MathDB

1

Part of 2021 China Team Selection Test

Problems(4)

Inequality with ordering

Source: 2021 China TST, Test 1, Day 1 P1

3/17/2021
Given positive integers mm and nn. Let ai,j(1≤i≤m,1≤j≤n)a_{i,j} ( 1 \le i \le m, 1 \le j \le n) be non-negative real numbers, such that ai,1≥ai,2≥⋯≥ai,n and a1,j≥a2,j≥⋯≥am,j a_{i,1} \ge a_{i,2} \ge \cdots \ge a_{i,n} \text{ and } a_{1,j} \ge a_{2,j} \ge \cdots \ge a_{m,j} holds for all 1≤i≤m1 \le i \le m and 1≤j≤n1 \le j \le n. Denote Xi,j=a1,j+⋯+ai−1,j+ai,j+ai,j−1+⋯+ai,1, X_{i,j}=a_{1,j}+\cdots+a_{i-1,j}+a_{i,j}+a_{i,j-1}+\cdots+a_{i,1}, Yi,j=am,j+⋯+ai+1,j+ai,j+ai,j+1+⋯+ai,n. Y_{i,j}=a_{m,j}+\cdots+a_{i+1,j}+a_{i,j}+a_{i,j+1}+\cdots+a_{i,n}. Prove that ∏i=1m∏j=1nXi,j≥∏i=1m∏j=1nYi,j. \prod_{i=1}^{m} \prod_{j=1}^{n} X_{i,j} \ge \prod_{i=1}^{m} \prod_{j=1}^{n} Y_{i,j}.
inequalities
So many angle bisectors

Source: 2021 China TST, Test 2, Day 1 P1

3/21/2021
A cyclic quadrilateral ABCDABCD has circumcircle Γ\Gamma, and AB+BC=AD+DCAB+BC=AD+DC. Let EE be the midpoint of arc BCDBCD, and F(≠C)F (\neq C) be the antipode of AA wrt Γ\Gamma. Let I,J,KI,J,K be the incenter of △ABC\triangle ABC, the AA-excenter of △ABC\triangle ABC, the incenter of △BCD\triangle BCD, respectively. Suppose that a point PP satisfies △BIC∼+△KPJ\triangle BIC \stackrel{+}{\sim} \triangle KPJ. Prove that EKEK and PFPF intersect on Γ.\Gamma.
geometryincenterangle bisector
convex polygon and quadrilaterals

Source: 2021ChinaTST test3 day1 P1

4/13/2021
Given positive integer n≥5 n \ge 5 and a convex polygon PP, namely A1A2...An A_1A_2...A_n . No diagonals of PP are concurrent. Proof that it is possible to choose a point inside every quadrilateral AiAjAkAl(1≤i<j<k<l≤n) A_iA_jA_kA_l (1\le i<j<k<l\le n) not on diagonals of PP, such that the (n4) \tbinom{n}{4} points chosen are distinct, and any segment connecting these points intersect with some diagonal of P.
combinatorial geometrycombinatoricsgraph theory
numbers always being local maximum in n*n matrix

Source: 2021ChinaTST test4 day1 P1

4/13/2021
Let n(≥2) n(\ge2) be a positive integer. Find the minimum m m , so that there exists xij(1≤i,j≤n)x_{ij}(1\le i ,j\le n) satisfying: (1)For every 1≤i,j≤n,xij=max{xi1,xi2,...,xij}1\le i ,j\le n, x_{ij}=max\{x_{i1},x_{i2},...,x_{ij}\} or xij=max{x1j,x2j,...,xij}. x_{ij}=max\{x_{1j},x_{2j},...,x_{ij}\}. (2)For every 1≤i≤n1\le i \le n, there are at most mm indices kk with xik=max{xi1,xi2,...,xik}.x_{ik}=max\{x_{i1},x_{i2},...,x_{ik}\}. (3)For every 1≤j≤n1\le j \le n, there are at most mm indices kk with xkj=max{x1j,x2j,...,xkj}.x_{kj}=max\{x_{1j},x_{2j},...,x_{kj}\}.
combinatoricsmatrix