Subcontests
(9)pawn can be moved from (x,y) to (x',y') iff |x - x'| = p and |y- y'| = q
Each square of an n×m board is assigned a pair of coordinates (x,y) with 1≤x≤m and 1≤y≤n. Let p and q be positive integers. A pawn can be moved from the square (x,y) to (x′,y′) if and only if ∣x−x′∣=p and ∣y−y′∣=q. There is a pawn on each square. We want to move each pawn at the same time so that no two pawns are moved onto the same square. In how many ways can this be done? largest real constant c, such p^2 + q^2 + 1 >= bp(q + 1(
(a) Prove that p2+q2+1>p(q+1) for any real numbers p,q, .
(b) Determine the largest real constant b such that the inequality p2+q2+1≥bp(q+1) holds for all real numbers p,q
(c) Determine the largest real constant c such that the inequality p2+q2+1≥cp(q+1) holds for all integers p,q. lQ(p_1)l = lQ(p_2)l = lQ(p_3)l = lQ(p_4 )l = 3, Q(x) = ax^3 + bx^2 + cx + d
Let p1,p2,p3,p4 be four distinct primes. Prove that there is no polynomial Q(x)=ax3+bx2+cx+d with integer coefficients such that ∣Q(p1)∣=∣Q(p2)∣=∣Q(p3)∣=∣Q(p4)∣=3. Austria-Poland 1997 Problem
Numbers 149,249,...,9749 are writen on a blackboard. Each time, we can replace two numbers (like a,b) with 2ab−a−b+1. After 96 times doing that prenominate action, one number will be left on the board. Find all the possible values fot that number. APMC 1997
Let P be the intersection of lines l1 and l2. Let S1 and S2 be two
circles externally tangent at P and both tangent to l1, and let T1
and T2 be two circles externally tangent at P and both tangent to l2.
Let A be the second intersection of S1 and T1,B that of S1 and T2,C that of S2 and T1, and D that of S2 and T2. Show that the points A,B,C,D are concyclic if and only if l1 and l2 are perpendicular.