MathDB

1

Part of 2017 Taiwan TST Round 2

Problems(3)

Very easy combinatoric problem

Source: 2017 Taiwan TST

4/13/2018
There is a 2n×2n2n\times 2n rectangular grid and a chair in each cell of the grid. Now, there are 2n22n^2 pairs of couple are going to take seats. Define the distance of a pair of couple to be the sum of column difference and row difference between them. For example, if a pair of couple seating at (3,3)(3,3) and (2,5)(2,5) respectively, then the distance between them is ∣3−2∣+∣3−5∣=3|3-2|+|3-5|=3. Moreover, define the total distance to be the sum of the distance in each pair. Find the maximal total distance among all possibilities.
combinatorics
Functional equation 007

Source: 2017 Taiwan TST Round 2, Day 3, Problem 1

4/19/2017
Determine all surjective functions f:Z→Z f: \mathbb{Z} \to \mathbb{Z} such that f(xyz+xf(y)+yf(z)+zf(x))=f(x)f(y)f(z) f\left(xyz+xf\left(y\right)+yf\left(z\right)+zf\left(x\right)\right)=f\left(x\right)f\left(y\right)f\left(z\right) for all x,y,z x,y,z in Z \mathbb{Z}
algebrafunctional equation
Geometry problem

Source: 2017 Taiwan TST

4/13/2018
Given a circle and four points B,C,X,YB,C,X,Y on it. Assume AA is the midpoint of BCBC, and ZZ is the midpoint of XYXY. Let L1,L2L_1,L_2 be lines perpendicular to BCBC and pass through B,CB,C respectively. Let the line pass through XX and perpendicular to AXAX intersects L1,L2L_1,L_2 at X1,X2X_1,X_2 respectively. Similarly, let the line pass through YY and perpendicular to AYAY intersects L1,L2L_1,L_2 at Y1,Y2Y_1,Y_2 respectively. Assume X1Y2X_1Y_2 intersects X2Y1X_2Y_1 at PP. Prove that ∠AZP=90o.\angle AZP=90^o.
Proposed by William Chao
geometry