Olaf Build Guide by Janitsu
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I know I'm probably going to get a lot of downvotes for this, but PLEASE LISTEN
AP Olaf is actually VIABLE. Here's how it works:
Undertow is on a really short cooldown, so you can constantly proc Lich Bane.
Vicious Strikes scales well off attack speed, and slightly off CDR since you can spam Undertow more, which is an auto-attack reset.
And finally, Ragnarok gives you a ton of free tenacity so that you don't even need to buy AD to do damage! Awesome!
Greater Quintessence of Attack Damage
Greater Mark of Attack Damage
Greater Seal of Armor
Greater Glyph of Magic Resist
Since Olaf doesn't do magic damage until he gets items, don't run any magic penetration runes.
Instead, run flat AD so you can be stronger early game and bully around your enemy laner.
Flat Armor is for in case you're against an AD assassin mid, but otherwise it's still good for mitigating auto-attack harass. Olaf is prone to this because of her short range.
You can switch out the Quints for Move Speed quints if you think you can abuse your mobility to bully certain lane opponents, like Syndra or Nidalee.
Take hybrid penetration in the offensive tree. Taking Spellsword is a MUST.
Anything else is up to you, really.
Boots of Speed and 4 Health Potions. These let you bully around those noobs who dare fight you in mid lane because you have the mobility advantage.
Doran's Blade is good too, but you're AP Olaf and you don't need AD.
Doran's Ring is stupid because Olaf has no problems and no ap scaling.
Next, you want to rush a Sheen. It synergizes very well off Undertow, and it's good for harass since it basically doubles your AD for one attack.
Your next item is Berserker's Greaves, because it gives Attack Speed and Vayne doesn't benefit off Spell Penetration. With increased movespeed and attack speed, you're able to bully the other laner with Vicious Strikes more easily.
The first big item is Nashor's Tooth. Not only does it give attack speed so you can pull off more Silver Bolts procs, BUT the passive that converts 15% of your AP into bonus on-hit damage is OP. I mean, if you do the math, 2000 AP gives you 300 damage! Who needs AD when you can go AP?
Next items: Lich Bane, Zhonya's Hourglass, and Rabadon's Deathcap.
Lich Bane first, because you can constantly proc it with Undertow for a lot of damage. Also, with the S3 change, it has a base value, so you don't need to stack 1000000 AP before it starts doing damage.
Zhonya's Hourglass next, because you can Ult > Tumble > Invis > Hourglass > Tumble > Invis for a super trolly 5 second untargetability against people without Vision Wards or Oracle's Elixirs. Talk about unkillable!
Rabadon's Deathcap is last, so your Lich Bane and Nashor's Tooth procs will start to hurt a lot!
Liandry's Torment - If the other team is stupid and they keep letting you pin their tanks into the wall, then buy this so while they're stunned you can shred their health really really fast :D
Twin Shadows - Are people still getting away from you? Then uninstall Have no fear, Twin Shadows makes you a complete chasing monster! The targetted 30% slow + Berserker Rage + increased movespeed makes you literally impossible to escape.
Tear of the Goddess/ Archangel's Staff/ Seraph's Embrace - If there's a lot of burst on the other team, consider buying an early Tear and charging it so that you have an available shield to protect you from burst. Olaf also charges it quickly because of the short cooldown and mana cost of Undertow!
Iceborn Gauntlet - I don't like this item, yeah the stats are nice but IMO it's better on tanks and Vayne has enough chasing power already. I guess this can replace Lich Bane if the other team is 5 AD.
Guinsoo's Rageblade/ Hextech Gunblade - These aren't good because they're hybrid items, but buy Guinsoos if you want more AS and Gunblade for more chasing power + damage + lifesteal/survivability.
Deathfire Grasp - You don't really do magic damage but the stats are nice. 120 AP is a sizable amount, and CDR to be an annoying stealthing ***** is nice too.
In mid, make sure you're against a champion who
1) Can't gapclose very well,
2) Has a lot of skillshots, or
3) Is controlled by an unskilled summoner (preferably Bronze or unranked)
Olaf's main strength is her mobility, so if you're against an Nidalee you can just Undertow away from her Spears then counterattack.
If you're against someone with all targetted abilities like Ryze, you're ****ed you might as well bend over and ask your jungler to camp.
Which he won't do, because you're AP Olaf.
You want to play passively because you aren't that strong early on, but if you see them commit to CS, Undertow towards them, resetting your AA timer, and you're able to get a free hit on them.
At level 6/7, your power spikes tremendously because you have obtained your ultimate. If you see them out of position, you want to Ragnarok > Undertow > Reckless Swing them into a wall, then keep auto-attacking them while trying to avoid their skillshots, and hopefully you force them out of lane via death or them being low on health.
Late game, remember that you're AP Olaf and your only real strength is being annoying with stealth + mobility, slight damage, and tank shredding Vicious Strikes. Use Reckless Swing to peel for your ADC, or you can use it to pin down priority targets to walls and hopefully you can finish them off with your team.
This is just a quick guide on the unconventional. I'm sorry if I didn't go too in-depth with stuff, but I only had about half an hour :/
REMEMBER TO TRY IT BEFORE GIVING FEEDBACK THANKS
Special thanks to TehAsian whose AP Vayne guide inspired me.