A sleepy Izuku and his incredibly determined brother dragged themselves out of bed to go eat breakfast, before continuing their game of chess from a few nights prior. A surprise to everyone, Katsuki was actually an incredibly powerful strategist, only thanks to Izuku, however struggled to keep his head cool in most situations, so you didn't normally see this.
Izuku was still superior in this, although Katsuki wasn't going to admit this until one of their deathbeds, as he had an incredible ability to analyse, interpret and predict. Both of them knew this, so games they had typically took a very long time, having had spent so much time together they knew the other better than the other knew themselves and so they were incredibly competitive.
Even without using his quirk to accelerate his brain, Izuku was able to perfectly lure Katsuki and take advantage of his aggressive nature to take out piece after piece. "Checkmate." Izuku said, sticking his hand out. Katsuki smirked, responded with a fist bump and simply said "I'll get you next time."
"Bring it." Izuku responded, before stifling a yawn. "So what are we gonna do today?"
"School's out, time to prepare for the entrance exams I guess."
"Alright, so sparring, general quirk training, and..." Izuku trailed off, looking at the news. It was a report about Takoba beach, once admired for the natural beauty of it, was currently overrun by trash, both washed ashore and illegally dumped.
His mutterings were interrupted by his brother tossing a pair of ankle and wrist weights to him, a smirk on his face. "Let's go clean it then." Katsuki stated, knowing that would be the question at the end of his brother's ramblings.
They arrived at the beach, seeing large amounts of small litter, hard rubbish, even what appeared to be fragments of a shipwreck. They helped each other put the weights on and established the schedule. They knew they had to train both their ability to use their quirks, their raw physical abilities, and their combat abilities.
They decided they would spend the first week developing their quirk amidst sparring and cleaning the rubbish. The final three weeks before the exam would be dedicated to strength as they cleared the rest of the rubbish, and once their muscles were sore, they would take a 'break' by focusing on improving their capacity to use their quirk. For Izuku this meant dedicating time to charging his quirk faster, whereas for Katsuki focused on using his quirk repetitively.
During their first week of sparring, Izuku spent most of his time trying to develop a way to target from a range. He knew he could fire through the air but that was difficult to control at the moment, and it was far too easy for his sparring partner to dodge. During one of, probably several dozen spars between the brothers, Katsuki had initiated by flying through the air with one of his explosions propelling him, and Izuku directed lightning to his feet and head as he propelled away with a dodge and started trying to come up with a plan.
That's when it hit him (along with a ranged attack from one of Katsuki's explosions). Directing his charged energy to his feet, Katsuki could see the arcs coming off of his legs, and as soon as it appeared, it disappeared. Leaving Izuku with a smirk on his face. Katsuki knew this meant his brother had a trump card ready to play and swiftly backed off.
Izuku had directed a single streak of his green lightning through the ground, the shock hitting Katsuki hard. He smirked and came over to help his brother up. It didn't take long for Katsuki to realise what had happened, and knew he had a new challenge to overcome. Izuku had a few words of advice, and it sent gears in Katsuki's head spinning. "Why do you make explosions from your hands?"
Katsuki and Izuku returned to their positions, both knowing Izuku's plan. Because he was sparring with his brother, he didn't worry so much about hiding his quirk and stopped hiding the charged energy, and it began to arc around him, and hair began to stand on end. Although it looked like more effort, it was less because he no longer had to suppress the energy inside him. It was just an aesthetic difference. However, he did this to hide his plans from his brother, because he knew that, after sparring for so long, Katsuki would immediately notice the change in the arc patterns when he activated it on a particular target.
Katsuki on the other hand was lost in thought by what Izuku said. Of course, to him, it made logical sense that explosions came from his hands. It was easier to direct and control. But with his brother attacking through the ground, he needed to be able to propel himself upwards whilst still being able to attack.
A smirk broadened on his face as he understood what his brother said. Focusing on his feet, he started making small explosions, carefully propelling himself upwards. It was difficult to control, and jumpy, but it served its purpose Izuku smirked as he knew his brother had just unlocked new potential.
They charged towards each other, continuing the sparring bouts. They had become less about knocking the other out, but more about landing a single blow as that was hard enough in itself.
As the day went on, they finished sparring and began work on moving the garbage laid on the beach. Bakugo trying to fly around and collecting the smaller rubbish, Izuku using his quirk to improve his strength to move the fridges and freezers. They daren't open a single one, for all they knew there could a body in it (maybe their excessive horror movie watching had made their imaginations somewhat overactive).
Their first week of training continued like this, although occasionally being scolded by their mother for their use of ice and heat packs after pushing themselves so hard. They had barely gotten a quarter of the way through the piles of rubbish, although they did spend a lot of time sparring.
Then came along the next few weeks, where the ankle and bracelet weights would become more important. Raw strength and quirk capacity. This was going to be rough.
They started with different routines, whilst one was moving the various large objects the other was pushing their quirks to the limit.
For Izuku, it involved him using his quirk whilst supressing the arcs, charging it at high speed and using it whilst still needing to be able to move afterwards. For his brother, he was straining the sweat glands around his body, rather than just those in his hands.
After about two weeks they were almost done, both of the boys exhausted. Katsuki was kept home by their mother after he insulted her one too many times, so Izuku was off working on the beach himself. He was fitting the bracelet on his left hand as he saw a flash of blonde hair.
He thought it was Katsuki but knew even he wouldn't dare go against their mother's direct will. Finishing fitting the bracelets and ankle weights, he ignored it and began to charge his quirk. With no-one around and him not fighting anyone he didn't bother supressing the quirk. The charge was low, so the arcs of lightning were less, but still undeniably there. His hair didn't stick up completely but wasn't the same floppy hair he had without it.
He started moving the fridges and freezers before he saw the hair again. So he definitely wasn't imagining it, or he was definitely going insane. Great. "Wait, were you waiting out the front of UA the other week?" Izuku, feigning confidence, called out in the general direction of the hair.
"Yeah I was waiting for my friend, she had accidentally flown off into town. How do you know though?" The person, who had now stepped through the wall of hedges to reveal himself as the cartoonish boy from the other day.
"Oh y-y-yeah, I-I-I think I, uhm, saw you! Yeah! Waiting outside the f-f-front of the g-gates," Izuku nervously stuttered, knowing his lie would probably be caught but just tried to sail through, "S-s-sorry, I'm, uhh, not great with new people."
"Nah it's all fine man, but I swear I've seen your quirk somewhere before," The blonde boy had his suspicions, but knew he had friends like that. Well. A friend like that. He gave a thumbs up before continuing, "Like, electrification is common but green, and the way yours forms more akin to lightning, if I'd seen it before I'd definitely recognise it... wait... are you-?"
"Sorry, gotta go, cya later!" Izuku cut him off, blitzing away with streaks of lightning behind him.
The blonde smiled nervously but, in his head, knew that was him. He smiled to himself. 'See you at UA lightning boy'.
Izuku meanwhile was on the way home, taking a slightly different route. In his panic, he had managed to get himself slightly lost.
He knew the streets, but he wasn't anticipating going out of his normal path. Wandering around slightly slower than normal as he glanced around, he haired a scream that was quickly muffled. Something wasn't right. He could feel it.
He still had some energy at least and quickly directed this to his legs and head in an attempt to single out the noise. He heard the same scream, this time muffled. He felt the energy in his legs flow as he ran towards the noise.
Eventually he made it to a shady alley way. He had a decent reserve of energy still, but his muscles were aching. Luckily, his brain was fine and the moment he saw what was happening he immediately started planning. He saw that the sludge guy had someone has a hostage, but after watching All Might fight them before he knew if he was fast enough he would be able to go through the sludge guy and grab its hostage, hopefully being able to immobilize it with the slime solidifying around him from friction.
Seconds after taking in the situation, Izuku dived in with his quirk running full power on his legs and arms. He felt as though time slowed as he started moving at an incredible speed. He felt himself slow down as he reached the sludge villain, but still was capable of maintaining enough speed to burst through and grab the victim. He grabbed the victim using his lightning to make sure his grip was enough and focusing more of his energy on his legs he ripped them out of the villain's grasp with a burst of speed.
Time started to speed up again as he heard the villain screeching in pain. The extreme speed he moved at was producing so much friction, it made so much heat that the villain was solidified in the place he had moved through. "YOU LITTLE BRAT I'M GONNA TEAR YOU FROM LIMB TO LIMB!" It screeched, but despite it's threat it was in too much agony to move. Izuku knew this, and swiftly jumped away with the sludge victim's victim in his hands.
Eventually he made it to the open street in a less shady part of town. Taking a moment to think, he assessed the situation. He called the cops and directed them to the area he was just at, and then moved to the park. He was near the hill that UA stood on, next to a niche café and a construction site in its very early stages. He looked down at the person he had just saved, who were finally coming to. He recognised her as the girl he had caught just a couple weeks ago, as his eyes widened. "Hado-senpai?"
"Oh hey, it's you again!" She responded, "I think you can put me down now."
"Oh, R-r-right. S-s-sorry about t-t-that," Izuku stuttered as he began reddening and putting the girl back on her feet, "Are you okay now Hado-senpai?"
"Yeah I'm fine but y-" she broke off with a yawn again. And then fell asleep. In his arms. Again. This time, Izuku once more pulled her onto his back, his face reddening, before making his way back towards the school gates. He wasn't meant to be here until next week! Wait the entrance exams were next week. Crap. He just ignored that for now, and focussed on charging his energy once more, in order to get out of there fast. Again.
Once more he made it to the front of the school, charging his quirk, numbing his legs further and in no more than a matter of seconds the girl with periwinkle hair was lying next to her friends and teacher after a flash of green.
"What the?!" the shy, emo looking boy called out. Once more had one of his best friends gotten into trouble, seemingly disappeared, before reappearing lying next to him and his other friend and teacher with a flash of green. His blonde friend seemed to be lost in thought, as Aizawa-sensei picked up the girl and carried her to the nurse's office. "Problem child," He muttered under his breath.
Izuku had made it home a little later than usual, but not so late as to arouse suspicion. His mum welcomed him home as Katsuki merely grunted (he was fairly restless) and motioned to the chessboard, which he had set up. Izuku absentmindedly moved the king's pawn, which his brother mirrored. It was a fairly standard game opening for them, so he was flying on autopilot. His mind was preoccupied by the events of the day.
On one hand he knew that he had definitely developed his strength and his quirk (he just remembered to take off the weights), on the other he couldn't get that girl out of his mind. His brother was too focussed on the game at hand, so Izuku was able to be left to his thoughts.
"Check." Katsuki called out with a smirk.
"Check." Izuku called back effortlessly, simultaneously taking the offending piece and countering. Katsuki's face went back to the serious, if eternally scowling, face that Izuku was accustomed to when the brothers were together.
Back in his head, Izuku still had the blue haired girl in his mind. He wondered if he'd see her again. I mean, it's happened twice so far, who knows what'll happen next?
On the other hand, the girl had just woken up and was trying to figure out the location of her green haired saviour, to no avail as the boys and teacher both just shrugged and were trying to figure out if she was either delusional, or had a secret second quirk. The latter seemed really likely to them, as they saw but a flash of green and suddenly, she reappeared, unharmed and unaccompanied.
She sighed in defeat as her usual energetic nature was drained by the events of the day. She mumbled something about going back to sleep and that she'd prove them wrong, before returning back to blissful sleep in the nurse's office.