MathDB

2014 Taiwan TST Round 3

Part of TST Round 3

Subcontests

(3)
3
1
2
1
1
3

Fill in a "magic" 6x6 square

Consider a 6×66 \times 6 grid. Define a diagonal to be the six squares whose coordinates (i,j)(i,j) (1≤i,j≤6)1 \le i,j \le 6) satisfy i−j≡k(mod6)i-j \equiv k \pmod 6 for some k=0,1,…,5k=0,1,\dots,5. Hence there are six diagonals.
Determine if it is possible to fill it with the numbers 1,2,…,361,2,\dots,36 (each exactly once) such that each row, each column, and each of the six diagonals has the same sum.

Another hexagon geometry -- angles on midpoints of sides

In convex hexagon ABCDEFABCDEF, AB∥DEAB \parallel DE, BC∥EFBC \parallel EF, CD∥FACD \parallel FA, and AB+DE=BC+EF=CD+FA. AB+DE = BC+EF = CD+FA. The midpoints of sides ABAB, BCBC, DEDE, EFEF are A1A_1, B1B_1, D1D_1, E1E_1, and segments A1D1A_1D_1 and B1E1B_1E_1 meet at OO. Prove that ∠D1OE1=12∠DEF\angle D_1OE_1 = \frac12 \angle DEF.

Esoteric Sign Calculation

Let R\mathbb R be the real numbers. Set S={1,−1}S = \{1, -1\} and define a function sign⁡:R→S\operatorname{sign} : \mathbb R \to S by sign⁡(x)={1if x≥0;−1if x<0. \operatorname{sign} (x) = \begin{cases} 1 & \text{if } x \ge 0; \\ -1 & \text{if } x < 0. \end{cases} Fix an odd integer nn. Determine whether one can find n2+nn^2+n real numbers aij,bi∈Sa_{ij}, b_i \in S (here 1≤i,j≤n1 \le i, j \le n) with the following property: Suppose we take any choice of x1,x2,…,xn∈Sx_1, x_2, \dots, x_n \in S and consider the values \begin{align*} y_i &= \operatorname{sign} \left( \sum_{j=1}^n a_{ij} x_j \right),   \forall 1 \le i \le n; \\ z &= \operatorname{sign} \left( \sum_{i=1}^n y_i b_i \right) \end{align*} Then z=x1x2…xnz=x_1 x_2 \dots x_n.