MathDB

7

Part of 2015 China Northern MO

Problems(2)

min xyz if x|(y^5+1),y|(z^5+1),z|(x^5+1)

Source: China Northern MO 2015 grade 10 p7 CNMO

5/5/2024
It is known that odd prime numbers x,yzx, y z satisfy x∣(y5+1),y∣(z5+1),z∣(x5+1).x|(y^5+1),y|(z^5+1),z|(x^5+1). Find the minimum value of the product xyzxyz.
number theory
infinite n such that C_n^{S_n} is even, where S_n =[1+1/2+..+1/n]

Source: China Northern MO 2015 grade 11 p7 CNMO

5/5/2024
Use [x][x] to represent the greatest integer no more than a real number xx. Let Sn=[1+12+13+...+1n],  (n=1,2,..,)S_n=\left[1+\frac12 +\frac13+...+\frac{1}{n}\right], \,\, (n =1,2,..,) Prove that there are infinitely many nn such that CnSnC_n^{S_n} is an even number.
PS. Attached is the original wording which forgets left [ . I hope it is ok where I put it.
number theory