A NICE USA OLYMPIAD: SOLVE FOR a+b=? | Algebra

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[@pas6295]

The common denominator is an. So the equation becomes b+a/ab. =1/11. Cross multiplying you get 11a+11b=ab. By bringing RHS to LHS side you get 11a+11b-ab=o. By squaring you get 121asquare plus 121bsquare-ab=0. Quadratic equation. So you have two roots one is a Ani the other b.

[@Alamaths]

Wow, thank you so much. Are you a professor of maths?

[@victornikolaev3557]

Решал задачу как и ты. Привет из Донецка, ДНР!

[@ManojGupta-bp5ws]

Not clear.?

[@icheichigo]

I heard a trick : 1. facterize 11=1*11 2. calculate 1/1 + 1/11 = 12/11 3. divide both sides in the above equation by 12, we get 1/12 +1/132= 1/11

[@Alamaths]

Thank you. Have you subscribed?

[@donsena2013]

Initial condition : a, b, are both integers

[@Alamaths]

Thank you.

[@rorydaulton6858]

Seeing the method of solution, the condition is that a and b are positive integers. If negative integers are allowed then a-11 and b-11, being factors of 121, could also be -1, -11, or -121. Checking those possibilities yields a = b = 0, which must be rejected as not satisfying the original equation, or a = -110 and b = 10 or the reverse. So an additional value for a + b is -100: if negative integers are allowed.

[@nasrullahhusnan2289]

@@donsena2013 : Two unknowns but only one equation is given. Usually additional info as the second equation is that the two unknowns are integers. That's why I ask that the given equation is a diophantine one. Surpraisingly the solutions are all integers despite no additional information.

[@nasrullahhusnan2289]

@@rorydaulton6858: a-11=-1 --> a=10 and b-11=-121 --> b=-110. Note that a≠0 and b≠0. Note also that (1/a)+(1/b)=(1/10)+[1/(-110)] =(1/10)-(1/110) =(11-1)/110 =10/110 =1/11 as the given equation

[@kcirebla]

Solución en los complejos a = (1+i raíz(43))/2, b= (1-i raíz(43))/2 Se llega al partir de que a+b = 1 y ab=11 Tenemos (a+b)²= 1 = a² + 2ab + b² a²+b² + 22 - 1 = 0 tenemos que b= 1-a a²+(1-a)²+22-1=0 2a²-2a+22+(1-1)=0 a²-a+11=0 Formula general se tiene la solución (1±i raíz(43))/2

[@Alamaths]

Wow 😮, thank you so much. Are you a professor of maths?

[@nasrullahhusnan2289]

(1/a)+(1/b)=1/11 • Note that there are 2 unknowns in the equation, but only one equation is given. Does it mean that it is s diophantine equation? • After getting (a-11)(b-11)=121, 121 is only factored as 1×121, 121×1, 11×11. All of them yield integer solution for a and b. Inded factoring 121 as (-11)×(-11) will yield a=b=0. But why -1×(-121) is not considered as a possibility? If a-11=-1 --> a=10 and b-11=-121 --> b=-110. As check (1/10)+(-1/110)=(1/10)[1-(1/11)] =1/11 a+b=-100

[@Alamaths]

Thank you so much for this comment. However, it can be considered as diophantine equation, but we just decided to go along with put integers z+ that must give us real solution. Here is another new video: OLYMPIAD PROBLEM. kzbin.info/www/bejne/Zqekn2ltotqEmassi=oCu8OglAm9R8O0l0

[@Alamaths]

By the way, are you professor of maths?

[@pinjalabhagiradha4897]

I think one more solution 1/a + 1/b = 1/11 (a+b)/ab = 1/11 a+b = 1. ......(1) ab = 11 b = 11/a Substitute in eq.(1) a+ 11/a = 1 a^2 + 11 = a a^2 - a + 11 = 0 Therefore a = (1 ± √(1-4.11))/2 = (1 ± i√43)/2 So a = (1 + i√43)/2 b = 1-a = 1- (1 + i√43)/2 = (1 - i√43)/2

[@Alamaths]

Wow, thank you so much 💖. Are you a professor of maths?

[@fouadhammout651]

كنت أتمنا لو كان لي منزلا خاص لوحدي ولكن وزارة التربية الوطنية بداخلها الدرك لم يريدان أصبحت كأنني أسكن بداخل محطة القطار الداخلون والخارجون كثر

[@Alamaths]

Wow, thank you so much 💖. Where are you watching from?

[@murdock5537]

This is amazing, many thanks, Sir!

[@Alamaths]

Glad you liked it!. Thank you

[@marinapiskun6423]

Це просто супер.Все ясно и дотепно.

[@Alamaths]

Thank you so much. Are you a professor of math?

[@davidtaran952]

11a+11b=ab 11(a+b)=ab since 11 is a prime number a=kx11, k!=0 11(kx11+b)=11kb kx11+b=kb 11k=b(k-1) if k=1, then a=0 -> not acceptable b=11k/(k-1), k!=0,1 a=11k or a=11k, k=-1, +-2, ... b=a/(k-1) a+b=11k+11k/(k-1)=11k(1+1/(k-1))=11k*k/(k-1)=11k^2/(k-1), k=-1, +-2, ... for example k=-1 a=-11 b=11/2 1/a+1/b=-1/11+2/11=1/11 a+b=11k^2/(k-1)=11/(-2)=-11/2 a+b=-11+11/2=-11/2

[@Alamaths]

Wow, thank you so much. Are you a professor of maths?

[@davidtaran952]

@@Alamaths You made me smile. Academic degrees have never interested me. I took a minimal pension thanks to Zimmer Biomet, where I worked as a worker on the two-port press for almost 6 years. Before that I worked as a cleaner, a dishwasher in restaurants, and before that as a bicycle mechanic. True, before immigrating, I single-handedly worked on solving burning high-tech, technical, software, algorithmic and military problems, and was always successful. But in this country, all my previous experience turned out to be of no use to anyone and I had the honor of learning the other side of life, about which I had very vague ideas.

[@jim2376]

Use the formula. a = 11 + 1 and b = (11 + 1) x 11. So 1/12 + 1/132 = 1/11. Obviously the commutative property applies to the LHS. If the author intends a = b then the problem is trivial and the answer is 1/22 + 1/22.

[@Alamaths]

Wow ❕, thank you. Are you a professor?

[@jim2376]

@@Alamaths No.

[@Alamaths]

Thank you

[@user-ji5su2uq9m]

What is the domain of a,b? integers? positive integers? in case of integers, we have two more cases -121 x -1 and -1 x -121 (-11 x -11 not allowed.) which means (a,b) = { (10,-110),(-110,10) } , a + b = -100.

[@Alamaths]

Yes, positive integers: 1×121 or 121×1 and 11×11

[@Alamaths]

Are you a professor of maths?

[@user-ji5su2uq9m]

@@Alamaths No. I had studied math at university 50 years ago.

[@Alamaths]

@@user-ji5su2uq9m Wow, that's good. Thank you so much.

[@zadacha-kz9954]

a=10, b=-110

[@Alamaths]

Wow, thank you so much. Are you a professor of maths?

[@zadacha-kz9954]

(a-11)*(b-11)=(11-a)*(11-b)

[@SrisailamNavuluri]

1/11=2/11=1/11+1/11 (a,b)=(11,11) 1/11=(11+1)/12×11=11/12×11+1/12+11 =1/12+1/132 (a,b)=(12,132) a+b=11+11=22 or 12+132=144

[@Alamaths]

Thank you. Are a professor?

[@SrisailamNavuluri]

@@Alamaths retired teacher.

[@Alamaths]

@@SrisailamNavuluri that is why. Thank you so much.

[@CharlesChen-el4ot]

When ab = 11, a+b = 1

[@Alamaths]

Wow, thank you so much 💖. Are you a maths teacher?

[@user-zj9vt9el3w]

1/a+1/b=1/11 : (1/a+1/b=2/22) แสดงว่าถ้าa=b :a=11,b=11

[@Alamaths]

Thank you so much.

[@user-xh3ih4ks9y]

a=12 b=12×11=132. 則 a+b=144

[@Alamaths]

Wow, thank you so much. Are you a maths teacher?

[@Alamaths]

Hello My Dear Family😍😍😍 I hope you all are well 🤗🤗🤗 If you like this video about How to solve this math problem Please like and subscribe to my channel as it helps me a lot，🙏🙏🙏🙏

[@Girl-dz8ck]

From now I am also a part of your family 😊💗💗⚡

[@ToanPham-wr7xe]

😮

[@Alamaths]

Thank you so much. Where are you watching from?

[@virendrasule3258]

The equation has no integral solutions since a, b are non zero.

[@Alamaths]

Thank you so much. Are you a professor of maths?

[@virendrasule3258]

@@Alamaths send such interesting problems. I m a cryptographer.

[@OLDHIPPO-s6i]

a 와 b가 다른수라는 조건이 없고, a와 b는 같은 수라고 한다면 a=b=22입니다 1/22 + 1/22 = 2/22 = 1/11 입니다 😅😅😅

[@Alamaths]

Wow, thank you so much 💖. Are you a professor of maths?

[@kennethkan3252]

a+b=22+22=44 2/22=1/11

[@Alamaths]

Wow , thank you so much. Are you a maths teacher?

[@kennethkan3252]

@Alamaths No, I only interest in math's.

[@jst8922]

1:38-2:18 to all who got lost here. 121 = 11* 11 11*a + 11* 11 = 11*(a-11)

[@Alamaths]

Thank you. Rewatch it you should understand ❤

[@kenjiosumi6471]

1/11 = (1/11)*(12/12) = 12/ (11*12) = (11+1)/(11*12) = 11/(11*12) + 1/(11*12) = 1/12+1/132

[@Alamaths]

Wow, excellently. Thank you so much. Are you a professor of maths?

[@nailabrain6714]

Another solution a=5.5 b=-11

[@Alamaths]

Are you a l Professor? Thank you so much.

[@jim2376]

No restriction to positive integers?

[@NessunoIncognito]

a=1/22 b=1/22

[@Alamaths]

Thank you. Are you a professor?

[@przemysaw9081]

121 is albo = -1×(-121)

[@przemysaw9081]

121=(1/2)*242

[@przemysaw9081]

121=3×121/3 and a lot of more

[@Alamaths]

Thank you so much 💖. I have also solved -1×-121 watch it til the ending

[@ManojGupta-bp5ws]

Question is wrong. At least it must be stated that solve for integers.

[@Alamaths]

Thank you so much. Are you a professor of maths?

[@jason-nt6pw]

44?

[@Alamaths]

Thank you so much. Are you a professor of maths?

[@dellagobaikal8205]

Sorry. General solution in my view can be found from 2 equations a+b= K a×b=11×K, Where K is a valnurable coefficient. Sorry, if Im wrong, not a professional math....

[@Alamaths]

It seems so anyway. Thank you so much. Are you a maths teacher?

[@dellagobaikal8205]

@@Alamaths no. Im a retired corporate lawyer, many years ago graduated from a soviet school. Many thanks for your interesting lessons. 👍

[@Alamaths]

@@dellagobaikal8205 thank you so much.