Such an asteroid would probably have a mass of about a trillion kg.
A year isn’t enough time to design, prototype and test a new rocket design - so we’d have to pick something tried and trusted.
The Saturn-V is one of largest rockets we’ve ever built.
It had a thrust force of about 35 million Newtons that it could sustain for 165 seconds.
…so acceleration of the asteroid is F/m which is 35 million N / 1 trillion kg - so the acceleration would be a decidedly un-exciting 3 millionths of a meter per second for 165 seconds - then we’re out of juice. So we could impart a lateral velocity that would displace the asteroid from its track by 1/18000′ths of a meter per second after the rocket shut off. But that lateral speed carries on until the asteroid hits - but even after a year (about 3 million seconds), we’d have pushed it about 1,700 meters off course!
Still - 1.7km ! Our “puny” Saturn-V definitely produced a deflection. Maybe we just need to build a whole lot of them?
If the asteroid is aimed at the center of the Earth, we’d need to produce enough lateral velocity to displace it’s arrival point by at least the radius of the Earth…over 6 million meters.
So I’m thinking that we’d need at least 3,500 Saturn-V rockets…since we’re building them in a rush, quite a few will fail on launch or fail on arrival…so better plan on building at least 4,000 of them.
Now, we first have to build those rockets - and their launch facilities…that’s an insane amount of effort…but if it’s needed to save the Earth…
Trouble is, if it takes you 6 months to launch them and get them to the asteroid - then you only have half the amount of time to displace it…which means that you now need twice as many rockets.
So could we build and launch 8,000 Saturn V moon rockets in 6 months?
Even if you think we could…the Saturn V needs ALMOST all of that thrust just to get itself up to orbit…by the time it gets there, the fuel tanks are empty.
So how will we get fuel for our 8,000 Saturn V’s up to the asteroid?
It turns out that 80% of a Saturn V is the fuel! So it would take 5 launches to get the fuel into orbit for one rocket to reach the asteroid. So now we need to build and launch 40,000 Saturn V’s within 6 months…and seriously ramp-up our production of RP-1 fuel and liquid oxygen.
So, I guess the answer is a very definite “NO CHANCE WHATEVER!”
Just in case you’re thinking that we might blow it up with nuclear warheads or something - that’s really just science fiction. We would have a very hard time making even a small dent in something so huge…and even if we did, we’d just be transforming one giant rock into a trillion tiny rocks - and the overall effect of them hitting the Earth would be as bad…if not worse.
This is why the ONLY viable defense against asteroids is to spot them when they are a hundreds years away from hitting us.