Unraveling the Mystery of a Math Olympiad Number Theory Question

Unraveling the Mystery of a Math Olympiad Number Theory Question
Are you ready to challenge yourself with some of the toughest Math Olympiad questions? In this video, we'll take a deep dive into a complex number theory problem and explore different techniques to solve it. Join us on this mathematical journey and improve your problem-solving skills!
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[@Morbius_Official]

Wow, this is an excellent test of logical thinking.

[@MathBooster]

Thanks 🙂

[@Abby-hi4sf]

Additional to what you get [ a, b, c ] = [ -2. -3. 6 ] . [ -3. -2. 6 ]

[@amandionobre3016]

I didn't understand how in the time from 4:00m to 4:23m ...+(a+b)^2 -2ab became ...+(a+b) -2 .Could someone explain to me?

[@alexey.c]

a³ + b³ = 1 - c² (a + b)(a² - ab + b²) = (1 - c)(1 + c) (1 - c)(a² - ab + b²) = (1 - c)(1 + c) 1. c = 1, a = -b 2. c ≠ 1 a² - ab + b² = 1 + c (a + b)² = a² + 2ab + b² = (1 - c)² = c² - 2c + 1 Subtracting the equations above: 3ab = c² - 3c, c² = 3(ab + c) ∴ c = 3m 3ab = 9m² - 9m, ab = 3(m² - m) ∴ a = 3n or b = 3n Let a = 3n 9nb = 9m² - 9m, nb = m² - m b = 1 - c - a = 1 - 3m - 3n nb = n - 3mn - 3n² = m² - m m² + (3n - 1)m + (3n² - n) = 0 m² + (3n - 1)m + n(3n - 1) = 0 d² = (3n - 1)² - 4n(3n - 1) = (3n - 1)(3n - 1 - 4n) (3n - 1)(-n - 1) = d² ≥ 0 (3n - 1)(n + 1) ≤ 0 -1 ≤ n ≤ 1/3, n = {-1, 0}, a = {-3, 0} 1. n = 0, a = 0: m² - m = 0, m = {0, 1}, c = {0, 3}, b = {1, -2} 2. n = -1, a = -3: m² - 4m + 4 = 0, m = 2, c = 6, b = -2 Case b = 3n is symmetric.

[@echandler]

Based on your start 3ab = c² - 3c, c is clearly divisible by 3. let c = 3m. Substituting gives ab = 3m(m-1). Also a+b = 1 - 3m. Given sum and product, form a quadratic equation and find the negative of its discriminant 3m² - 6m - 1 ≤ 0. Given its roots, m = 0,1,2. m = 0; c = 0; a+b = 1; ab = 0 by inspection (a,b,c) = (0,1,0),(1,0,0) m = 1; c = 3; a+b = -2; ab = 0 by inspection (a,b,c) = (-2,0,3), (0,-2,3) m = 2; c = 6; a+b = -5; ab = 6 by inspection (a,b,c) = (-2,-3,6),(-3,-2,6) Observe symmetry in an and b, which permute. There may be an even shorter solution as we know c is a multiple of 3 and ab is a multiple of 6. ab = 3m(m-1). m and m-1 are consecutive integers. Hence their product is even.

[@Creamcheese65234]

I reached the relationship a²+b²+a+b-ab+2=0 but i wasn't able to figure out next, thanks for teaching me that sometimes quadratic formula can help instead of factorisation like in this case Edit:- dear math booster please upload more questions related to geometry specially from simmillarity and circles

[@MyOneFiftiethOfADollar]

You prefer the term relationship to the more common and standard term “equality”?

[@Creamcheese65234]

@@MyOneFiftiethOfADollar yes it feels really rational

[@dmtri1974]

A nice question and well demonstrated!!!. You may add at the end that "after substituting in the original 2 equations we see that these are really solution". (otherwise the proof is incomplete).

[@MathBooster]

No, we don't need to substitute the values at the end to check that answer is correct or not. You should believe on your method, if you are using correct method then you can't get wrong answer.

[@dmtri1974]

@@MathBooster to have faith or not to have..

[@AliHassan-hb1bn]

a= -1 , b= 1 and c= + - 1

[@sandaccz9228]

What an eloquent title

[@Subhash0705]

All 6 solutions I got sir

[@negriisha]

Solve it, please: a,b,c are not divisible by each of them. abc+1 is divisible by ab-b+1. Which is bigger, c or b? Russian math Olympiad, 2023

[@MathBooster]

I also have a lot of questions like this to solve. But these questions don't have good thumbnail. So, very few viewers clicks on that videos, that decreases the reach of the channel. But I will try to make a video on your question.

[@sachindamani]

I think 0,0,1 is also a solution

[@vijayannair2316]

Nice

[@MathBooster]

Thanks 🙂

[@idvbane8580]

math😍🥰

[@idvbane8580]

🥰🥰🥰🥰🥰🥰🥰

[@idvbane8580]

v ggood explain😜🤪

[@MathBooster]

Thanks 🙂

[@idvbane8580]

sir🤪