MathDB

1

Students and problems with constraints, find number of students

n \ge 2

be given. Let Laura be a student in a class of more than

n+2

students, all of which participated in an olympiad and solved some problems. Additionally, it is known that:
[*] for every pair of students there is exactly one problem that was solved by both students; [*] for every pair of problems there is exactly one student who solved both of them; [*] one specific problem was solved by Laura and exactly

n

other students.
Determine the number of students in Laura's class.

P7

1

Determine the existence of an n-gon with given points

Let

n \ge 3

points be given in the plane, no three of which lie on the same line. Determine whether it is always possible to draw an

n

-gon whose vertices are the given points and whose sides do not intersect. Remark. The

n

-gon can be concave.

P6

1

Colored grid with many constraints

Let

ABCD

be a rectangle consisting of unit squares. All vertices of these unit squares inside the rectangle and on its sides have been colored in four colors. Additionally, it is known that:
[*] every vertex that lies on the side

AB

has been colored in either the

1.

or

2.

color; [*] every vertex that lies on the side

BC

has been colored in either the

2.

or

3.

color; [*] every vertex that lies on the side

CD

has been colored in either the

3.

or

4.

color; [*] every vertex that lies on the side

DA

has been colored in either the

4.

or

1.

color; [*] no two neighboring vertices have been colored in

1.

and

3.

color; [*] no two neighboring vertices have been colored in

2.

and

4.

color.
Notice that the constraints imply that vertex

A

has been colored in

1.

color etc. Prove that there exists a unit square that has all vertices in different colors (in other words it has one vertex of each color).

P5

1

Combi game on a row of squares

Alice and Bob play a game on a numbered row of

n \ge 5

squares. At the beginning a pebble is put on the first square and then the players make consecutive moves; Alice starts. During a move a player is allowed to choose one of the following:
[*] move the pebble one square forward; [*] move the pebble four squares forward; [*] move the pebble two squares backwards.
All of the possible moves are only allowed if the pebble stays within the borders of the square row. The player who moves the pebble to the last square (a.k.a

n\text{-th}

) wins. Determine for which values of

n

each of the players has a winning strategy.

P4

1

Functional inequality with size constraint

Let

f:\mathbb{R}\rightarrow\mathbb{R}

be a function that satisfies

\sqrt{2f(x)}-\sqrt{2f(x)-f(2x)}\ge 2

for all real

x

. Prove for all real

x

: (a)

f(x)\ge 4

; (b)

f(x)\ge 7.

P3

1

General Fibonacci sequence with a property

Let

a_1,a_2,...

be an infinite sequence of integers that satisfies

a_{n+2}=a_{n+1}+a_n

for all

n \ge 1

. There exists a positive integer

k

such that

a_k=a_{k+2018}

. Prove that there exists a term of the sequence which is equal to zero.

P2

1

System of equations in real numbers

Find all ordered pairs

(x,y)

of real numbers that satisfy the following system of equations:

\begin{cases} y(x+y)^2=2\\ 8y(x^3-y^3) = 13. \end{cases}

P1

1

Max and min value with constraint

Let

p_1,p_2,...,p_n

be

n\ge 2

fixed positive real numbers. Let

x_1,x_2,...,x_n

be nonnegative real numbers such that

x_1p_1+x_2p_2+...+x_np_n=1.

Determine the (a) maximal; (b) minimal possible value of

x_1^2+x_2^2+...+x_n^2

.

P16

1

Infinitely many consecutive pairs of square-free integers

Call a natural number simple if it is not divisible by any square of a prime number (in other words it is square-free). Prove that there are infinitely many positive integers

n

such that both

n

and

n+1

are simple.

P15

1

Choose n to get 2018 solutions to a system of equations

Determine whether there exists a positive integer

n

such that it is possible to find at least

2018

different quadruples

(x,y,z,t)

of positive integers that simultaneously satisfy equations

\begin{cases} x+y+z=n\\ xyz = 2t^3. \end{cases}

P14

1

Sequence of biggest prime divisors

Let

a_1,a_2,...

be a sequence of positive integers with

a_1=2

. For each

n \ge 1

,

a_{n+1}

is the biggest prime divisor of

a_1a_2...a_n+1

. Prove that the sequence does not contain numbers

5

and

11

.

P13

1

Expression of primes never an integer

Determine whether there exists a prime

q

so that for any prime

p

the number

\sqrt[3]{p^2+q}

is never an integer.

P12

1

Parallelogram with arbitrary segments, another parallelity needed

Let

ABCD

be a parallelogram. Let

X

and

Y

be arbitrary points on sides

BC

and

CD

, respectively. Segments

BY

and

DX

intersect at

P

. Prove that the line going through the midpoints of segments

BD

and

XY

is either parallel to or coincides with line

AP

.

P11

1

Given angles of triangle, prove third order segment equality

Let

ABC

be a triangle with angles

\angle A = 80^\circ, \angle B = 70^\circ, \angle C = 30^\circ

. Let

P

be a point on the bisector of

\angle BAC

satisfying

\angle BPC =130^\circ

. Let

PX, PY, PZ

be the perpendiculars drawn from

P

to the sides

BC, AC, AB

, respectively. Prove that the following equation with segment lengths is satisfied

AY^3+BZ^3+CX^3=AZ^3+BX^3+CY^3.

P10

1

Obtuse triangle with symmetric points

Let

ABC

be an obtuse triangle with obtuse angle

\angle B

and altitudes

AD, BE, CF

. Let

T

and

S

be the midpoints of

AD

and

CF

, respectively. Let

M

and

N

and be the symmetric images of

T

with respect to lines

BE

and

BD

, respectively. Prove that

S

lies on the circumcircle of triangle

BMN

.

P9

1

Triangle configuration with line intersection on circumcircle

Acute triangle

\triangle ABC

with

AB<AC

, circumcircle

\Gamma

and circumcenter

O

is given. Midpoint of side

AB

is

D

. Point

E

is chosen on side

AC

so that

BE=CE

. Circumcircle of triangle

BDE

intersects

\Gamma

at point

F

(different from point

B

). Point

K

is chosen on line

AO

satisfying

BK \perp AO

(points

A

and

K

lie in different half-planes with respect to line

BE

). Prove that the intersection of lines

DF

and

CK

lies on

\Gamma

.

2018 Latvia Baltic Way TST

Subcontests

Students and problems with constraints, find number of students

Determine the existence of an n-gon with given points

Colored grid with many constraints

Combi game on a row of squares

Functional inequality with size constraint

General Fibonacci sequence with a property

System of equations in real numbers

Max and min value with constraint

Infinitely many consecutive pairs of square-free integers

Choose n to get 2018 solutions to a system of equations

Sequence of biggest prime divisors

Expression of primes never an integer

Parallelogram with arbitrary segments, another parallelity needed

Given angles of triangle, prove third order segment equality

Obtuse triangle with symmetric points

Triangle configuration with line intersection on circumcircle