MathDB

2

Part of 2024 India Regional Mathematical Olympiad

Problems(2)

RMO 2024 Q2

Source: RMO 2024 Q2

11/3/2024
For a positive integer nn, let R(n)R(n) be the sum of the remainders when nn is divided by 1,2,⋯ ,n1,2, \cdots , n. For example, R(4)=0+0+1+0=1,R(4) = 0 + 0 + 1 + 0 = 1, R(7)=0+1+1+3+2+1+0=8R(7) = 0 + 1 + 1 + 3 + 2 + 1 + 0 = 8. Find all positive integers such that R(n)=n−1R(n) = n-1.
number theory
RMO KV 2024 Q2

Source: RMO KV 2024 Q2

11/3/2024
Show that there do not exist non-zero real numbers a,b,ca,b,c such that the following statements hold simultaneously: ∙\bullet the equation ax2+bx+c=0ax^2 + bx + c = 0 has two distinct roots x1,x2x_1,x_2; ∙\bullet the equation bx2+cx+a=0bx^2 + cx + a = 0 has two distinct roots x2,x3x_2,x_3; ∙\bullet the equation cx2+ax+b=0cx^2 + ax + b = 0 has two distinct roots x3,x1x_3,x_1. (Note that x1,x2,x3x_1,x_2,x_3 may be real or complex numbers.)
algebra