École Polytechnique is on the Saclay plateau south of Paris. To get there from Paris, the only reasonable route is to take RER line B. Get off at Massy-Palaiseau and take bus 91-06 or 91-10 to Polytechnique. You can also get off a little further at Lozère and walk, but it's a steep hill from the train, and since some trains terminate at Massy, there's little advantage in walking.
It takes about 1 hour to get to the center of Paris. The buses and trains run every few minutes throughout the day. Allow for delays as both the roads and the trains are congested.
Use the RATP website or the app to get routes and schedules, and real-time information for some lines (I don't know if real-time information is available for 91-06 and 91-10).
If you want to conveniently go to Paris, you'll pretty much need to be close to RER B. Some trains skip stations; all trains to Massy-Palaiseau serve all stations in Paris as well as Bougr-la-Reine, La Croix de Berny and Antony. However, with events lasting from 8am to 10pm, you'll really want to be close to École Polytechnique. École Polytechnique itself has some housing for visitors, but it's limited. The last bus to Polytechnique departs from Massy-Palaiseau at 0:00 (beware that there may be fewer buses in July and August).
Unfortunately, Paris is both a major tourist city and a major business city, so accommodation tends to be very expensive. 80€ per night is very cheap for Paris. If you qualify for some student accommodation, it'll be cheaper than that, but there isn't much of it except during academic holidays (and even then you'd better book in advance and hope you're lucky). Otherwise it may condemn you to a hostel bed and a long commute.