The decision for a Christian to be involved in this type of event would be marginal at best. Phil 1:27 says,
"Only conduct yourselves in a manner worthy of the gospel of Christ"  and Eph 4:17-19 instructs us to

"walk no longer just as the Gentiles also walk , in the futility of their mind, being darkened in their understanding, excluded from the life of God because of the ignorance that is in them, because of the hardness of their heart; and they, having become callous, have given themselves over to sensuality for the practice of every kind of impurity with greediness."

Notice the word used here; "sensuality". While a person might argue that there's nothing wrong with a beauty contest, I don't think anyone could really argue that they are anything but sensual. And that is a troubling question if a Christian is really trying to live for Christ. Consider also

Mark 7:21-23
"For from within, out of the heart of men, proceed the evil thoughts, fornications, thefts, murders, adulteries,  deeds of coveting and wickedness, as well as deceit, sensuality , envy, slander, pride and foolishness.  All these evil things proceed from within and defile the man."

2 Peter 2:2
"Many will follow their sensuality, and because of them the way of the truth will be maligned".

Now, if your question is, "does the Bible forbid beauty contests", the answer is no. However, the appropriate consideration might not be "is it permissable", but rather "is it the best choice". As Paul writes in 1 Cor 10:23
"All things are lawful, but not all things are profitable . All things are lawful, but not all things edify."