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Moore birdied five of the first seven holes to set a cracking pace, posting a seven-under 65 to head resurgent Jordan Spieth by one stroke with half the field back in the clubhouse.
But most eyes were on Woods in his first competitive round since missing the cut at the PGA Championship two weeks ago.
The 43-year-old needs one more victory to match Sam Snead's record 82 wins on the PGA Tour, though perhaps more importantly the Memorial offers Woods a chance to assess his form heading to the U.S. Open at Pebble Beach in two weeks' time.
After winning the Masters in April, Woods requires three more major titles to match the record of 18 held by Jack Nicklaus, this week's tournament host.
"One of those days (where) it could have easily flipped, I feel like I could have got a few more out of it," Woods said of his score.
"It's so soft out there it's definitely gettable and look at the scores, the guys are doing it."
Leader Moore said a par at the first hole had set a positive tone that primed him to run off five birdies in rapid-fire fashion.
"On the first hole didn't hit a good tee shot and hacked something out short of the green, chipped to five to six feet and rolled that one in," said the five-times PGA Tour winner.
"It was nice to start out that way. Those are the putts that haven't been going in as much for me recently. Momentum went from there.
"I hit some great mid-irons today. It was good solid golf but mostly just capitalising and making the ones I should make."
Spieth was tied for the lead until bogeying his penultimate hole, but nonetheless it was another promising performance on the back of his recent revival after a slow start to the year.
The three-times major winner tied for third at the PGA Championship at Bethpage two weeks ago and last week finished equal eighth at Colonial Country Club in Fort Worth.
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