MathDB

1

Part of 2018 Saint Petersburg Mathematical Olympiad

Problems(3)

Distance from point to line

Source: St Petersburg Olympiad 2018, , Grade 11, P1

6/21/2018
Let ll some line, that is not parallel to the coordinate axes. Find minimal dd that always exists point AA with integer coordinates, and distance from AA to ll is ≤d\leq d
analytic geometrygeometrynumber theory
Destroyer of the roads

Source: St Petersburg Olympiad 2018, Grade 10, P1

6/22/2018
Misha came to country with nn cities, and every 22 cities are connected by the road. Misha want visit some cities, but he doesn`t visit one city two time. Every time, when Misha goes from city AA to city BB, president of country destroy kk roads from city BB(president can`t destroy road, where Misha goes). What maximal number of cities Misha can visit, no matter how president does?
combinatoricsgraph theory
Almost binary

Source: St Petersburg Olympiad 2018, Grade 9, P1

7/13/2018
Prove, that for every natural NN exists kk, such that N=a020+a121+...+ak2kN=a_02^0+a_12^1+...+a_k2^k, where a0,a1,...aka_0,a_1,...a_k are 11 or 22
number theory