MathDB

1

union of mutually disjoint 3-element subsets >= n-5

n \ge 6

be an integer and

F

be the system of the

3

-element subsets of the set

\{1, 2,...,n \}

satisfying the following condition: for every

1 \le i < j \le n

there is at least

\lfloor \frac{1}{3} n \rfloor -1

subsets

A\in F

such that

i, j \in A

. Prove that for some integer

m \ge 1

exist the mutually disjoint subsets

A_1, A_2 , ... , A_m \in F

also, that

|A_1\cup A_2 \cup ... \cup A_m |\ge n-5

(Poland)
PS. just in case my translation does not make sense, I leave the original in Slovak, in case someone understands something else

5

1

two binomials leave the same remainder when divided by 2

Let all positive integers

n

satisfy the following condition: for each non-negative integers

k, m

with

k + m \le n

, the numbers

\binom{n-k}{m}

and

\binom{n-m}{k}

leave the same remainder when divided by

2

.
(Poland)
PS. The translation was done using Google translate and in case it is not right, there is the original text in Slovak

2

1

x_n = ax_{n-1} + b, m | n => x_m | x_n

For the positive integers

a, b, x_1

we construct the sequence of numbers

(x_n)_{n=1}^{\infty}

such that

x_n = ax_{n-1} + b

for each

n \ge 2

. Specify the conditions for the given numbers

a, b

and

x_1

which are necessary and sufficient for all indexes

m, n

to apply the implication

m | n \Rightarrow x_m | x_n

.
(Jaromír Šimša)

4

1

concurrent lines by a triangle ,a circle and 3 products

Let

ABC

be a triangle, and let

P

be the midpoint of

AC

. A circle intersects

AP, CP, BC, AB

sequentially at their inner points

K, L, M, N

. Let

S

be the midpoint of

KL

. Let also

2 \cdot | AN |\cdot |AB |\cdot |CL | = 2 \cdot | CM | \cdot| BC | \cdot| AK| = | AC | \cdot| AK |\cdot |CL |.

Prove that if

P\ne S

, then the intersection of

KN

and

ML

lies on the perpendicular bisector of the

PS

.
(Jan Mazák)

3

1

perpendiculars on a convex ABCD with 2 angles 135^o

Given is a convex

ABCD

, which is

|\angle ABC| = |\angle ADC|= 135^\circ

. On the

AB, AD

are also selected points

M, N

such that

|\angle MCD| = |\angle NCB| = 90^ \circ

. The circumcircles of the triangles

AMN

and

ABD

intersect for the second time at point

K \ne A

. Prove that lines

AK

and

KC

are perpendicular.
(Irán)

1

trigonometric existence of triangle given equation w sides

Prove that if the positive real numbers

a, b, c

satisfy the equation

a^4 + b^4 + c^4 + 4a^2b^2c^2 = 2 (a^2b^2 + a^2c^2 + b^2c^2),

then there is a triangle

ABC

with internal angles

\alpha, \beta, \gamma

such that

\sin \alpha = a, \qquad \sin \beta = b, \qquad \sin \gamma= c.

2014 Czech-Polish-Slovak Match

Subcontests

union of mutually disjoint 3-element subsets >= n-5

two binomials leave the same remainder when divided by 2

x_n = ax_{n-1} + b, m | n => x_m | x_n

concurrent lines by a triangle ,a circle and 3 products

perpendiculars on a convex ABCD with 2 angles 135^o

trigonometric existence of triangle given equation w sides

2014 Czech-Polish-Slovak Match

Subcontests

union of mutually disjoint 3-element subsets &gt;= n-5

two binomials leave the same remainder when divided by 2

x_n = ax_{n-1} + b, m | n =&gt; x_m | x_n

concurrent lines by a triangle ,a circle and 3 products

perpendiculars on a convex ABCD with 2 angles 135^o

trigonometric existence of triangle given equation w sides

union of mutually disjoint 3-element subsets >= n-5

x_n = ax_{n-1} + b, m | n => x_m | x_n