I found some more info about the International QRP Day (June 17th) in G-QRP Clubs Members Handbook. G-QRP has a Club Trophy called "The International QRP Day Plaque".
Here are the rules from the members handbook:
Annually, on 17th June, International QRP Day. Power and bands as for Chelmsley Trophy*. Operation for a maximum of 6 hours in not more than two periods. Contacts are with Region 1 Country; normal QSOs and no serial numbers.
SCORING: Each Region 1 country contacted on each band counts 1 point. The total score is the total of IARU Region 1 countries contacted all bands used. Only one contact per country per band is allowed irrespective of mode. Stations contacted may be QRO.
ENTRIES: Give name, address, call, power, equipment, and time/call/band for each contact. Summary to give band and overall scores. Logs to Communications Manager by 17th July. Trophy and book token winner plus runner-up certificates.
* All authorized bands 1.8 to 28 MHz Modes CW and SSB. Power not to exceed 5W RF output (cw) or 10W PEP (SSB)
JARTS WW RTTY contest
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