MathDB

p1. There are eight different dominoes in the box (fig.), but the boundaries between them are not visible. Draw the boundaries. https://cdn.artofproblemsolving.com/attachments/6/f/6352b18c25478d68a23820e32a7f237c9f2ba9.png
p2. The teacher drew a quadrilateral

ABCD

on the board. Vanya and Vitya marked points

X

and

Y

inside it, from which all sides of the quadrilateral are visible at equal angles. What is the distance between points

X

and

Y

? (From point

X

, side

AB

is visible at angle

AXB

.)
pЗ. Several identical black squares, perhaps partially overlapping, were placed on a white plane. The result was a black polygonal figure, possibly with holes or from several pieces. Could it be that this figure does not have a single right angle?
p4. The bus ticket number consists of six digits (the first digits may be zeros). A ticket is called lucky if the sum of the first three digits is equal to the sum of the last three. Prove that the sum of the numbers of all lucky tickets is divisible by

13

.
p5. The Meandrovka River, which has many bends, crosses a straight highway under thirteen bridges. Prove that there are two neighboring bridges along both the highway and the river. (Bridges are called river neighbors if there are no other bridges between them on the river section; bridges are called highway neighbors if there are no other bridges between them on the highway section.)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.

Problem Statement