MathDB

2.2

Part of 2011 Saudi Arabia Pre-TST

Problems(2)

2(a^3 + b^3 + c^3 + abc) >= (a+b)(b + c)(c + a) for a,b,c>0

Source: 2011 Saudi Arabia Pre-TST November 2.2

1/1/2022
Prove that for any positive real numbers a,b,ca, b, c, 2(a3+b3+c3+abc)≥(a+b)(b+c)(c+a)2(a^3 + b^3 + c^3 + abc) \ge (a+b)(b + c)(c + a).
algebrainequalities
s_m = x_0 + x_1 + x_2 + ... + x_m is power of 2 , x_n = 2^n-n

Source: 2011 Saudi Arabia Pre-TST February 2.2

1/1/2022
Consider the sequence xn=2n−nx_n = 2^n-n, n=0,1,2,...n = 0,1 ,2 ,.... Find all integers m≥0m \ge 0 such that sm=x0+x1+x2+...+xms_m = x_0 + x_1 + x_2 + ... + x_m is a power of 22.
power of 2Sumnumber theory