempt is let a=x^x, b=y^x, c=x^y d=y^y, thus a^2+b^2=26, c^2+d^2=5, ac+bd=7, now we are going to find ad-bc....😅😅😅

@PreMath Жыл бұрын

No worries. We are all lifelong learners. That's what makes our life exciting and meaningful! Thanks for sharing! Cheers!

@AL_saoud_the_century_of_satan Жыл бұрын

simpler method using complexes we consider u=x^x+i.y^x and v=y^y+i.x^y we note that |u|²=26 and |v|²=5 we put A = x^x.y^y - x^y.y^x and B = x^(x+y) + y^(x+y) =7 we search for A=? We observe that u.v = A +iB = A +7i we then |u.v|²=A²+B² and so A²= |u.v|² - B² = |u|²|v|² - B² = 26.5 - 7² = 81 so A=+9 or -9

@Abby-hi4sf Жыл бұрын

Thank you! I thought subtitution can make it easy, but I find out how I missed the [x^(x+y].[y^(x+y)} can be manipulated -

@bigm383 Жыл бұрын

Thanks, Professor!❤🥂

@colinchalmers3467 Жыл бұрын

What are x and y to make the three original equations feasible within the domain of real numbers?

@samuelbenet007 Жыл бұрын

Elle était bien corsée celle-là ^^

@pralhadraochavan5179 Жыл бұрын

Good morning sir

@waheisel Жыл бұрын

I graphed the first 3 equations on Desmos and the 3 curves do not all intersect (there is a unique value of x and y that solves any combination of 2 of the equations, but not all 3). So there are no real values of x and y that satisfy all 3 equations. Can anyone tell if there are any imaginary solutions for the equations for x and y?