MathDB

2

Take the gcd!

n\geq 2

for which there exists a positive integer

m

such that

mn

divides

m^{2023} + n^{2023} + n

.

Heavy counting of routes

On the coast of a circular island there are eight different cities. Initially there are no routes between the cities. We have to construct five straight two-way routes, which do not intersect, so that from each city there are one or two routes. In how many ways can this happen?

1

Some training in systems of equations

Determine all triples

(x,y,z)

of real numbers such that

x^4 + y^3z = zx

,

y^4 + z^3x = xy

and

z^4 + x^3y = yz

.

4

1

Subsets of 3 without exactly one common element

Given is a set of

n\ge5

people and

m

commissions with

3

persons in each. Let all the commissions be nice if there are no two commissions

A

and

B

, such that

\mid A\cap B\mid=1

. Find the biggest possible

m

(as a function of

n

).

3

Diophantine Equation with prime numbers and bonus conditions

Find all natural numbers

a

,

b

,

c

and prime numbers

p

and

q

, such that:

\blacksquare

4\nmid c

\blacksquare

p\not\equiv 11\pmod{16}

\blacksquare

p^aq^b-1=(p+4)^c

Playing with balls in boxes

There are infinitely many boxes - initially one of them contains

n

balls and all others are empty. On a single move we take some balls from a non-empty box and put them into an empty box and on a sheet of paper we write down the product of the resulting amount of balls in the two boxes. After some moves, the sum of all numbers on the sheet of paper became

2023

. What is the smallest possible value of

n

?

Hidden devil properties of incircle and excircle

Let

ABC

be a non-isosceles triangle with circumcircle

k

, incenter

I

and

C

-excenter

I_C

. Let

M

be the midpoint of

AB

and

N

be the midpoint of arc

\widehat{ACB}

on

k

. Prove that

\angle IMI_C + \angle INI_C = 180^{\circ}

.

2023 Bulgaria JBMO TST

Subcontests

Take the gcd!

Heavy counting of routes

Some training in systems of equations

Subsets of 3 without exactly one common element

Diophantine Equation with prime numbers and bonus conditions

Playing with balls in boxes

Hidden devil properties of incircle and excircle