Germany - Math Olympiad Question

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Germany - Math Olympiad Question | The BEST Trick

Рет қаралды 23,825

29 күн бұрын

You should know this approach. Solution
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Пікірлер: 40

@VictorPensioner 9 күн бұрын

Find a from source equation (1) a = (22 - b) / (1 + 2b) (a > 0, b > 0) => b = 1 or (22 - b) / (1 + 2b) >= 1 (0 < b = 1 + 2b or 21 >= 3b or b

@ManojkantSamal 22 күн бұрын

a=4, b=2 As per question a+2ab+b=22 Multiply 2 on both the sides3 2a+4ab+2b=44 Add 1 on both the sides3 2a+4ab+2b+1=44+1=45 2a(1+2b)+1(1+2b)=45 (1+2b)(2a+1)=9×5 or 15×3 Let's take 9×5 So, (1+2b)(2a+1)=9×5 So, 1+2b=9 and2 2a+1=5 2b=9-1 and 2a=5_1 2b=8 and 2a =4 b=4 and a=2( also vice versa)

@prime423 28 күн бұрын

The key is the solutions are integers. Simply solve for a in terms of b.No fancy manipulation necessary. Always look for simple solutions first. In addition,thats applicable to all linear Diophantine equations.

@jamesharmon4994 6 күн бұрын

With Diophantine problems, I start by plugging in 1, 2, and 3 for one of the variables. With this problem, a solution was found when I plugged in 2. That this is the only solution is proven by plugging in 3, which results in 3 + 7b = 22. Here, b must be less than 3. (Keeping in mind that either a or b could be 2)

@roberthayter157 28 күн бұрын

Excellent, as usual. What I learn is that I need to learn to be more creative. 😊

@giuliofalco9816 28 күн бұрын

I solved in this way: a + 2ab + b = 22 -> b(2a +1) = 22 - a -> b = (22 - a) / (2a +1). b must be integer so I can compute rhis table a | 2a +1 | 22 -a -------------------------------- 1 3 21 x (22-a divide 2a +1) b=7 2 5 20 x b= 4 3 7 19 4 9 18 x the same solutions switched 5 11 17 6 13 16 7 15 15 x 8 17 14 9 19 13 10 21 12

@harrymatabal8448 28 күн бұрын

Good work from you and Mr Giulio. Thanks

@bookert2373 28 күн бұрын

a+b = 22-2ab, so a+b is divisible by 2. Therefore, both are even or both odd. a+b positive, so 2ab < 22 and ab < 11. There are very few positive pairs of (a,b) to test that are both odd or both even that and satisfy ab

@user-nr6tt5he7g 28 күн бұрын

Continuing your reasoning. Let a≡a' and b≡b' (mod 4), where 0

@tombufford136 27 күн бұрын

At a quick glance if a = 4 and b = 2 then 4+16 + 2 = 22. This gives two solutions: a = 4 and b = 2. a = 2 and b = 4.

@giannaleoci2328 24 күн бұрын

Eccellente soluzione!

@yogamulyadi2046 14 күн бұрын

(2a+1)(2b+1)=45 (a;b)={(1;7),(7;1),(2;4),(4;2)}

@LinhTran-om6qh 28 күн бұрын

How did you come up with tricks like multiplying both sides with 2 and then adding 1 to both sides? I mean, it seemed so random what you did and yet effective. Is there any signal of when to use such tricks and which tricks to use?

@neon1300 28 күн бұрын

Same question here.

@immersivalgames2696 28 күн бұрын

What I think he did is that he factored a+2ab and got a(1+2b) or a(2b+1) and since there is still one b left over he matched the factor by multiplying both sides by 2 to get the 2b and added 1 to finally match the factor so he can get 2a(2b+1) + 1(2b+1) = 45 And simplified it to (2b+1)(2a+1)=45. This can also be done to 2ab + b but for the video it was done as a + 2ab.

@Cagouille79 28 күн бұрын

a(1+2b)+b=22 a=(22-b)/(1+2b) For each value of b you have a value for a except if 1+2b=0 (if b=-1/2) So, number of solutions = Infinite. Isn't it ?

@SubhroMandal-r8k 27 күн бұрын

Bro! Only positive integers are asked.

@TheFrewah 22 күн бұрын

You could do a(1+2b) +b =22. And now you see you want the single b to be 2b+1 so you multiply by 2 and add 1 and then you can factorise.

@Walter_Carnielli 7 күн бұрын

You didn't say at the beginning that you're looking for integer solutions. This simplifies everything!

@sheepy3958 7 күн бұрын

You can see it at the very beginning. Just watch closely 😊

@Akenfelds1 4 күн бұрын

He did say it at the beginning, both in words and in writing.

@Apollorion 3 күн бұрын

He wrote & spoke of positive integers at the beginning. Don't you think this implies integers?

@EduardoDuvane-mc7dm 28 күн бұрын

It's big a deal👏🏻👏🏻👏🏻 I have a math olympiad question, can I send it?

@maryjay7833 26 күн бұрын

Please send it here, we will either help or learn something😊

@jameslouder Күн бұрын

Surely I'm very thick-headed, but how can 2a(2b+1) + (2b+1) be simplified to (2b+1)(2a+1) ?

@DebdasBandyopadhyay-yq5jg 28 күн бұрын

Is it called integer Solution?

@nikolayguzman331 28 күн бұрын

There are more answers: a=22 b=0 and simetrical a=0 b=22 The solve idea is very nice and creative.

@johnlv12 28 күн бұрын

a and b must be positive

@gogo201158 24 күн бұрын

This is too complicated,it is because a and b are positive integer,just try a equal to 1 to 6, then you can find all answers

@olegg9726 20 күн бұрын

a=2 and b=4...Just be common sense!

@keithwillenson8314 16 күн бұрын

2a + 1 >= 3 is not true if a (or b) = 0, you missed the 1, 45 pair and 45, 1 pair or (0, 22) and (22, 0) as long as your solution is only non-negative integers

@marcogallardo-my7hb 10 күн бұрын

Error. a and b positive!!!!

@Mofiac 27 күн бұрын

(1,7), (7,1), (0, 22), (22,0) are the only solutions. Your answers don't work with the original equation.

@maryjay7833 26 күн бұрын

Sorry my friend , but you are wrong, a and b must be positive. He solved it nicely

@user-ul3ms1og2f 27 күн бұрын

a = 1 and b = 7 ?

@Mofiac 27 күн бұрын

And 22, 0. His answers don't work in the original equation .

@quantumbuddha777 26 күн бұрын

@@Mofiaczero is not a positive integer

@richardslater677 25 күн бұрын

You have used totally unexplained steps such as multiplying by two and adding one to both sides. These steps seem picked out of the air with no logical reasoning. You do this a lot on your channel and I’ve commented as such in precious videos. It would be very useful for me if you could explain the reasons why you choose each step please.