不等式への招待 第10章

81 ：１３２人目の素数さん：2019/04/03(水) 07:14:05.62 ID:/TkvX91f.net

>>74
crux/v44/n3/Problems_44_3

4321.
Find the greatest positive real number k such that
　(aa + bb + cc + dd + ee)^2 ≧ k(a^4 + b^4 + c^4 + d^4 + e^4)
for all real numbers a,b,c,d & e satisfying a+b+c+d+e=0.

　k = 20/13,　等号成立は {1,1,1,1,-4} のとき。　

4325.
Solve in real numbers the system of equations:
　x^4 -2y^3 -x^2 +2y = -1 +2√5,
　y^4 -2x^3 -y^2 +2x = -1 -2√5,

　x = (1+√5)/2 = φ = 1.61803399
　y = (1-√5)/2 = -1/φ = - 0.61803399
　(x, y) ≒ (1.8　1.5)　付近では接触しないようでござる。

4327.
Prove the following inequality for all x>0:
arctan(x) arctan(1/x) < π/{2(xx+1)}.

4330.
Let a & b be integers such that aa -20b +24 = 0.
Find the complete set of solutions of the following equation over integers:
　5xx + axy + byy = 11.

a = 2(5n+7), b = 5nn+14n+11
　(x, y) = (n, -1)　(-n, 1)　(3n+4, -3)　(-3n-4, 3)

a = 2(5n-7), b = 5nn-14n+11
　(x, y) = (n, -1)　(-n, 1)　(3n-4, -3)　(-3n+4, 3)