TKOF MADE WITHOUT CLRNC AT QUITO; ECUADOR. RADIO PROBS WERE A FACTOR.

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PUSHED BACK ACFT FOR ENG START AND TAXI. WE WERE IN QUITO; ECUADOR. WE CALLED GND FOR TAXI AND CLRNC. WE WERE UNABLE TO CONTACT GND ON RADIO #1. WE ALSO TRIED RADIO #2 WITH SAME RESULTS. WE FINALLY CONTACTED GND ON #2 RADIO; BUT WAS STILL GARBLED. WE WERE GIVEN TAXI INTO POS AND HOLD WHICH WE DID. AFTER 30 SECONDS; WE WERE GIVEN TKOF CLRNC. I ACKNOWLEDGED CLRED FOR TKOF AND WE TOOK OFF. THE CAPT DID NOT HEAR THE CLRED FOR TKOF CLRNC. I TOLD HIM I DID AND ACKNOWLEDGED IT. AFTER TKOF; TWR SAID WE DID NOT HAVE TKOF CLRNC. I BELIEVE THE RADIOS HAD A PROB WHICH LED TO THIS PROB. I HEARD THE TKOF CLRNC AND ACKNOWLEDGED IT. MAYBE MY XMITTER WAS NOT WORKING. THIS ACFT HAS A HISTORY OF RADIO PROBS. SUPPLEMENTAL INFO FROM ACN 493619: WE WERE UNABLE TO DETERMINE IF IT WAS A STUCK MIKE OR THE TWR HAD A RADIO PROB. I DID NOT HEAR THE CTLR SAY CLRED FOR TKOF. I SAID 'I DIDN'T HEAR THAT ARE WE CLRED FOR TKOF?' THE FO SAID 'YES.' I ASSUMED WITH THE RADIO PROBS WE HAD ON TAXI I HAD MY VOLUME DOWN TOO LOW. MY CONCERN WAS THAT SOMEONE MAY HAVE HAD A HAND-HELD RADIO XMITTING AT THE END OF THE RWY AND JAMMING THE FREQS.