MathDB

p1. The following infinite pyramid is filled with the odd numbers as follows: https://cdn.artofproblemsolving.com/attachments/6/0/c7194dec2d5ab15d6a958758ea69b618667391.png Determine from what level the sum of all the numbers that appear in that level and previous levels is greater than

2020

.
p2. Considered the numbers

x_n =\overline{abaabaaab...b}\overline{\underbrace{a...a}_{n\,\, times}{b}}

where

a

and

b

are digits. Determine for how many integers

n

with

1\le n\le 2020

x_n

is divisible by

11

regardless of the values of

a

and

b

.
p3. In the following figure, segments

BC

and

AD

are parallel. Calculate the length of segment

AD

. https://cdn.artofproblemsolving.com/attachments/b/d/bace12767e8fd53400e55f609ec38abced3c7b.png
p4. Determine if there are real numbers

x, y,z

other than

0

, such that the numbers

a, b, c

defined by

a = \frac{y-z}{2}

b = \frac{z-x}{2}

and

c=\frac{x- y}{2}

satisfy the equality

(a + b + c)^2 = 2abc- 1

p5. Armando and Mauricio play Schwarz. Schwarz is a turn-based game played on one page of paper marked with 202020 points placed arbitrarily such that there are no 3 aligned. On Each turn the player must select two pairs of points that have not been previously joined and join each of them with a line segment, if a triangle is formed, at the end of the turn, they are eliminated the points that make it up and the line segments connected to these points and continue with the other player's turn. If in a turn it is only possible to join a couple of points, they must be joined. The loser is the one who eliminates the last three points. If you start Armando, determine who has the winning strategy.

Problem Statement