They decided to meet the next day at Takoba beach to start Midoriya's training.
"What time we gonna meet?" Bakugou questioned.
"'We?' Are you coming too young Bakugou?" All Might questioned, slightly confused.
Bakugou scoffed. "Of course I'm coming. You think I'm going to let this idiot get ahead of me by having extra training? Hell no. I'm coming too," He tilted his head in thought. "Also you're probably gonna need my help. It's harder than you'd think keeping this dumbass from killing himself."
Midoriya rolled his eyes. "I'm fine! No need to baby me."
"Trust me, you'll be grateful that I'm coming by the end."
All Might was.
By the time they had finished with Midoriya's first day of training, he had died no less than 17 times which was 17 more times than All Might had expected. Although to be fair that was probably his own mistake considering Midoriya had died three times the day prior but I digress.
It seemed that there was a slight problem with One For All that caused Midoriya to be unable to use it at less than 100% and that meant he was constantly exploding himself.
All though he hadn't believed Bakugou at the time, All Might was starting to feel a little desensitized to the constant death.
"Midoriya…" All Might said as the boy woke up for the 17th time that day. "I'm beginning to think we need a new approach."
Bakugou snorted. "Ya think?"
"Yeah, that's probably a good idea. I don't seem to be getting much more control of this quirk. What do you suggest?"
"Well originally I was going to have you train your body to become a better vessel for One For All, perhaps we should go back to that?"
In the end, it was decided that All Might would continue to train Midoriya by helping him to strengthen his body. And Bakugou, not being one to relax, decided that he too would prepare for the upcoming entrance exam by pushing his body to the limits. Thus the two boys each worked hard for the next many months in hopes of achieving the same goal; being accepted into UA's hero course and becoming the best possible heroes.
Now if you are so inclined, I would urge you to imagine a particularly powerful training montage in which you may see Midoriya clearing the beach with All Might, lifting weights as he studies for the exam, and perhaps most importantly, attempting to not use 100% of his power 100% of the time. This was much easier said than done considering the boy was now in the possession of not one, but TWO quirks that seemed to only allow him to go over his body's limits and evidently would not allow him to settle for anything less. Despite this, he managed to get to the point where he could use 5% of One For All with minimum repercussions. This was also due to the fact that he had quite a bit of prior experience with limiting his strength and as such was confident that he would get a handle on the two quirks during his time at UA.
If you are to continue this fictitious montage you may see Bakugou who has also thrown himself completely into training. Not only training his body but also paying particular attention to his quirk which, although not similar to Midoriya's in most ways, also put quite a strain on him due to its, shall I say, explosive nature. As if this wasn't enough, he had also decided that it would be in his best interest to study for the written exam. (no he is NOT a nerd and you should be ashamed of thinking so.)
That would conclude the training montage that I do hope you have imagined in far greater detail than my words could ever express. My reasoning for having you imagine this scene is that I quite frankly do not have the energy to describe it for you. Although I am beginning to realize that to some extent I have done just that. Alas.
This would lead us to the day of the entrance exam where Bakugou and Midoriya were walking towards UA, predictably arguing.
"... all I'm saying is that depending on the hero their personality can actually increase civilian approval! It's not enough to be efficient, as a hero you have to make sure to garner the support of the general public. Take Endeavor for example. His cold demeanor has a certain type of appeal for some fans, but only because the persona is well crafted and all encompassing. If he were to start acting more aloof and out of character his ratings would plummet, but since he-"
"I cannot BELIEVE you just compared me to that shitstain." Bakugou glared at Midoriya.
"Yeah well, that 'shitstain' has more popularity than you ever will with that flaming pile of trash you call a personality."
"I don't NEED a personality. I have BOMBS for HANDS!"
"Oh my goodness. I can't," Midoriya rolled his eyes and threw his hands in the air, exasperated. "Look, you want to be number one? I can't wait to see how well that will gooOOOOO-" It was at that moment that Midoriya found himself falling, having tripped over either his own two feet or perhaps something invisible that lay on the ground. Most likely his own feet.
Midoriya had, like most people, tripped many times in his life. However, in his case, it was slightly different as if he was to catch himself it would most likely result in shattering his arms. He knew this and, not wanting to go to the exam with two broken arms, acted accordingly, closing his eyes and bracing for impact but kept his hands at his side.
This of course meant that he slammed into the ground with enough force to give him a bloody nose, the sight of which caused Bakugou to laugh.
Not even a second after he decided to become acquainted with the cement he felt a slap to the back. This was odd considering that Midoriya was fairly certain he hadn't slapped himself on the back and Bakugou was still doubled over laughing at his predicament.
It was strange, although Midoriya was sure that he had been lying on the pavement only moments before, he now seemed to be pinwheeling through the air completely weightless. His vision was rather blurry from the spinning but he thought he saw a girl around his age flailing her arms and Bakugou wheezing next to her.
"WHOA," Was all Midoriya said.
"Oh-no-I-am-so-sorry-" The girl said. "I saw you falling and I tried to catch you but I guess I was too late oh, I should probably put you down I'm so sorry-" She rambled without taking a single breath before touching her fingers together.
Weight returned to Midoriya and he fell on his head on the floor.
"Oh gosh, I'm so sorry!"
"Wow!" Midoriya scrambled back to his feet, unfazed. "Is that your quirk? That's so neat! Five-point contact right? Can you make anything weightless or does the size matter? How long can you hold it? Can you make yourself weightless?"
"Chill Izu, you look like a complete maniac, you're scaring the poor girl!" Bakugou rolled his eyes.
Midoriya took a second to realize that there was still blood trickling down his face and had bombarded this stranger with intense questions about her quirk.
"Oops," He said. "Um- don't worry! I'm totally fine!'
The girl looked on the verge of crying. "You're bleeding!"
"Oh. Yeah well, not a big deal."
"I'm so embarrassed!" She buried her face in her hands. "I'm just going to run away now!"
She then proceeded to do just that, racing into the building without a second glance.
"Wow. Quite the charmer aren't you?" Bakugou said sarcastically.
"Hey! It wasn't that bad!"
"It was."
"And what about you! You just stood around laughing at me!"
Bakugou dug through his bag and tossed a handkerchief to Midoriya. "You would too if you had seen it. You looked HILARIOUS."
"Rude!" Midoriya took the handkerchief and tried his best to wipe the blood off his face.
"Oh give it here," Bakugou said, roughly taking it from him. "You're just smearing it all over."
Bakugou dabbed at Midoriya who squirmed.
"Just leave it! We're gonna be late for the exam!"
"You are NOT going to the exams with blood on your face! What kinda impression would that leave?!"
Approximately 18 feet away (or 6 meters for literally anyone who does not live in the US) Koji Koda, a UA hopeful, stood watching the entire exchange. Because he had seen the whole situation he decided to change that 18 feet into 33 as he made a very wide circle around the strange children.
He shuddered and decided he very much did not want to associate with Bakugou and Midoriya. Which, in my opinion, was probably smart.