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International Contests
Czech-Polish-Slovak Junior Match
2014 Czech-Polish-Slovak Junior Match
2014 Czech-Polish-Slovak Junior Match
Part of
Czech-Polish-Slovak Junior Match
Subcontests
(6)
6
1
Hide problems
min max of F =(y-x)(x+4y) when x^2y^2 + xy + 1 = 3y^2
Determine the largest and smallest fractions
F
=
y
−
x
x
+
4
y
F = \frac{y-x}{x+4y}
F
=
x
+
4
y
y
−
x
if the real numbers
x
x
x
and
y
y
y
satisfy the equation
x
2
y
2
+
x
y
+
1
=
3
y
2
x^2y^2 + xy + 1 = 3y^2
x
2
y
2
+
x
y
+
1
=
3
y
2
.
5
2
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largest possible sum of internal angles of n polygons a square is divdided
A square is given. Lines divide it into
n
n
n
polygons. What is he the largest possible sum of the internal angles of all polygons?
contructing positive integers when (a + b + 1) divides (a^2 + b^2 + 1) ?
There is the number
1
1
1
on the board at the beginning. If the number
a
a
a
is written on the board, then we can also write a natural number
b
b
b
such that
a
+
b
+
1
a + b + 1
a
+
b
+
1
is a divisor of
a
2
+
b
2
+
1
a^2 + b^2 + 1
a
2
+
b
2
+
1
. Can any positive integer appear on the board after a certain time? Justify your answer.
4
2
Hide problems
find smallest t such that 11 | a_t , where a_n=12345678091011...n
The number
a
n
a_n
a
n
is formed by writing in succession, without spaces, the numbers
1
,
2
,
.
.
.
,
n
1, 2, ..., n
1
,
2
,
...
,
n
(for example,
a
11
=
1234567891011
a_{11} = 1234567891011
a
11
=
1234567891011
). Find the smallest number t such that
11
∣
a
t
11 | a_t
11∣
a
t
.
perpendicular wanted when 2 triangles have same areas, circle related
Point
M
M
M
is the midpoint of the side
A
B
AB
A
B
of an acute triangle
A
B
C
ABC
A
BC
. Circle with center
M
M
M
passing through point
C
C
C
, intersects lines
A
C
,
B
C
AC ,BC
A
C
,
BC
for the second time at points
P
,
Q
P,Q
P
,
Q
respectively. Point
R
R
R
lies on segment
A
B
AB
A
B
such that the triangles
A
P
R
APR
A
PR
and
B
Q
R
BQR
BQR
have equal areas. Prove that lines
P
Q
PQ
PQ
and
C
R
CR
CR
are perpendicular.
3
2
Hide problems
3x2 tiles in a 7x7 boards
We have
10
10
10
identical tiles as shown. The tiles can be rotated, but not flipper over. A
7
×
7
7 \times 7
7
×
7
board should be covered with these tiles so that exactly one unit square is covered by two tiles and all other fields by one tile. Designate all unit sqaures that can be covered with two tiles. https://cdn.artofproblemsolving.com/attachments/d/5/6602a5c9e99126bd656f997dee3657348d98b5.png
|n^3 - 4n^2 + 3n - 35| and |n^2 + 4n + 8| are primes
Find with all integers
n
n
n
when
∣
n
3
−
4
n
2
+
3
n
−
35
∣
|n^3 - 4n^2 + 3n - 35|
∣
n
3
−
4
n
2
+
3
n
−
35∣
and
∣
n
2
+
4
n
+
8
∣
|n^2 + 4n + 8|
∣
n
2
+
4
n
+
8∣
are prime numbers.
2
2
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a + b + 4 = 4\sqrt{a\sqrt{b}}
Solve the equation
a
+
b
+
4
=
4
a
b
a + b + 4 = 4\sqrt{a\sqrt{b}}
a
+
b
+
4
=
4
a
b
in real numbers
circumcenter equidistant from 2 points wanted, parallelogram related
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram with
∠
B
A
D
<
9
0
o
\angle BAD<90^o
∠
B
A
D
<
9
0
o
and
A
B
>
B
C
AB> BC
A
B
>
BC
. The angle bisector of
B
A
D
BAD
B
A
D
intersects line
C
D
CD
C
D
at point
P
P
P
and line
B
C
BC
BC
at point
Q
Q
Q
. Prove that the center of the circle circumscirbed around the triangle
C
P
Q
CPQ
CPQ
is equidistant from points
B
B
B
and
D
D
D
.
1
2
Hide problems
equal segments wanted, starting with intersecting circles and a parallel
On the plane circles
k
k
k
and
ℓ
\ell
ℓ
are intersected at points
C
C
C
and
D
D
D
, where circle
k
k
k
passes through the center
L
L
L
of circle
ℓ
\ell
ℓ
. The straight line passing through point
D
D
D
intersects circles
k
k
k
and
ℓ
\ell
ℓ
for the second time at points
A
A
A
and
B
B
B
respectively in such a way that
D
D
D
is the interior point of segment
A
B
AB
A
B
. Show that
A
B
=
A
C
AB = AC
A
B
=
A
C
.
max of gcd of 3 products of elements of 3 disjoint partitions of {1,2,...,63}
The set of
{
1
,
2
,
3
,
.
.
.
,
63
}
\{1,2,3,...,63\}
{
1
,
2
,
3
,
...
,
63
}
was divided into three non-empty disjoint sets
A
,
B
A,B
A
,
B
. Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be the product of all numbers in each set
A
,
B
,
C
A,B,C
A
,
B
,
C
respectively and finally we have determined the greatest common divisor of these three products. What was the biggest result we could get?