MathDB

3

Part of 1978 Romania Team Selection Test

Problems(4)

painting 3n points

Source: Romanian TST 1978, Day 2, P3

9/30/2018
Let A1,A2,...,A3n A_1,A_2,...,A_{3n} be 3n≥3 3n\ge 3 planar points such that A1A2A3 A_1A_2A_3 is an equilateral triangle and A3k+1,A3k+2,A3k+3 A_{3k+1} ,A_{3k+2} ,A_{3k+3} are the midpoints of the sides of A3k−2A3k−1A3k, A_{3k-2}A_{3k-1}A_{3k} , for all 1≤k<n. 1\le k<n. Of two different colors, each one of these points are colored, either with one, either with another.
a) Prove that, if n≥7, n\ge 7, then some of these points form a monochromatic (only one color) isosceles trapezoid. b) What about n=6? n=6?
geometrytrapezoidColoring
polynomial connection to geometry

Source: Romanian TST 1978, Day 1, P3

9/28/2018
Let P[X,Y] P[X,Y] be a polynomial of degree at most 2. 2 . If A,B,C,A′,B′,C′ A,B,C,A',B',C' are distinct roots of P P such that A,B,C A,B,C are not collinear and A′,B′,C′ A',B',C' lie on the lines BC,CA, BC,CA, respectively, AB, AB, in the planar representation of these points, show that P=0. P=0.
algebrapolynomialanalytic geometrygeometry
3D analytic geometry (skew lines)

Source: Romanian TST 1978, Day 3, P3

9/30/2018
a) Let D1,D2,D3 D_1,D_2,D_3 be pairwise skew lines. Through every point P2∈D2 P_2\in D_2 there is an unique common secant of these three lines that intersect D1 D_1 at P1 P_1 and D3 D_3 at P3. P_3. Let coordinate systems be introduced on D2 D_2 and D3 D_3 having as origin O2, O_2, respectively, O3. O_3. Find a relation between the coordinates of P2 P_2 and P3. P_3.
b) Show that there exist four pairwise skew lines with exactly two common secants. Also find examples with exactly one and with no common secants.
c) Let F1,F2,F3,F4 F_1,F_2,F_3,F_4 be any four secants of D1,D2,D3. D_1,D_2, D_3. Prove that F1,F2,F3,F4 F_1,F_2, F_3, F_4 have infinitely many common secants.
analytic geometrygeometryskew lines3D geometry3D analytic geometry
Partitions.

Source: Romanian TST 1978, Day 4, P3

10/1/2018
Let p p be a natural number and let two partitions A={A1,A2,...,Ap},B={B1,B2,...Bp} \mathcal{A} =\left\{ A_1,A_2,...,A_p\right\} ,\mathcal{B}=\left\{ B_1,B_2,...B_p\right\} of a finite set M. \mathcal{M} . Knowing that, whenever an element of A \mathcal{A} doesn´t have any elements in common with another of B, \mathcal{B} , it holds that the number of elements of these two is greater than p, p, prove that ∣M∣≥12(1+p2). \big| \mathcal{M}\big|\ge\frac{1}{2}\left( 1+p^2\right) . Can equality hold?
set theorypartitions