Subcontests
(7)Large and small number
For any two coprime positive integers p,q, define f(i) to be the remainder of p⋅i divided by q for i=1,2,…,q−1. The number i is called a large number (resp. small number) when f(i) is the maximum (resp. the minimum) among the numbers f(1),f(2),…,f(i). Note that 1 is both large and small. Let a,b be two fixed positive integers. Given that there are exactly a large numbers and b small numbers among 1,2,…,q−1, find the least possible number for q.Proposed by usjl OH parallel to BC
Let ABC be a triangle with circumcenter O and orthocenter H such that OH is parallel to BC. Let AH intersects again with the circumcircle of ABC at X, and let XB,XC intersect with OH at Y,Z, respectively. If the projections of Y,Z to AB,AC are P,Q, respectively, show that PQ bisects BC.Proposed by usjl