When inequality turns to equality

Geometry Level 3

Find the number of ordered 5-tuple ( u , v , l , m , n ) (u,v,l,m,n) where u , v , l , m , n ∈ [ 1 , 11 ] u,v,l,m,n \in\left[1,11\right] which satisfy the following inequation. ( 2 sin ⁡ 2 u + 3 cos ⁡ 2 v ) ( 3 sin ⁡ 2 l + cos ⁡ 2 m ) ( 5 cos ⁡ 2 n ) ≥ 720 \LARGE\left({2}^{\sin ^{ 2 }{ u }+3\cos^{2}{v}}\right)\left({3}^{\sin ^{ 2 }{ l }+\cos^{2}{m}}\right)\left({5}^{\cos^{2}{n}}\right)\ge720

Problem is not original.

54 432 108 216

Rohit Ner by Rohit Ner , 22, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

At first we realise its an equality. Then we check limits and these can be written (modifying) as[π/2,3π/2]. Now from factorisation of720 and comparing powers we get (sin(u))^2=(sinl)^2=1 and(cosv)^2=(cosm)^2=(cosn)^2=1. Now we solve (sinx)^2=1 which gives 4solutions in given limit and for the cos parts we get 3solutions. So at last number of unordered pairs=4×3×4×3×3=432 which is the required answer.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...