MathDB

p1. Determine the unit digit of the number resulting from the following sum

2013^1 + 2013^2 + 2013^3 + ... + 2013^{2012} + 2013^{2013}

2. Every real number a can be uniquely written as

a = [a] +\{a\}

, where

[a]

is an integer and

0\le \{a\}<1

. For example, if

a = 2.12

, then

[2.12] = 2

and

\{2.12\} = 0.12

. Given the:

x + [y] + \{z\} = 4.2

y + [z] + \{x\} = 3.6

z + [x] + \{y\} = 2.0

Determine the value of

x - y + z

.
p3. Determine all pairs of positive integers

(x, y)

such that

2 (x + y) + xy = x^2 + y^2

.
p4. Consider the following arrangement of dots on the board in the figure. Determine the number of ways three of these points can be selected, to be the vertices of a right triangle whose legs are parallel to the sides of the board. https://cdn.artofproblemsolving.com/attachments/3/1/a9025e6e6f41f6a8d745a1c695b61640e9c691.png
p5. In an acute triangle

ABC

\angle A=30^o

. Let

D

and

E

be the feet of the altitudes drawn from

B

and

C

, respectively. Let

F

and

G

be the midpoints of sides

AC

and

AB

, respectively. Prove that segments

DG

and

EF

are perpendicular.

Problem Statement