MathDB

2

Part of 2008 Mathcenter Contest

Problems(3)

f(xy^2)+f(x^2y)=y^2f(x)+x^2f(y), f(2008) =f(-2008)

Source: Mathcenter Contest / Oly - Thai Forum 2008 R1 p2 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/10/2022
Find all the functions f:R→Rf:\mathbb{R}\to\mathbb{R} which satisfy the functional equation f(xy2)+f(x2y)=y2f(x)+x2f(y)f(xy^2)+f(x^2y)=y^2f(x)+x^2f(y) for every x,y∈Rx,y\in\mathbb{R} and f(2008)=f(−2008)f(2008) =f(-2008)
(nooonuii)
algebrafunctional equationfunctional
polynomial wanted, If P(a)=0 then P(a|a|)=0

Source: Mathcenter Contest / Oly - Thai Forum 2008 R2 p2 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/11/2022
Find all polynomials P(x)P(x) which have the properties: 1) P(x)P(x) is not a constant polynomial and is a mononic polynomial. 2) P(x)P(x) has all real roots and no duplicate roots. 3) If P(a)=0P(a)=0 then P(a∣a∣)=0P(a|a|)=0
(nooonui)
algebrapolynomial
AX=AY wanted, touchpoints of incircle related

Source: Mathcenter Contest / Oly - Thai Forum 2008 R3 p2 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/9/2022
In triangle ABCABC (AB≠ACAB\not= AC), the incircle is tangent to the sides of BCBC ,CACA , ABAB at DD ,EE, FF respectively. Let ADAD meet the incircle again at point PP, let EFEF and the line passing through the point PP and perpendicular to ADAD intersect at QQ. Let AQAQ intersect DEDE at XX and DFDF at YY. Prove that AX=AYAX=AY.
(tatari/nightmare)
geometryequal segmentsincircle