MathDB

p1. There are two glasses, glass

A

contains

5

red balls, and glass

B

contains

4

red balls and one white ball. One glass is chosen at random and then one ball is drawn at random from the glass. This is done repeatedly until one of the glasses is empty. Determine the probability that the white ball is not drawn.
p2. For any real number

x

, define as

[x]

the largest integer less than or equal to

x

. Find the number of natural numbers

n \le 1,000,000

such that

\sqrt{n}-[\sqrt{n}] <\frac{1}{2013}

p3. A natural number

n

is said to be valid if

1^n + 2^n + 3^n +... + m^n

is divisible by

1 + 2 + 3 +...+ m

for every natural number

m

. a) Show that

2013

is valid. b) Prove that there are infinitely many invalid numbers.
[url=https://artofproblemsolving.com/community/c4h2685414p23296970]p4. Prove that for all positive real numbers

a, b, c

where

a + b + c\le 6

holds

\frac{a+2}{a(a+4)}+\frac{b+2}{b(b+4)}++\frac{c+2}{c(c+4)} \ge 1

[url=https://artofproblemsolving.com/community/c6h2371585p19388892]p5. Given an acute triangle

ABC

. The longest line of altitude is the one from vertex

A

perpendicular to

BC

, and it's length is equal to the length of the median of vertex

B

. Prove that

\angle ABC \le 60^o

Problem Statement