Spain | A Nice Algebra Problem | Math Olympiad

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[@yuki2go]

(abc)² = 100 * 200 * 300 = 6000000 abc = ±1000√6 a + b + c = abc/bc + abc/ac + abc/ab = abc(1/200 + 1/300 + 1/100) = abc/100(1/2 + 1/3 + 1) = ±(55√6)/3

[@Yuri_Kravets]

@yuki2go дал ЭЛЕГАНТНОЕ аналитическое решение, которое правда не дает индивидуальных значений a, b и c. Чего, впрочем и не требовалось. Для их нахождения потребовались бы дополнительные шаги. Позволю объяснить решение по шагам, надеюсь будет полезно школьникам, подобным мне. 1. (abc)² = 100 * 200 * 300 = 6000000 Это следует из перемножения всех трех уравнений: (ab)(bc)(ca) = 100 * 200 * 300 2. abc = ±1000√6 Это квадратный корень из 6000000. Знак ± появляется, так как квадратный корень может быть положительным или отрицательным. 3. a + b + c = abc/bc + abc/ac + abc/ab Это ключевой шаг. Мы выражаем a, b и c через их произведение и попарные произведения. 4. = abc(1/200 + 1/300 + 1/100) Здесь мы заменяем bc, ac и ab их значениями из исходных уравнений. 5. = abc/100(1/2 + 1/3 + 1) Упрощаем выражение, вынося общий множитель. 6. = ±(55√6)/3 Подставляем значение abc = ±1000√6 и упрощаем: ±1000√6/100 * (1/2 + 1/3 + 1) = ±10√6 * (11/6) = ±(55√6)/3

[@user-rw7qb1ef1o]

@@Yuri_Kravets лишние действия в ответе. ±110/√6 Таки красивше))

[@MarcoPolo-xu9te]

It's a nice problem, but I guess you could have solved it faster if you had first isolated b , this way: b = 100/a; b = 200/c; c = 2a. I found a = 5 sqrt 6; c = 10 sqrt 6; b = (10/3) sqrt 6; a+b+c = (55/3) sqrt 6; but I did not have to deal with big numbers, like root of 6050. I wonder if my solution has something wrong, but I think it's easier. Anyway, thanks for the problem.

[@BruceLee-io9by]

I solved the problem in the same way. Final result: +- 110/sqrt6 giving the same result as +- 55 sqrt6/3.

[@QWERTYQWERTY-iv7pi]

agree with just see i know that c^2=300*2 c=10sqare6 so dont life to hard

[@sggua]

absolutely agree 👍🏻

[@user-klepikovmd]

I solved the same way

[@nenedillats2999]

After <a href="#" class="seekto" data-time="194">3:14</a> (the value of a^2, b^2 and c^2), you can continue by : ab > 0, bc > 0 and ac > 0 so a, b and c are positives. And now you know the value of a, b and c i.e the value of a + b + c. It's faster. Your way after <a href="#" class="seekto" data-time="194">3:14</a> is beautiful, I appreciate your idea.

[@ericrobin9554]

AB=100 B=100/A so from BC=200 we have 100C/A=200 ie C=2A Then from CA=300 we have 2A^2=300 ie A=sqrt(150) We can deduce that: C=2sqrt(150) since C=2A and B=200/(2sqrt(150))=100sqrt(150)/150

[@ЧувакИзКосмоса]

But if you solve it isolating every value, you'll get only the positive answer.

[@joenorsworthy]

I was imagining a longer possible approach, but I could not think of one.

[@soshakobyan3123]

Olympiad problem? Is this a joke?

[@mukeshmishra665]

Yes, it's nothing but indeed a joke

[@user-klepikovmd]

Probably its Olympiad for a 5 year school student. For them this is very hard

[@meurdesoifphilippe5405]

Or you multiply everything to get (abc)^2 then you deduce abc and a by dividing by bc and so on.

[@ebEliminator]

I started with, [ab = 100, bc = 200], [200 = 2 * 100], [bc = 2ab], cancel out the b's so that [c = 2a], then plug it into ca = 300 so that 2a^2 = 300 and solve for a.

[@AntonioTiagoNunes-kf6ml]

Beautiful.

[@Yuri_Kravets]

Исходные уравнения: 1. a *b = 100 2. b *c = 200 3. c *a = 300 Шаг 1: Найдем b через a b = 100/a Шаг 2: Подставим b в уравнение (2) 100/a *c = 200 100c = 200a c = 2a Шаг 3: Подставим c = 2a в уравнение (3) c *a = 300 (2a) *a = 300 2a^2 = 300 a^2 = 150 a = √150= 5√6 Шаг 4: Найдем b через a b = 100/a = 100/5√6= 20√6 Шаг 5: Найдем c через a c = 2a = 2 *5√6= 10√6 Итак, решение: a = 5√6 b = 20√6 c = 10√6

[@user-se5wr3sd5y]

b -?

[@user-rw7qb1ef1o]

1) a = 30/√6 b = 20/√6 (!!!) c = 60/√6 2) a = --30/√6 b = --20/√6 (!!!) c = --60/√6 a+b+c = ± (30+20+60)/√6 = ±110/√6

[@user-rw7qb1ef1o]

Да... И кстати, все исходные могут быть либо положительными, либо отрицательными. Произведения отрицательных также дают положительный результат.

[@jopefon]

Muy interesante.

[@XfeeXg]

ab=100 ; bc=200 So bc=2ab or c=2a ca=300=2a.a=2a*2 a*2=150 a= 5.(6)^2 b=100/(5×6^2) b=(10×6^2)/3 c=10×6^2 a+b+c=(5+10/3+10)×6^2 a+b+c=(55×6^2)/3

[@onkotonkoblu]

A nice algebra problem

[@user-nv7om8bk7j]

A= 150/ кор.150 B= 100/ кор.150 С= 2× кор.150 Сумма 550/кор.150

[@philipbao3725]

a^2=150 b^2=200/3 a=5*sqrt(6) b=10/3*sqrt(6) c=10*sqrt(6)

[@emmanueltanguy4670]

bc/ab=200 /100 c=2a ac=a*2a 2a^2=300 a=5√6 c=10√6 ca/ab=300/100 c=3b b=(10/3)√6

[@user-yw7qg6zz5i]

Nossa 😅 bc/ab=c/a=2 ca*(c/a)=c^2=600 c=✓600=10✓6 a=300/10✓6=5✓6 b=100/5✓6=10✓6/3 😅😅😅

[@SOMNATHGOPE234]

Thanks

[@steveneagan3953]

I get a= (3b)/2 b=√66.67 c=3b I had to use a calculator to check the answer and it came surprisingly close. ab=100.001 bc=200.001 ca=300.001

[@alestee4241]

a=±5*sqrt(6); b=±(10/3)*sqrt(6); c=±10*sqrt(6).

[@Interl7g]

1,100 2,50 4,25 5,20(100)

[@lecombustor3571]

A, b, c are positive, so sum can't be negative.

[@kurtmager1626]

Not necessarily. ab, bc, and ca are positive, but (-a)(-b)=ab (-b)(-c)=bc (-c)(-a)=ca

[@icilyann1000]

By this way, you will need one week to solve an exams, Faster sir,

[@SidneiMV]

(bc)/(ab) = 2 => c = 2a ac = 300 = 2a² => a = ± √150 => c = ± 2√150 ab = 100 => b = ± 100/√150 a + b + c = ± (3√150 + 100/√150) a + b + c = ± (550/√150) a + b + c = ± [550/(5√6)] a + b + c = ± (110/√6) a + b + c = ± (110√6/6) *a + b + c = ± (55√6/3)*

[@user-rw7qb1ef1o]

a + b + c = ± 110/√6 Зачем что-то ещё? Здесь два действия, а потом уже аж три.

[@user-iw2zn2mj3c]

Легко

[@user-ql2wu5uc4p]

A=√150 B=100÷√150 C =2√150 Solved in mind without pen papaer😂😂

[@SongMaiHor]

Why ab×ca/bc

[@unidok_]

Bec

[@andryvokubadra2644]

ab = 100 bc = 200 ca = 300 =========== Metode I ab*bc*ca = 100*200*300 = 100*6*10000 (abc)² = 100*6*10000 abc = ±10*√6*100 = ±1000√6 a = ab*ca/abc = 100*300/(±1000√6) a = ±30/√6 b = ab*bc/abc = 100*200/(±1000√6) b = ±20/√6 c = bc*ca/abc = 200*300/(±1000√6) c = ±60/√6 a + b + c = ±(30+20+60)/√6 a + b + c = ±110/√6 ========== Methode 2 ab/bc = a/c = 100/200 = 1/2 a/c = 1/2 --> c = 2a bc/ca = b/a = 200/300 = 2/3 b/a = 2/3 --> b = 2a/3 ca = 300 (2a)a = 300 2a² = 300 a² = 150 a = ±5√6 a + b + c = a + 2a/3 + 2a a + b + c = 11a/3 a + b + c = 11(±5√6)/3 a + b + c = ±(55√6)/3 a + b + c = ±110/√6 ========= Method 3 As video ========= • • • Do we other method? Comment below 🙏🏻🙏🏻🙏🏻

[@ZeroGravityDog]

Your methods give the right answer (method 2 seems simplest), but the convention seems to be to put the sqrts upstairs so (55/3)*sqrt(6) is preferable to 110/sqrt(6)

[@andryvokubadra2644]

​@@ZeroGravityDog thank you. . . You're true. . .. 2nd methode is simplest than other one I know 🍵🍵🍵

[@nenedillats2999]

See my answer for another way.

[@andryvokubadra2644]

​@@nenedillats2999you're true. . . 👌🏼👌🏼👌🏼

[@user-ye6go4ft7q]

Как-то это не убедительное для экзаменов ??? Простая трата времени находить ответ в таком виде !!! Атеперь встав и проверь-правильно-ли получил ???