Legal Priorities Project — program: Legal Priorities Summer Institute (473 applications, 35 admitted from 18 countries in 2022). Law & Longtermism Network (~300 members). Writing Competition with prizes $2,500-$30,000.

Note: The source text confirms that Legal Priorities Project ran these three programs in 2022 (Legal Priorities Summer Institute, Writing Competition, and Law & Longtermism Network are all mentioned). However, the excerpt provided does not contain the specific numerical details claimed: the number of applications (473), admissions (35), countries represented (18), network membership (~300), or prize amounts ($2,500-$30,000). The source mentions 'hundreds of applications from students and academics' for events generally, but does not specify the exact figures for the Summer Institute. The claim is partially confirmed in terms of program existence but unverifiable regarding the specific statistics.