MathDB

3

Part of 2017 Thailand TSTST

Problems(3)

CR = 2XY wanted, AQ = AC, AR = CP, circumcircle realted

Source: 2016 Thailand October Camp 1.3

10/15/2020
In △ABC\vartriangle ABC with AB>ACAB > AC, the tangent to the circumcircle at AA intersects line BCBC at PP. Let QQ be the point on ABAB such that AQ=ACAQ = AC, and AA lies between BB and QQ. Let RR be the point on ray APAP such that AR=CPAR = CP. Let X,YX, Y be the midpoints of AP,CQAP, CQ respectively. Prove that CR=2XYCR = 2XY .
geometrycircumcircleequal segments
f(x-f(x))=f(x)-f(f(x))

Source: 2016 Thailand October Camp 2.3

2/26/2022
Let ff be a function on a set XX. Prove that f(X−f(X))=f(X)−f(f(X)),f(X-f(X))=f(X)-f(f(X)), where for a set SS, the notation f(S)f(S) means {f(a)∣a∈S}\{f(a) | a \in S\}.
functionfunctional equationalgebra
Weird Inequality

Source: 2016 Thailand October Camp 3.3

2/28/2022
Let a,b,c∈R+a, b, c \in\mathbb{R}^+. Prove that ∑cycab(12a+c+12b+c)<∑cyca3+b3c2+ab.\sum_{cyc}ab\left(\frac{1}{2a+c}+\frac{1}{2b+c}\right)<\sum_{cyc}\frac{a^3+b^3}{c^2+ab}.
inequalities