MathDB

4

Part of 2008 Mathcenter Contest

Problems(3)

(a^2+b^2+c^2)/(a+b+ c) is an integer if (a\sqrt{3}+b)/(b\sqrt3+c) is rational

Source: Mathcenter Contest / Oly - Thai Forum 2008 R1 p4 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/10/2022
Let a,ba,b and cc be positive integers that a3+bb3+c\frac{a\sqrt{3}+b}{b\sqrt3+c} is a rational number, show that a2+b2+c2a+b+c\frac{a^2+b^2+c^2}{a+b+ c} is an integer.
(Anonymous314)
number theoryalgebrarationalInteger
AD + DX - (BC + CX) = 8 , trapezoid ABCD

Source: Mathcenter Contest / Oly - Thai Forum 2008 R2 p4 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/11/2022
The trapezoid ABCDABCD has sides ABAB and CDCD that are parallel DAB^=6∘\hat{DAB} = 6^{\circ} and ABC^=42∘\hat{ABC} = 42^{\circ}. Point XX lies on the side ABAB , such that AXD^=78∘\hat{AXD} = 78^{\circ} and CXB^=66∘\hat{CXB} = 66^{\circ}. The distance between ABAB and CDCD is 11 unit . Prove that AD+DX−(BC+CX)=8AD + DX - (BC + CX) = 8 units.
(Heir of Ramanujan)
geometrytrapezoid
1/(p^n+q^n+1) + 1/(q^n+r^n+1)+ 1/(r^n+p^n+ 1) <=1 if pqr=1

Source: Mathcenter Contest / Oly - Thai Forum 2008 R3 p4 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/9/2022
Let p,q,r∈R+p,q,r \in \mathbb{R}^+ and for every n∈Nn \in \mathbb{N} where pqr=1pqr=1 , denote 1pn+qn+1+1qn+rn+1+1rn+pn+1≤1 \frac{1}{p^n+q^n+1} + \frac{1}{q^n+r^n+1} + \frac{1}{r^n+p^n+ 1} \leq 1
(Art-Ninja)
algebrainequalities