MathDB

1

A hard problem with triangle inequality

x, y, z

be positive real numbers such that

\sqrt {x}, \sqrt {y}, \sqrt {z}

are sides of a triangle and

\frac {x}{y}+\frac {y}{z}+\frac {z}{x}=5

. Prove that

\frac {x(y^2-2z^2)}{z}+\frac {y(z^2-2x^2)}{x}+\frac {z(x^2-2y^2)}{y}\geqslant0

7

1

Minimal number of students

In the round robin chess tournament organized in a school every two students played one match among themselves. Find the minimal possible number of students in the school if each girl student has at least 21 wins in matches against boy students and each boy student has at least 12 wins in matches against girl students.

5

1

a+2018 is a perfect square

Let

a_1, a_2, ... , a_{1000}

be a sequence of integers such that

a_1=3, a_2=7

and for all

n=2, 3, ... , 999

a_{n+1}-a_n=4(a_1+a_2)(a_2+a_3) ... (a_{n-1}+a_n)

. Find the number of indices

1\leq n\leq 1000

for which

a_n+2018

is a perfect square.

4

1

Number of possible ball distributions

n\geq3

boxes are placed around a circle. At the first step we choose some boxes. At the second step for each chosen box we put a ball into the chosen box and into each of its two neighbouring boxes. Find the total number of possible distinct ball distributions which can be obtained in this way. (All balls are identical.)

2

1

Sum of divisors

Two distinct positive integers are called "relatively consistent" if the larger one can be written as a sum of some distinct positive divisors of the other one. Show that there exist 2018 positive integers such that any two of them are "relatively consistent"

1

3 eqaul numbers and prove that x=1

Let

a, b, c

be distinct real numbers and

x

be a real number. Given that three numbers among

ax^2+bx+c, ax^2+cx+b, bx^2+cx+a, bx^2+ax+c, cx^2+ax+b, cx^2+bx+a

coincide, prove that

x=1

.

3

1

orthocenter, intersection of 2 circles, 2 midpoints of arcs, are concyclic

Let

H

be the orthocenter of an acute angled triangle

ABC

. Circumcircle of the triangle

ABC

and the circle of diameter

[AH]

intersect at point

E

, different from

A

. Let

M

be the midpoint of the small arc

BC

of the circumcircle of the triangle

ABC

and let

N

the midpoint of the large arc

BC

of the circumcircle of the triangle

BHC

Prove that points

E, H, M, N

are concyclic.

6

1

concyclic circumcenters, starting with a point that sees in equal angles in a #

A point

E

is located inside a parallelogram

ABCD

such that

\angle BAE = \angle BCE

. The centers of the circumcircles of the triangles

ABE,ECB, CDE

and

DAE

are concyclic.

2018 JBMO TST-Turkey

Subcontests

A hard problem with triangle inequality

Minimal number of students

a+2018 is a perfect square

Number of possible ball distributions

Sum of divisors

3 eqaul numbers and prove that x=1

orthocenter, intersection of 2 circles, 2 midpoints of arcs, are concyclic

concyclic circumcenters, starting with a point that sees in equal angles in a #