Shalom, Can a Jewish man, according to halacha, participate in mixed gender cycling race around Jerusalem?

Shalom, Thank you for your question. The issue of mixed gender events is addressed from several different fronts in halachic sources. As you are surely aware there are several different opinions and approaches to the issue. All agree that mixed events that involve or lead to improper behavior or improper thoughts are forbidden. Mixed dancing, mixed swimming in less than modest swimwear, and so on, are forbidden according to all opinions. Certainly events where there is physical contact between the sexes, such as contact sport, are forbidden. Events where these issues are less problematic, but none the less involve a mixing of the genders – such as a mixed orchestra, lectures, hikes or picnics, are a matter of debate. In some communities these are avoided, and in others they are approved of. This debate has sources that can be found in the early Rabbis (Rishonim) (see Rav Y.H. Henkin, Equality Lost, p 76-86). For example, many Germanic communities had mixed seating at weddings, while others had separate rooms for men and women. In relation to your question. Even though I imagine that the women cyclists are not dressed in a manner approved of by Jewish law, the male cyclists can certainly focus their eyes on the road, and avoid looking at them, as they (should) do when riding their bicycles on a daily basis anywhere where there are immodestly clad people. The race itself does not bring the sexes into improper behavior. That being so, in my opinion, a person will have to decide themselves if they feel it appropriate that they take part or not. Those who are accustomed to totally separate events for men and women should probably avoid the mixed event – whilst those from communities that follow the more lenient opinion will not find the situation troubling. With all that said - if a particular man knows himself that attending such an event will lead him to improper thoughts or cause his eyes to stray, he should avoid the event, and strengthen himself with Torah learning. Blessings.