MathDB

[url=https://artofproblemsolving.com/community/c1068820h2917798p26064010]p1. Consider three circles of radius one tangent to each other. Find the radius of each of the circles tangent to the three given circles.
p2. A three-digit number is called balanced if any of its numbers is the average of the others. For example

528

is balanced since

5 =\frac{2+8}{2}

. How many balanced three-digit numbers are there?
[url=https://artofproblemsolving.com/community/c1068820h2917801p26064051]p3.

AB = 3

AC = 8

A_1

midpoint of

AC

BA_1 \parallel B_1A_2 \parallel B_2A_3

, ...

AB\perp AC

A_1B_1\perp AC

A_2B_2\perp AC

A_3B_3\perp AC

... https://cdn.artofproblemsolving.com/attachments/8/3/5fdbdaec663ce80a031c4ebbaa1418085d6de7.jpg Find

2^9 (BA_1 + B_1A_2 + B_2A_3 + ... + B_8A_9)

[url=https://artofproblemsolving.com/community/c1068820h2917799p26064022]p4. Let

ABC

be a triangle right at

C

a

and

b

the lengths of its legs and

r

the radius of the circle that is tangent to the legs and that has the center

O

at the hypotenuse. Prove that

\frac{1}{r}=\frac{1}{a}+\frac{1}{b}

.
[url=https://artofproblemsolving.com/community/c1068820h2917800p26064029]p5.

ABCD

is a square,

E

is the midpoint of

BC

F \in DE

such that

AF \perp DE

. Prove that

\angle CDE = \angle BFE

.
p6. Let

x

be a real number,

[x]

its integer part,

\{x\} = x-[x]

. Find the largest number

C

such that for all x the inequality

x^2\ge C \cdot [x] \cdot \{x\}

holds.

Problem Statement