MathDB

5

Part of 2021 China Team Selection Test

Problems(4)

Geometry with Weird Conditions

Source: 2021 China TST, Test 1, Day 2 P5

3/17/2021
Given a triangle ABCABC, a circle Ω\Omega is tangent to AB,ACAB,AC at B,C,B,C, respectively. Point DD is the midpoint of ACAC, OO is the circumcenter of triangle ABCABC. A circle Γ\Gamma passing through A,CA,C intersects the minor arc BCBC on Ω\Omega at PP, and intersects ABAB at QQ. It is known that the midpoint RR of minor arc PQPQ satisfies that CR⊥ABCR \perp AB. Ray PQPQ intersects line ACAC at LL, MM is the midpoint of ALAL, NN is the midpoint of DRDR, and XX is the projection of MM onto ONON. Prove that the circumcircle of triangle DNXDNX passes through the center of Γ\Gamma.
geometrycircumcircle
Average of consecutive terms at least 1

Source: China TST 2021, Test 2, Day 2 P5

3/22/2021
Let nn be a positive integer and a1,a2,…a2n+1a_1,a_2,\ldots a_{2n+1} be positive reals. For k=1,2,…,2n+1k=1,2,\ldots ,2n+1, denote bk=max⁡0≤m≤n(12m+1∑i=k−mk+mai)b_k = \max_{0\le m\le n}\left(\frac{1}{2m+1} \sum_{i=k-m}^{k+m} a_i \right), where indices are taken modulo 2n+12n+1. Prove that the number of indices kk satisfying bk≥1b_k\ge 1 does not exceed 2∑i=12n+1ai2\sum_{i=1}^{2n+1} a_i.
inequalitiesalgebracombinatorics
functional equation

Source: 2021ChinaTST test3 day2 P2

4/13/2021
Determine all f:R→R f:R\rightarrow R such that f(xf(y)+y3)=yf(x)+f(y)3 f(xf(y)+y^3)=yf(x)+f(y)^3
algebraTSTfunction
polygon's area doesn't add much when combined with its centric symmetry

Source: 2021ChinaTST test4 day2 P2

4/14/2021
Find the smallest real α\alpha, such that for any convex polygon PP with area 11, there exist a point MM in the plane, such that the area of convex hull of P∪QP\cup Q is at most α\alpha, where QQ denotes the image of PP under central symmetry with respect to MM.
geometrycombinatorial geometrycombinatoricsareaConvex hullconvex polygonsymmetry