MathDB

3

Part of 2021 Korea Winter Program Practice Test

Problems(2)

inequality

Source: 2021 Korea Winter Program Practice Test

2/8/2021
n≥2n\ge2 is a given positive integer. i≤ai≤ni\leq a_i \leq n satisfies for all 1≤i≤n1\leq i\leq n, and SiS_i is defined as a1+a2+...+ai(S0=0)a_1+a_2+...+a_i(S_0=0). Show that there exists such 1≤k≤n1\leq k\leq n that satisfies ak2+Sn−k<2Sn−n(n+1)2a_k^2+S_{n-k}<2S_n-\frac{n(n+1)}{2}.
algebrainequalities
R,D,F is collinear

Source: 2021 Korea Winter Program Test2 Day1 #3

2/13/2021
The acute triangle ABCABC satisfies AB‾<BC‾<CA‾\overline {AB}<\overline {BC}<\overline {CA}. Let HH a orthocenter of ABCABC, DD a intersection point of AHAH and BCBC, EE a intersection point of BHBH and ACAC, and MM a midpoint of segment BCBC. A circle with center EE and radius AEAE intersects the segment ACAC at point FF(≠A\neq A), and circumcircle of triangle BFCBFC intersects the segment AMAM at point SS. Let PP(≠D\neq D), QQ(≠F\neq F) a intersection point of circumcircle of triangle ASDASD and DFDF, circumcircle of triangle ASFASF and DFDF respectively. Also, define RR as a intersection point of circumcircles of triangle AHQAHQ and AEPAEP. Prove that RR lies on line DFDF.
geometry