Italy | A Nice Algebra Problem | Math Olympiad

math olympiad
olympiad math
math olympiad question
math olympiad questions
math olympiad problem
math olympiad problems
olympiad math problems
algebra olympiad problems
math
maths
algebra
algebra problem
nice algebra problem
a nice algebra problem
equation
exponential equation
a nice exponential equation
radical equation
a nice radical equation
math tricks
imo
Find the value of x and y?
How to solve x^y - y^x = 17

In this video, we'll show you How to Solve Math Olympiad Question A Nice Exponential Equation x^y - y^x = 17 in a clear , fast and easy way. Whether you are a student learning basics or a professtional looking to improve your skills, this video is for you. By the end of this video, you'll have a solid understanding of how to solve math olympiad exponential equations and be able to apply these skills to a variety of problems.
#matholympiad #maths #math #algebra

[dardoburgos]

X= 18, Y= 1. Tiene infinitas soluciones. Eso y mucho ➕ lo aprendí en la escuela pública.

[sy]

Thank you for explaining. There are other solutions. They are (x, y)=(18, 1) and (x, y)=(1, -16) .
I wrote with miss-typing, so I changed to correct sentences.

[SGuerra]

A questão é bonita, mas é necessário especificar que x e y são números naturais, de acordo com a solução apresentada. Na verdade há mais de uma solução para essa questão. Brasil Outubro de 2024.

[nasrullahhusnan]

(x^y)-(y^x)=17
Trivial solution: (x^y)-(y^x)=18¹-1¹⁸
(x, y)=(18, 1)
Another solution:
17=(x^y)-(y^x)
=[(x^(½y)]²-[(y^(½x)]

[x^(½y)]+[y^(½x}]=17
[x^(½y)]-[y^(½x)]=1
Summing: 2x^(½y)=18
x^(½y)=9
x^y=81
--> (x, y)={(9, 2), (3, 4)}
Differencing: 2y^(½x)=16
y^(½x)=8 --> y^x=64
=4³
(x, y)=(3, 4)
Therefore (x, y)=(3, 4)

[Najibullah_Hassani]

You missed some pairs of answers as others said because you assumed that x and y are even and a and b are integers. I meant if x and y are odd then x/2 and y/2 are not integers so you have to assume 17 as a product of 2 non-integer numbers to reach other answers. (One pair is 3-4) Also you must assume 17 as (-1)*(-17).

[MananArhamOP]

Why are these equations named after countries??