Asphalt Domination
Collect all Asphalt trophies.
Trophy type: Platinum
Congratulations on your Platinum!

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First First!
Finish in 1st place!
Trophy type: Bronze
Easy trophy and likely to be your first one. As the trophy states, you simply need to win any race. I got this on Los Angeles, Normal Race in "League Qualifiers."

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Three in a Row!
Finish in 1st place 3 times in a row!
Trophy type: Bronze
Just win three events in a row, (a 5-star rating is not required for this trophy.) I got this in the first league, "League Qualifiers." Just finish the race in first place on Los Angeles (Normal Race,) Cape Town (Elimination) and Chamonix (Normal Race) and the trophy will be yours!

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Contender
Win 10 multiplayer races.
Trophy type: Silver
See "Legend"

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Legend
Win 100 multiplayer races.
Trophy type: Gold
This trophy will likely take the most time to achieve as well as being your last one. I suggest playing on the Nassau or Moscow tracks and choosing the best car that you have. I averaged those tracks at around 2:15 and 2:07 per run respectively, so these would be your best choice if you wish to finish a race slightly quicker, but it's your choice and you can play any track that you wish.

If you seem to be having a hard time winning races, remember to always press the acceleration button just before the "Go!" which will grant you an extra boost bar, allowing you to start off with a 3x Nitro Boost. You should always be starting off with Nitro if you can.

If you are still having difficulty with this, then I suggest finding a boosting partner. The amount of people online is rather scarce and opponents have a tendency to disconnect if they're losing which will not award you with a win.

The boosting thread can be found here: Trophy Boosting Thread (Read First Post!)

Tip: Both boosting partners do NOT need to finish the race. After the first player wins the race, he/she can safely exit to the lobby and the game will count it as both a win and completion for them.

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Photo finish
Finish 1st in a MP race less than 1s ahead of the 2nd-place finisher.
Trophy type: Silver
This trophy is actually easier than it sounds, but it requires a bit of luck. It might come naturally whilst playing online so I wouldn't suggest specifically going for it until you have completed a minimum of around 50 races. I personally got it whilst boosting and this is how I did it:

Just before the end of Lap 3, I would stop just at the finish line and wait for my opponent to approach. As soon as the opponent was close enough, I would triple-tap the Nitrous button and finish just in front of him. I had finished 1.001 second before my opponent on one attempt and had not gotten the trophy, however on my next try, I finished just 0.004 seconds in front of him and the trophy unlocked. Just be persistent!

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Persistant
Complete 50 races.
Trophy type: Bronze
Self-explanatory, all you need to do is race 50 times and the trophy is yours. Will come naturally over time and you should not concern yourself about it.

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Veteran
Complete 100 multiplayer races.
Trophy type: Silver
See "Legend"

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Perfectionist
Obtain the maximum amount of stars for all events in a league.
Trophy type: Silver
This is actually a rather easy trophy, and all you need to do is achieve a 5-star rating in only one of the 20 leagues available. I suggest going for the first league, "League Qualifiers" as soon as you start the game so you can get it out of the way. In order to achieve a 5-star rating on the tracks within the "League Qualifiers" event, you need to meet the following criteria in these game modes:

Normal Race - Finish in first place.
Elimination - Finish in first place.
Cop Chase - Crash less than a specified number of times.
Beat 'em All - Obtain a set number of knockdowns.
You will also need to complete the two secondary objectives for each track.

Secondary objectives are shown during the loading screen under "Extra Bonus," but if you happen to miss them, you can always pause the game using the start button and look under "Objectives" to see them. The requirements for the secondary objectives most often consist of "Get X amount of drift points" and "Get X amount of takedowns." Don't worry if you don't get a 5-star rating on first try, as you can always retry later on and they are not too difficult.

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Accomplished
Finish every race in Career Mode.
Trophy type: Gold
As the trophy states, you need to finish every single one of the 100 races in Career Mode with a minimum of a 1-star rating, which is considered a pass. There are 20 leagues and each league consists of 5 events.

In order to obtain at least a 1-star rating in an event, you must reach the following criteria:

Normal Race - Finish in third place or higher.
Elimination - You need to stay in a high position to avoid being knocked out of the race.
Cop Chase - Do not crash more than a set amount of times and finish the race. Simply finishing counts as a pass.
Beat em' All - Perform a specific number of takedowns within a given time period.
Under Pressure - Finish the race without crashing more than the specified amount of times.
Urban Havoc - Crash into various objects on the sidewalks such as street lamps or benches (traffic in the road also counts.) Getting the minimum required (is shown on the top-right corner) will net you a "pass."
No track should prove too difficult, just frustrating for some of them. Persistence is key here - if you happen to be stuck on a track, then proceed further and come back later with a better car. This helped me a few times. This trophy will most likely be your last before moving onto the Online portion of the game.

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Lickety Split
Collect 25 Nitro power-ups in a single race.
Trophy type: Bronze
See "Velocius."

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Velocius
Collect 35 Nitro power-ups in a single race.
Trophy type: Silver
Nitro power-ups are the blue and orange bubbles that come in rows of three (two blue and one orange) and you should have no problems spotting any of them. Bigger tracks such as Reykjavik have 12 Nitro power-ups per lap. Once you reach the three lap races in your Career Mode playthrough, aim to collect every single power up and the trophy should be yours.

Tip: Don't use shortcuts as these tend to lack the Nitro power-ups more often than not, or only have one orange power-up.

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Collection Master
Collect more than half of the purple cash items in a Collector race.
Trophy type: Bronze
Quite self-explanatory - all you need to do is enter a Collector race and collect more than half of the big purple circles with a $ symbol in them located in the middle of the track. These items also determine your star rating for the Collector event and you can see the amount you've collected in the top-right corner. You should obtain the trophy once the race is finished.

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Brute
Knock down 10 opponents in a race.
Trophy type: Bronze
See "CARnage!"

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CARnage!
Knock down 15 opponents in a race.
Trophy type: Silver
The title is self-explanatory - all you need to do is take down 15 opponents in a single race by shunting, rubbing or slamming them. I got this one naturally during my Career Mode playthrough. The easiest way to knock down an opponent is to either "touch" them whilst the Level 3 Nitro is active (pressing the Nitro button three times,) or during the Adrenaline Mode.

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1, 2, 3
Perform 3 knockdowns in 10 seconds.
Trophy type: Bronze
See "Berserk"

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Berserk
Perform 5 knockdowns in 20 seconds.
Trophy type: Silver
This trophy is rather luck based mainly due to the knockdown cam. Once you knock down an opponent, your car goes into "autopilot" which usually tries to avoid other cars regardless of whether you're in Adrenaline Mode. This can result in your car gaining a few extra positions after the knockdown rather than attacking a group of other opponents.

I got this during the "Under Pressure" event in League 19, but you can attempt and achieve it much earlier than this. Under Pressure is your best bet in my opinion due to the fact that there is an infinite amount of opponents and they spawn rather quickly and often, plus they try to take you down which lets you attempt knockdowns more often.

My strategy would be either not to use Adrenaline for knockdowns and use the Level 3 Nitro instead, or if you are using Adrenaline, just push the brakes once after each knockdown to come out of the mode and re-enter Level 3 Nitro. By using this method, 20 seconds should be more than enough as long as its an "Under Pressure" event.

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Harasser
Perform at least 3 knockdowns on the same opponent.
Trophy type: Bronze
This trophy should not prove to be too difficult and should come naturally as you play through the Career Mode. However, if you need to specifically aim for this trophy, then I would suggest taking down an opponent, braking, waiting for him to pass you and then taking him down twice more.

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Shakedown
Knock down each opponent at least once in a single race.
Trophy type: Gold
As the trophy description states, you need to knock down each opponent at least once. I got this on my way to completing the Career Mode but if you are trying to get it out of the way, I would suggest the earlier tracks whilst memorising which opponent you have knocked down already.

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Drift Warrior
Knock down an opponent while drifting.
Trophy type: Gold
This trophy can be slightly luck based as the knockdown system is not always very accurate in this game. I would suggest a long corner on Reykjavik or Los Angeles. Grind at your opponent of choice whilst drifting, and if you happen to take him down, then the trophy is yours. In order to drift, press the left trigger, keep on accelerating and whilst doing so, try to push your opponent against a wall.

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Drifter
Earn over 2000 drift points in a single race.
Trophy type: Bronze
See "Tire-Greater"

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Tire-Greater
Earn over 3000 drift points in a single race.
Trophy type: Silver
This trophy might come naturally by simply playing the game, however if you wish to get it out of the way quickly, just enter a Drift or Under Pressure event and continue on drifting. Reykjavik is a good track for this as it is quite long with many curves and few straights, which is perfect for racking up those drift points.

Please note, that it is not the Drift $ that counts towards this trophy, but the actual distance travelled during the drift, as shown at the end of the track on the results screen.

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Around the world
Earn over 1000 drift points in a single drift.
Trophy type: Gold
This trophy can be obtained in pretty much any event. Just find a big corner, (i.e. Reykjavik,) and try to maintain your drift chain. The turn after the massive jump towards the end of the track is a good place to try for this. Another good corner is the one locate just before the factory, near the beginning of the track.

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Drift to the beat
Earn over 500 drift points in a Beat em All race.
Trophy type: Bronze
This is a very easy trophy to achieve. The principle of drifting is to press the right trigger and keep on accelerating whilst turning. It is actually harder to brake in this game than it is to drift, so this shouldn't give you any problems.

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Bouncing Bucket
Jump 1000m in a single race.
Trophy type: Bronze
See "Catapulting Clunker"

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Catapulting Clunker
Jump 1500m in a single race.
Trophy type: Silver
This trophy is unlikely to come until you unlock one of the faster cars. I suggest trying for this trophy on the Los Angeles track and using Nitro before every big jump. Please refer to the video below for more information.

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Safety First
Complete a Normal Race without crashing once.
Trophy type: Bronze
Another easy trophy which you should attempt at the beginning of the Career Mode. You might as well get it out of the way during the "League Qualifiers" Normal Race event in Los Angeles, which is the first race. Just try not to drift too recklessly into corners at high speeds and you should be fine.

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Spotless
Complete a Normal Race without touching a wall.
Trophy type: Bronze
One of the trophies that can be slightly frustrating, but as long as you drive carefully, you should be fine. Touching an opponent does not void the trophy, as it is only the walls that you can't touch during the whole race. I'd suggest the earlier levels of the Career Mode for this, as the races there consist of only two laps. I recommend the "Rookie League," Nassau, Normal Race event.

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Get Away From Me!
Complete a Normal Race without touching an opponent.
Trophy type: Bronze
One of the more frustrating trophies. My strategy was to try to get in front as soon as possible and not allow the opponents to catch up, thus preventing them from coming into contact with me. Please note that touching the walls will not void this trophy. I recommend the "Rookie League," Nassau, Normal Race event for this trophy as well.

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Survivor
Win an Under Pressure race without crashing once.
Trophy type: Bronze
As stated before, the earlier levels of the Career Mode are your best bet for this. Only Under Pressure events will count - other events will not net you this trophy. I'd recommend the "Gold League" Under Pressure event in Moscow, as you can race through the track rather quickly and the opponents should not have many chances to knock you down.

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Air Time
Hit at least 1 jump in each lap in a single race.
Trophy type: Bronze
All you need to do is to hit one jump per lap, which will be counted when the words 'Air Time!' appear on the screen. Los Angeles in "League Qualifiers" is a particularly good track for this. See "Catapulting Clunker" if you're unsure of the jump locations, as I have posted a video which shows a lot of the jumps being performed.

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Blowout
Finish each lap in first place in a single race.
Trophy type: Bronze
The description is self-explanatory, and this trophy should come naturally as you play through the game. After each lap, you should receive a Nitro Boost as well as a notification at the top of the screen entitled "Lap Leader!" if you are in first place. You should perform knockdowns on opponents whenever you have the chance in order to obtain extra Nitro Boost and just use your Nitro whenever Adrenaline is available. If you do this, then the trophy should pose no problems.

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Show-off
Cross the finish line while drifting.
Trophy type: Bronze
Extremely easy to achieve, all you need to do is press the left trigger to initiate a drift just before you cross the finish line.

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Too Good
Win a race without using Nitro.
Trophy type: Bronze
I unlocked this during a drift event, so you don't need to try for it in a traditional race as the description appears to suggest. Just refrain from using Nitro and the trophy will be yours at the end of the event.