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International Contests
Czech-Polish-Slovak Junior Match
2021 Czech-Polish-Slovak Junior Match
2021 Czech-Polish-Slovak Junior Match
Part of
Czech-Polish-Slovak Junior Match
Subcontests
(6)
6
1
Hide problems
p x q x s(r) = p x s(q) x r = s (p) x q x r, 2 digits numbers, s= sum of digits
Let
s
(
n
)
s (n)
s
(
n
)
denote the sum of digits of a positive integer
n
n
n
. Using six different digits, we formed three 2-digits
p
,
q
,
r
p, q, r
p
,
q
,
r
such that
p
⋅
q
⋅
s
(
r
)
=
p
⋅
s
(
q
)
⋅
r
=
s
(
p
)
⋅
q
⋅
r
.
p \cdot q \cdot s(r) = p\cdot s(q) \cdot r = s (p) \cdot q \cdot r.
p
⋅
q
⋅
s
(
r
)
=
p
⋅
s
(
q
)
⋅
r
=
s
(
p
)
⋅
q
⋅
r
.
Find all such numbers
p
,
q
,
r
p, q, r
p
,
q
,
r
.
3
2
Hide problems
max no of 6 pieces crosses in a 6x11 sheet of paper
A cross is the figure composed of
6
6
6
unit squares shown below (and any figure made of it by rotation). https://cdn.artofproblemsolving.com/attachments/6/0/6d4e0579d2e4c4fa67fd1219837576189ec9cb.png Find the greatest number of crosses that can be cut from a
6
×
11
6 \times 11
6
×
11
divided sheet of paper into unit squares (in such a way that each cross consists of six such squares).
gcd (a,b)=50! , lcm(a,b)=(50!)^2
Find the number of pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of positive integers with the property that the greatest common divisor of
a
a
a
and
b
b
b
is equal to
1
⋅
2
⋅
3
⋅
.
.
.
⋅
50
1\cdot 2 \cdot 3\cdot ... \cdot50
1
⋅
2
⋅
3
⋅
...
⋅
50
, and the least common multiple of
a
a
a
and
b
b
b
is
1
2
⋅
2
2
⋅
3
2
⋅
.
.
.
⋅
5
0
2
1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2
1
2
⋅
2
2
⋅
3
2
⋅
...
⋅
5
0
2
.
1
2
Hide problems
equal areas fro quadrilaterals in a trapezoid, AD=PC, BC=PD, <CMD =90^o
Consider a trapezoid
A
B
C
D
ABCD
A
BC
D
with bases
A
B
AB
A
B
and
C
D
CD
C
D
satisfying
∣
A
B
∣
>
∣
C
D
∣
| AB | > | CD |
∣
A
B
∣
>
∣
C
D
∣
. Let
M
M
M
be the midpoint of
A
B
AB
A
B
. Let the point
P
P
P
lie inside
A
B
C
D
ABCD
A
BC
D
such that
∣
A
D
∣
=
∣
P
C
∣
| AD | = | PC |
∣
A
D
∣
=
∣
PC
∣
and
∣
B
C
∣
=
∣
P
D
∣
| BC | = | PD |
∣
BC
∣
=
∣
P
D
∣
. Prove that if
∣
∠
C
M
D
∣
=
9
0
o
| \angle CMD | = 90^o
∣∠
CM
D
∣
=
9
0
o
, then the quadrilaterals
A
M
P
D
AMPD
A
MP
D
and
B
M
P
C
BMPC
BMPC
have the same area.
sum of products of numbers in rows and columns in 2x2 table is 2021
You are given a
2
×
2
2 \times 2
2
×
2
array with a positive integer in each field. If we add the product of the numbers in the first column, the product of the numbers in the second column, the product of the numbers in the first row and the product of the numbers in the second row, we get
2021
2021
2021
. a) Find possible values for the sum of the four numbers in the table. b) Find the number of distinct arrays that satisfy the given conditions that contain four pairwise distinct numbers in arrays.
4
2
Hide problems
min of x^4 + y^4 - x^2y - xy^2 when x,y>0 and x + y <= 1
Find the smallest value that the expression takes
x
4
+
y
4
−
x
2
y
−
x
y
2
x^4 + y^4 - x^2y - xy^2
x
4
+
y
4
−
x
2
y
−
x
y
2
, for positive numbers
x
x
x
and
y
y
y
satisfying
x
+
y
≤
1
x + y \le 1
x
+
y
≤
1
.
2 different numbers with product perfect square from {70, 71, 72,... 70 + n}
Find the smallest positive integer
n
n
n
with the property that in the set
{
70
,
71
,
72
,
.
.
.
70
+
n
}
\{70, 71, 72,... 70 + n\}
{
70
,
71
,
72
,
...70
+
n
}
you can choose two different numbers whose product is the square of an integer.
5
2
Hide problems
<PDG =? in regular hexagon ABCDEFG , AB and CE meet at P
A regular heptagon
A
B
C
D
E
F
G
ABCDEFG
A
BC
D
EFG
is given. The lines
A
B
AB
A
B
and
C
E
CE
CE
intersect at
P
P
P
. Find the measure of the angle
∠
P
D
G
\angle PDG
∠
P
D
G
.
x/y + y/z + z/x = x/z + z/y + y/x , x^2 + y^2 + z^2 = xy + yz + zx + 4
Find all three real numbers
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
satisfying the system of equations
x
y
+
y
z
+
z
x
=
x
z
+
z
y
+
y
x
\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{x}{z}+\frac{z}{y}+\frac{y}{x}
y
x
+
z
y
+
x
z
=
z
x
+
y
z
+
x
y
x
2
+
y
2
+
z
2
=
x
y
+
y
z
+
z
x
+
4
x^2 + y^2 + z^2 = xy + yz + zx + 4
x
2
+
y
2
+
z
2
=
x
y
+
yz
+
z
x
+
4
2
2
Hide problems
AF _|_ DE wanted, projections on external angle bisectors
An acute triangle
A
B
C
ABC
A
BC
is given. Let us denote by
D
D
D
and
E
E
E
the orthogonal projections, respectively of points
B
B
B
and
C
C
C
on the bisector of the external angle
B
A
C
BAC
B
A
C
. Let
F
F
F
be the point of intersection of the lines
B
E
BE
BE
and
C
D
CD
C
D
. Show that the lines
A
F
AF
A
F
and
D
E
DE
D
E
are perpendicular.
4 divides n if x_1x_2 +x_2x_3+... +x_{n-1}x_n +x_nx_1=0 with x_i \in {-1,1}
Let the numbers
x
i
∈
{
−
1
,
1
}
x_i \in \{-1, 1\}
x
i
∈
{
−
1
,
1
}
be given for
i
=
1
,
2
,
.
.
.
,
n
i = 1, 2,..., n
i
=
1
,
2
,
...
,
n
, satisfying
x
1
x
2
+
x
2
x
3
+
.
.
.
+
x
n
−
1
x
n
+
x
n
x
1
=
0.
x_1x_2 + x_2x_3 +... + x_{n-1}x_n + x_nx_1 = 0.
x
1
x
2
+
x
2
x
3
+
...
+
x
n
−
1
x
n
+
x
n
x
1
=
0.
Prove that
n
n
n
is divisible by
4
4
4
.