Can Muslim men compete in bodybuilding shows?

Usually not in the standard format. Most bodybuilding shows require posing in small trunks that expose the awrah in front of an audience, and most scholars say that isn't allowed. A category with longer shorts is the only realistic option, together with a humble intention, no drugs and no extreme dehydration.

The posing trunks problem

The awrah for men is from the navel to the knee, according to most scholars. Standard stage wear in bodybuilding exposes much of the thighs, in front of judges, other competitors and a crowd. The Prophet ﷺ said, "A man should not see the private parts of another man" (Sahih Muslim 338), so backstage changing and shared posing areas can also become a problem. See What is the awrah for men in sports?

What about shorts categories?

Some federations have physique categories where competitors wear board shorts instead of trunks. These shorts are often still short, so check the rulebook, ask whether longer shorts are allowed and make sure they reach the knee. Even then, the other points below still apply. If you can't compete covered, it's better to keep training for your own health and strength. See Is bodybuilding haram?

Showing off and intention

Wanting to look good isn't a sin, but pride is a serious danger. The Prophet ﷺ said that whoever has the weight of a mustard seed of pride in his heart will not enter Paradise, and explained that pride is disdaining the truth and holding people in contempt (Sahih Muslim 91). The Quran warns against arrogance too:

وَلَا تُصَعِّرْ خَدَّكَ لِلنَّاسِ وَلَا تَمْشِ فِى ٱلْأَرْضِ مَرَحًا "And do not turn your nose up to people, nor walk pridefully upon the earth." Quran 31:18

Ask yourself honestly why you want to compete: discipline, a challenge and personal growth are fine goals. Craving admiration or comparing yourself with others is where the heart can go wrong.

Harm and drugs

Competition prep often involves very low body fat, water and salt manipulation and pressure to use banned drugs. Islam forbids harming yourself. See Are anabolic steroids haram in Islam?

What about prize money?

Prize money paid by organisers or sponsors is generally fine. It is only a problem when competitors' own stakes make up the prize pot. See Is prize money from sports competitions halal?

Key points

Standard posing trunks expose the awrah, so most scholars say no.
Only a category where you stay covered from navel to knee could be considered.
Guard your intention against pride and showing off.
Avoid steroids and extreme dehydration, and ask a scholar about your specific case.

Tip: Before entering any show, email the organisers and ask exactly what competitors wear on stage. Then take their answer to a local imam or scholar.