MathDB

p1. A number of

n

students sit around a round table. It is known that there are as many male students as female students. If the number of pairs of

2

people sitting next to each other is counted, it turns out that the ratio between adjacent pairs of the same sex and adjacent pairs of the opposite sex is

3:2

. Find the smallest possible

n

.
p2. Let

a, b

, and

c

be positive integers so that

c=a+\frac{b}{a}-\frac{1}{b}.

Prove that

c

is the square of an integer.
[url=https://artofproblemsolving.com/community/c6h2370981p19378649]p3. Let

\Gamma_1

and

\Gamma_2

be two different circles with the radius of same length and centers at points

O_1

and

O_2

, respectively. Circles

\Gamma_1

and

\Gamma_2

are tangent at point

P

. The line

\ell

passing through

O_1

is tangent to

\Gamma_2

at point

A

. The line

\ell

intersects

\Gamma_1

at point

X

with

X

between

A

and

O_1

. Let

M

be the midpoint of

AX

and

Y

the intersection of

PM

and

\Gamma_2

with

Y\ne P

. Prove that

XY

is parallel to

O_1O_2

.
[url=https://artofproblemsolving.com/community/c4h2686086p23303294]p4. Let

a, b, c

be positive real numbers with

\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3

. Prove that

a+b+c+\frac{4}{1+(abc)^{2/3}}\ge 5

p5. On a chessboard measuring

200 \times 200

square units are placed red or blue marbles so that each unit square has at most

1

marble. Two marbles are said to be in a row if they are in the same row or column. It is known that for every red marble there are exactly

5

blue marbles in a row and for every blue marble there are exactly

5

red marbles in a row. Determine the maximum number of marbles possible on the chessboard.

Problem Statement