MathDB

4

Part of 2022 China Team Selection Test

Problems(4)

A special incenter configuration

Source: 2022 China TST, Test 1, P4

3/24/2022
Let ABCABC be an acute triangle with ∠ACB>2∠ABC\angle ACB>2 \angle ABC. Let II be the incenter of ABCABC, KK is the reflection of II in line BCBC. Let line BABA and KCKC intersect at DD. The line through BB parallel to CICI intersects the minor arc BCBC on the circumcircle of ABCABC at E(E≠B)E(E \neq B). The line through AA parallel to BCBC intersects the line BEBE at FF. Prove that if BF=CEBF=CE, then FK=ADFK=AD.
geometryincenter
Minimize a function with absolute values

Source: 2022 China TST, Test 2, P4

3/29/2022
Given a positive integer nn, find all nn-tuples of real number (x1,x2,…,xn)(x_1,x_2,\ldots,x_n) such that f(x1,x2,⋯ ,xn)=∑k1=02∑k2=02⋯∑kn=02∣k1x1+k2x2+⋯+knxn−1∣ f(x_1,x_2,\cdots,x_n)=\sum_{k_1=0}^{2} \sum_{k_2=0}^{2} \cdots \sum_{k_n=0}^{2} \big| k_1x_1+k_2x_2+\cdots+k_nx_n-1 \big| attains its minimum.
algebrainequalitiesfunctionabsolute value
Jigsaw puzzle on a Cartesian plane

Source: 2022 China TST, Test 3 P4

4/30/2022
Find all positive integer kk such that one can find a number of triangles in the Cartesian plane, the centroid of each triangle is a lattice point, the union of these triangles is a square of side length kk (the sides of the square are not necessarily parallel to the axis, the vertices of the square are not necessarily lattice points), and the intersection of any two triangles is an empty-set, a common point or a common edge.
combinatoricsgeometrycombinatorial geometrylattice points
Diophantine equation is back

Source: 2022 China TST, Test 4 P4

4/30/2022
Find all positive integers a,b,ca,b,c and prime pp satisfying that 2apb=(p+2)c+1. 2^a p^b=(p+2)^c+1.
Diophantine equationnumber theory