MathDB

p1. A king, his daughter and his son were locked at the top of a tower. The Monarch weighed

91

Kg, the daughter

42

Kg and the son

49

Kg. They had a pulley with a rope that reached exactly from the top of the tower to the ground with a basket at each end, and also had a huge rock

35

Kg. How did they manage to get off, if the weight difference between the two baskets could not be greater than

7

Kg because otherwise the rope would break?
p2. With three different digits, six different three-digit numbers are formed. If these six numbers are added the result is

4218

. The sum of the three largest numbers minus the sum of the smallest three equals

792

. Find the three digits.
p3. Let

a_n

(n is natural) be the unit digit of

2009^1+2009^2+...+2009^{n^2}

. Find the sum

a_1+a_2+...+a_n

.
p4.

ABC

is a triangle such that

AB=AC

and

\angle A=40^o

AD

is altitude,

E

is a point on side

AB

such that the

\angle ACE =10^o

F

is the intersection of

AD

with

CE

. Prove that

CF=BC

. https://cdn.artofproblemsolving.com/attachments/a/7/39412da50b291d1dbea540301dfd956ac61060.png
p5. Determine the values of natural

n

less than or equal to

100

for which the following expression is a natural number

\sqrt{\frac{1}{\sqrt1+\sqrt2}+\frac{1}{\sqrt2+\sqrt3}+...+\frac{1}{\sqrt{n-1}+\sqrt{n}}}

Problem Statement