MathDB

1

Part of 2022 Germany Team Selection Test

Problems(2)

Common refinement of two integer factorizations

Source: German TST 2022, exam 2, problem 1

3/11/2022
Let a1,a2,…,ana_1, a_2, \ldots, a_n be nn positive integers, and let b1,b2,…,bmb_1, b_2, \ldots, b_m be mm positive integers such that a1a2⋯an=b1b2⋯bma_1 a_2 \cdots a_n = b_1 b_2 \cdots b_m. Prove that a rectangular table with nn rows and mm columns can be filled with positive integer entries in such a way that
* the product of the entries in the ii-th row is aia_i (for each i∈{1,2,…,n}i \in \left\{1,2,\ldots,n\right\});
* the product of the entries in the jj-th row is bjb_j (for each i∈{1,2,…,m}i \in \left\{1,2,\ldots,m\right\}).
number-theoryalgebrafactorizationcombinatorics
Concurrent perpendicular bisectors from three circles in a triangle

Source: German TST 2022, probably on exam 6-7, again proposed by me

7/19/2022
Given a triangle ABCABC and three circles xx, yy and zz such that A∈y∩zA \in y \cap z, B∈z∩xB \in z \cap x and C∈x∩yC \in x \cap y.
The circle xx intersects the line ACAC at the points XbX_b and CC, and intersects the line ABAB at the points XcX_c and BB. The circle yy intersects the line BABA at the points YcY_c and AA, and intersects the line BCBC at the points YaY_a and CC. The circle zz intersects the line CBCB at the points ZaZ_a and BB, and intersects the line CACA at the points ZbZ_b and AA. (Yes, these definitions have the symmetries you would expect.)
Prove that the perpendicular bisectors of the segments YaZaY_a Z_a, ZbXbZ_b X_b and XcYcX_c Y_c concur.
geometryperpendicular bisectorTriangle GeometryTST