ENRTE AT FL350 IN CANADIAN AIRSPACE; A B767 CREW HAVE A SERIOUS CONFLICT WITH A B777 OPPOSITE; ALSO AT FL350.

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HAD BEEN CRUISING AT FL350 FOR AT LEAST AN HR; AND CHKED IN TO EACH CTR WITH OUR FLT LEVEL AND RECEIVED ACKNOWLEDGEMENT EACH TIME. AT APPROX XA20Z; WE HEARD A GARBLED AND UNREADABLE XMISSION. WE HEARD A SECOND; ALSO UNREADABLE XMISSION. AT THIS POINT; WE QUERIED CTR BUT THEY DID NOT RESPOND. ABOUT THIS TIME; WE RECEIVED A TCASII TA SHORTLY FOLLOWED BY AN RA; WHICH THE PF RESPONDED TO. WE DSNDED AS INSTRUCTED AND THE OTHER ACFT (ACR Y) CLBED. AT XING; SEPARATION WAS APPROX 700-1000 FT. WE LEVELED AT FL340 CLR OF CONFLICT. WE ASKED THE CTR ON 121.5 TO GIVE US A USABLE CTR FREQ. WE HEARD THEIR 121.5 XMISSION LOUD AND CLR AND SWITCHED TO THE NEW FREQ. WE CHKED IN ON THAT FREQ AND HEARD CTR CLRLY. WE HAD BEEN ACTIVELY MONITORING 121.5 THROUGHOUT THE ENTIRE FLT PER SOP. SUPPLEMENTAL INFO FROM ACN 605287: FLYING AS FO WHILE GIVING RTE QUALIFICATION; PF. AFTER COMING BACK FROM BREAK; WE WERE CRUISING AT FL350 FOR 2+ HRS AFTER CLBING AFTER COAST-IN. UNDER RADAR CONTACT. NOTICED ACFT AT OUR ALT ON EHS. I COMMENTED TO FO THAT 'THIS IS TOO CLOSE; WE'RE ONLY 5 MI APART.' REALIZED ACFT WAS ON COLLISION COURSE AND PREPARED FOR TCASII WARNING. RESPONDED TO RA IMMEDIATELY. DSNDED TO FL344. FLT ACR Y CLBED TO FL354 ON TCASII. ACFT FLEW DIRECTLY OVER US. STARTED TO CLB BACK TO FL350; BUT ATC CLRED US TO FL340. CLRED ACR Y TO FL360. DURING CONFLICT; I NEVER SAW THE ACFT AS I WAS 'INSIDE' FOLLOWING TCASII GUIDANCE. FO DID HAVE ACFT IN SIGHT. SUPPLEMENTAL INFO FROM ACN 605564: I WAS THE CAPT RECEIVING AN IOE FOR INITIAL OVERWATER/ETOPS. I WAS ON MY BREAK IN THE FIRST CLASS CABIN. THE RELIEF PLT WAS IN THE L SEAT AND THE LCA WAS IN THE R. FLT WAS AT CRUISE AT FL350; SHORTLY AFTER COASTING INTO CANADA; AND WAS TALKING TO AND IN RADAR CONTACT WITH MONTREAL CTR. AS RPTED TO ME; THEY RECEIVED SEVERAL SOUNDS OF CARRIER-ONLY OVER THE RADIO; THEN RECEIVED A TCASII TFC WARNING. A FEW SECONDS LATER; AN RA 'DSND' WAS SOUNDED. WE DSNDED APPROX 400 FT AND THE OTHER ACFT (ACR Y) CLBED ABOUT THE SAME. THE ACR Y ACFT WAS COMING OUT OF THE SUN AND WAS NEVER SEEN UNTIL PASSING OVERHEAD AT LESS THAN 1000 FT. IN THE CABIN IT WAS FELT AS SMOOTH BUT DEFINITELY AN ACFT MANEUVER. SEVERAL PAX RAISED THEIR WINDOW SHADES TO LOOK OUT. AFTER THE MANEUVER WAS COMPLETE AND BACK TO ALT; MONTREAL'S ONLY RESPONSE WAS TO ISSUE A FREQ CHANGE. I WAS NOTIFIED UPON RETURNING TO THE FLT DECK. THANK YOU FOR THE TECHNOLOGY; THIS WOULD HAVE BEEN A COLLISION. SUPPLEMENTAL INFO FROM ACN 605289: LOCATION: 175 MI S VIRSO ON NCA 30. ENRTE; OBSERVED CONVERGING ACFT SAME ALT. FO ATTEMPTED TO CONTACT CTR -- UNABLE. TA; THEN RA. CLBED 500 FT. OTHER ACFT (ACR Y) DSNDED 500 FT. PASSED ALMOST DIRECTLY OVER ACR X. STILL UNABLE TO CONTACT CTR. FINALLY TALKED TO DIFFERENT CTLR ON DIFFERENT FREQ. DO NOT KNOW NATURE OF CTR'S RADIO PROB; BUT WE HAD BEEN AT FL350 SINCE ZBW -- WELL OVER AN HR EARLIER.