MathDB

1

Part of 2018 Saudi Arabia GMO TST

Problems(2)

1 -1/2^{2^{n-1}}<sum 1/x+i<1/2^{2^{n}} if x_{n+1} = x_n^2 - x_n + 1, x_1 = 2

Source: 2018 Saudi Arabia GMO TST I p1

7/31/2020
Let {xn}\{x_n\} be a sequence defined by x1=2x_1 = 2 and xn+1=xn2−xn+1x_{n+1} = x_n^2 - x_n + 1 for n≥1n \ge 1. Prove that 1−122n−1<1x1+1x2+...+1xn<1−122n1 -\frac{1}{2^{2^{n-1}}} < \frac{1}{x_1}+\frac{1}{x_2}+ ... +\frac{1}{x_n}< 1 -\frac{1}{2^{2^n}} for all nn
Sequencealgebrainequalitiesrecurrence relationreciprocal
gcd(a_1^n + a-1a_2 ...a_n, a_2^n +a_1a_2 ...a_n, ... , a_n^n +a_1a_2 ...a_n)

Source: 2018 Saudi Arabia GMO TST II p1

7/31/2020
Let nn be an odd positive integer with n>1n > 1 and let a1,a2,...,ana_1, a_2,... , a_n be positive integers such that gcd (a1,a2,...,an)=1(a_1, a_2,... , a_n) = 1. Let dd = gcd (a1n+a1⋅a2⋅⋅⋅an,a2n+a1⋅a2⋅⋅⋅an,...,ann+a1⋅a2⋅⋅⋅an)(a_1^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, a_2^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, ... , a_n^n + a_1\cdot a_2 \cdot \cdot \cdot a_n). Show that the possible values of dd are d=1,d=2d = 1, d = 2
number theorygreatest common divisorGCD