MathDB

1.4

Part of 2011 Saudi Arabia Pre-TST

Problems(2)

BT = 2PT if <BAT = < BCS = 10^o in 40^o-70^o-70^o triangle

Source: 2011 Saudi Arabia Pre-TST November 1.4

12/31/2021
Let ABCABC be a triangle with AB=ACAB=AC and ∠BAC=40o\angle BAC = 40^o. Points SS and TT lie on the sides ABAB and BCBC, such that ∠BAT=∠BCS=10o\angle BAT = \angle BCS = 10^o. Lines ATAT and CSCS meet at PP. Prove that BT=2PTBT = 2PT.
equal anglesgeometryequal segments
sum 2^{k-1}/f_k <C for fermat numbers

Source: 2011 Saudi Arabia Pre-TST February 1.4

1/1/2022
Let fn=22n+1f_n = 2^{2^n}+ 1, n=1,2,3,...n = 1,2,3,..., be the Fermat’s numbers. Find the least real number CC such that 1f1+2f2+22f3+...+2n−1fn<C\frac{1}{f_1}+\frac{2}{f_2}+\frac{2^2}{f_3}+...+\frac{2^{n-1}}{f_n} <C for all positive integers nn
number theoryinequalitiesFermat numberalgebra