Germany - Math Olympiad Question | The BEST Trick

You should know this approach. Solution

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[ManojkantSamal]

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)

[YoungUkJo-sl]

That solution is so hard to think. It is wise that just put randomly some numbers seemed proper

[prime]

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.

[VictorPensioner]

Find a from source equation
(1) a = (22 - b) / (1 + 2b) (a > 0, b > 0) => b <= 22
Obviously
a >= 1
or
(22 - b) / (1 + 2b) >= 1 (0 < b <=22)
or
22 - b >= 1 + 2b
or
21 >= 3b
or
b <= 7
Therefore
1 <= b <= 7

Verify all b values by (1)

b = 1, a = (22 - 1) / (1 + 2*1) = 7 is solution
b = 2, a = (22 - 2) / (1 + 2*2) = 4 is solution
b = 3, a = (22 - 3) / (1 + 2*3) = 19/7 is no solution
b = 4, a = (22 - 4) / (1 + 2*4) = 2 is solution
b = 5, a = (22 - 5) / (1 + 2*5) = 17/11 is no solution
b = 6, a = (22 - 6) / (1 + 2*6) = 16/13 is no solution
b = 7, a = (22 - 7) / (1 + 2*7) = 1 is solution
All solutions are
a = 7, b = 1;
a = 4, b = 2 ;
a = 2, b = 4;
a = 1, b = 7

[roberthayter]

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

[madurappankalyanaraman]

Two answers are possible

If a is 2 then b is 4
If a is 4 then b is 2

Accordingly
2+2.2.4+4
2+16+4 = 22.

4+2.4.2+2
4+16+2=22

[jeanlemire]

If you are to try all the possibilities of numbers that give 45 when multiplied together, why not simply set a value for a the solve for b. For example, let a=1, the equation becomes 1 + 2b + b = 22 or 3b = 21 so b = 21/3 or b = 7. Let set a = 2, the equation becomes 2 + 4b + b = 22 or 5b = 20 or b = 20/5 or b = 4. With a = 3, b is not integer. With a = 4, the equation becomes 4 + 8b + b = 22 or 9b = 18 or b = 2. With a = 5 or 6, b is not an integer and with a = 7, b is calculated as 1.

[LinhTran-omqh]

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?

[PATRIKKALLBACK]

What was missing in the beginning of the video was the restrain, two positive integers.

[gogo]

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

[bookert]

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<11.

Or, you could examine ab < 11 for the cases where both are odd and both even. Either way, there are very few cases to test to get answers.

Or, you could spend valuable test time trying to find the ‘trick’ shown in the video.

[Walter_Carnielli]

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

[giuliofalco]

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

[tombufford]

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.

[EduardoDuvane-mcdm]

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

[yogamulyadi]

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

[nikolayguzman]

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

[jameslouder]

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

[keithwillenson]

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

[Mofiac]

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