MathDB

p1. Manuel writes a list with the positive integers from

1

2018

. Next, he places a sign negative in front of all powers of

2

and a positive sign in front of all other numbers, that is to say:

-1 - 2 + 3 - 4 + 5 + 6 + 7 - 8 +... + 2018

Calculate the result that Manuel obtains when carrying out the indicated operations.
p2. Determine the number of whole numbers

n

between

1

and

2018

, inclusive, such that the product

\left(1+\frac12\right)\left(1+\frac13\right)...\left(1+\frac{1}{n}\right)

is also an integer.
p3. Determine all pairs

(x, y)

of real numbers that satisfy the equation

(x + y)^2 = (x + 2018) (y - 2018)

p4. In triangle

ABC

, the angle

\angle ACB =90^o

. The internal bisectors of the angles

\angle BAC

and

\angle ABC

intersect the sides

BC

and

CA

P

and

Q

, respectively. Points

M

and

N

are the feet of the perpendiculars from

P

and

Q

to side

AB

. Calculate the measure of the angle

\angle MCN

.
p5. Carlos takes turns with Rodrigo on the next

5 \times 6

board. https://cdn.artofproblemsolving.com/attachments/8/e/388ee57b67289ac9b50565d1444ba5d3e1d958.png A move consists of placing a piece of one of the following three types on the board: https://cdn.artofproblemsolving.com/attachments/b/4/aa96fcb2b5acde281ca0b8b7fa1e8bd00e0083.png Parts can be rotated before being located and not allowed to overlap with those in place previously on the board. Whoever finishes covering the board wins. If Carlos moves first, determine the player who can secure the victory and the strategy that must continue to win.

Problem Statement