Hungarian Junior Math Olympiad Problem - Algebra

Hungarian Junior Math Olympiad Problem - Algebra

[adandap]

There's also a nice geometric approach to this problem. Take 2D vectors A = (a, b) and X = (x, y). Then we know that |A| = 9 and |X| = 11. The dot product is ax+by = A.X = |A| |X| cos(t) where t is the angle between them. But that gives cos(t) = 99/(9x11) = 1, so the vectors are co-linear. Thus their cross product must be zero, so ay - bx = 0.

[floresastuvilca]

Friend, I have seen like 30 videos of yours, I always failed when I tried them, or sometimes I gave up, but today, THIS PROBLEM CAME OUT, I am very happy, I am only 17 years old, but with the education I have, it makes me think that everything is possible, thanks,

[田村博志-zy]

By Cauchy-Schwartz inequality, we have
( ax + by )^2 <= ( a^2 + b^2 )( x^2 + y^2 ).
In the case of the problem the equality holds
99^2 = 81・121.
So we have by the equality condition
a : b = x : y,
ay = bx.
Hence we obtain ay - bx = 0.

In the case of 3 dimensions
a : b : c = x : y : z
is equivalent to the equalities
ay - bx = bz - cy = az - cx = 0.

[prabhudasmandal]

Amazing problem nicely solved. Thanks. Both having value of 99 square are equal, without subtracting we can say so.

[yongningcao]

I use trigonometry to solve it. Formula of sine and cosine of the sum of two angles.

[codergold]

One possible value for each of them:
X=0
a=0
Y=11
b=9

[gioacchinomanduano]

Hi i find another solution but i don't know if is right:
a^2=81-b^2=(9+b)(9-b)
9+b=9-b because a is a square and we have b=0 and i do the same thing with the second equation and we have a=9 and x=11 and b=0 and y=0

[fouadhammout]

Saraha ma3andich zhar la f tssahib la f zwaj walit tangoul tawahad maynawad tawahda Matahmal

[SpennyBoi]

quick question what level of maths is this?

[stickmanbattle]

I'm sleepy but I got the same solution noice

[sidiabdallamouemel]

It's so easy, we want problems hards

[falahalfadhel]

tan z = x / y, tan z = a/b, x/y = a/b, ,,,.ay = bx - bx = 0