MathDB

4

Part of 2023 Durer Math Competition Finals

Problems(3)

1 piece of 1 , 2 pieces of 1 , 3 pieces of 3 , . . . , 50 piecies of 50

Source: (2022-) 2023 XVI 16th Dürer Math Competition Finals Day 2 E4

5/25/2024
Benedek wrote down the following numbers: 11 piece of one, 22 pieces of twos, 33 pieces of threes, ...... , 5050 piecies of fifties. How many digits did Benedek write down?
algebra
infinite number of n-sided polygonal numbers = sum of 2 other polygonal numbers

Source: (2022 -) 2023 XVI Dürer Math Competition Finals Day 1 E4

5/25/2024
Prove that for all n≥3n \ge 3 there are an infinite number of nn-sided polygonal numbers which are also the sum of two other (not necessarily different) nn-sided polygonal numbers!
The first nn-sided polygonal number is 11. The kth n-sided polygonal number for k≥2k \ge 2 is the number of different points in a figure that consists of all of the regular nn-sided polygons which have one common vertex, are oriented in the same direction from that vertex and their sides are ℓ\ell cm long where 1≤ℓ≤k−11 \le \ell \le k - 1 cm and ℓ\ell is an integer.
In this figure, what we call points are the vertices of the polygons and the points that break up the sides of the polygons into exactly 11 cm long segments. For example, the first four pentagonal numbers are 1,5,12, and 22, like it is shown in the figure. https://cdn.artofproblemsolving.com/attachments/1/4/290745d4be1888813678127e6d63b331adaa3d.png
combinatoricscombinatorial geometrynumber theory
Stable pyramid

Source: Dürer Competition Finals 2023/E+ 4

3/8/2023
For a given integer n≥2n\geq2, a pyramid of height nn if defined as a collection of 12+22+⋯+n21^2+2^2+\dots+n^2 stone cubes of equal size stacked in nn layers such that the cubes in the kk-th layer form a square with sidelength n+1−kn+1-k and every cube (except for the ones in the bottom layer) rests on four cubes in the layer below. Some of the cubes are made of sandstone, some are made of granite. The top cube is made of granite, and to ensure the stability of the piramid, for each granite cube (except for the ones in the bottom layer), at least three out of four of the cubes supporting it have to be granite. What is the minimum possible number of granite cubes in such an arrangement?
geometry3D geometrypyramid