Portugal | Math Olympiad Equation | You Should Know the Trick

Portugal | Math Olympiad Equation | You Should Know the Trick | Can you solve this Index Problem?

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Portugal | Math Olympiad Equation | You Should Know the Trick | Can you solve this Index Problem?
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Пікірлер: 6

@VictorSalendu Ай бұрын

maybe someday people recognize your good videos

@superacademy247 Ай бұрын

Thanks 👍 💯 😊 for your support

@ioannismichalopoulos6936 Ай бұрын

Would a substitution of a=x-4 and b=y-4 where a+b = x-4+y-4=(x+y)-(4+4)= 13-8=5 simplify the calculations? The original equation would be transformed to 3^a+3^b=244 and then this in turn would be transformed to 3^a+3^(5-a)=244. Needless to say that 244 could also be transformed to 243+1=3^5+3^0 and by equating the to parts the solution would become evident.

@davidshen5916 21 күн бұрын

A=3^（X-4），B=3^(Y-4), A+B=244, A*B=3^(X-4+Y-4)=3^5=243, A B is roots of T^2-244T+243=0, T=(1,243), (A,B)=(1,243),(243,1), (X,Y)=(4,9),(9,4)

@tunneloflight Ай бұрын

Working through the solution it is straight forward to find (x,y) = (4,9); (9,4). Then notice that the sum of digits of 244 is not divisible by 3, hence 244 is not divisible by 3. So one of the two terms being summed must NOT be divisible by 3. That only happens at 3^0 = 1. And 1 plus 3^5 = 244. Ah! So the problem is easily solved by inspection without going through the difficulty of deriving the solution analytically. Simply examining the boundary conditions solves the problem.

@prollysine Ай бұрын

y=13-x , -> 3^(2x) - 3^(x)*81*244 + 3^13 = 0 , let u=3^x , u^2 - 19764*u + 3^13 = 0 , u=(19764 |+/-| 19602)/2 , u= 81 , 19683 , 3^x=81 , 3^x=3^4 , x=4 , 3^x=19683 , x=ln19683/ln3 , x=9 , y=13-x , y=13-4 and y=13-9 , if x=4 then y=9 and if x=9 then y=4 , solu. x=4 , y=9 , and x=9 , y=4 , test , 3^(4-4) + 3^(9-4) = 3^0 + 3^5 , 1+243 =244 , and , 3^(9-4) + 3^(4-4) = 3^5 + 3^0 , 243+1=244 , OK It was a crazy task... ,