MathDB

p1. Consider the sequence of digits obtained from writing the consecutive natural numbers from

1

100,000

1234567891011121314...9999899999100000

Determine how many times the

2016

block appears in this sequence.
[url=https://artofproblemsolving.com/community/c4h1845779p12426933]p2. For an equilateral triangle

\triangle ABC

, determine whether or not there is a point

P

inside

\triangle ABC

so that any straight line that passes through

P

divides the triangle

\triangle ABC

in two polygonal lines of equal length.
p3. On a

1000 \times 1000

squared board, place domino pieces (

2\times 1

1\times 2

), so that each piece of domino covers exactly two squares of the board. Two domino pieces are not allowed to be adjacent, and they are allowed to be touch in a vertex. Determine the maximum number of domino pieces that can be put following these rules.
[url=https://artofproblemsolving.com/community/c4h2945587p26369677]p4. The product

\frac12 \cdot \frac24 \cdot \frac38 \cdot \frac{4}{16} \cdot ... \cdot \frac{99}{2^{99}} \cdot \frac{100}{2^{100}}

is written in its most simplified form. What is the last digit of the denominator?
[url=https://artofproblemsolving.com/community/c4h2917781p26063806]p5. Let

\vartriangle ABC

be an is osceles triangle with

AC = BC

. Let

O

be the center of the circle circumscribed to the triangle and

I

the center of the inscribed circle. If

D

is the point on side

BC

such that

OD

is perpendicular to

BI

. Show that

ID

is parallel to

AC

.
p6. Beto plays the following solitaire: initially a machine chooses at random

26

positive integers between

1

and

2016

, and writes them on a blackboard (there may be numbers repeated). At each step, Beto chooses some of the numbers written on the blackboard, and they subtract from each of them the same non-negative integer number k with the condition that the resulting numbers remain non-negative. The objective of the game is to achieve that in sometime the

26

numbers are equal to

0

, in which case the game ends and Beto win. Determine the fewest steps that guarantee Beto victory, without import the

26

numbers initially chosen.

Problem Statement