MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Polish Junior Math Olympiad
2024 Polish Junior Math Olympiad
2024 Polish Junior MO Finals
2024 Polish Junior MO Finals
Part of
2024 Polish Junior Math Olympiad
Subcontests
(5)
5
1
Hide problems
A huge number with digits 1 and 9 is a multiple of 19
Let
S
=
111
…
1
⏟
19
999
…
9
⏟
19
S=\underbrace{111\dots 1}_{19}\underbrace{999\dots 9}_{19}
S
=
19
111
…
1
19
999
…
9
. Show that the
2
S
2S
2
S
-digit number
111
…
1
⏟
S
999
…
9
⏟
S
\underbrace{111\dots 1}_{S}\underbrace{999\dots 9}_{S}
S
111
…
1
S
999
…
9
is a multiple of
19
19
19
.
4
1
Hide problems
An isosceles triangle and a parallelogram
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
C
=
B
C
AC=BC
A
C
=
BC
. Let
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
be points on the sides
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
of the triangle such that
C
Q
P
R
CQPR
CQPR
is a parallelogram. Show that the reflection of
P
P
P
over
Q
R
QR
QR
lies on the circumcircle of
A
B
C
ABC
A
BC
.
3
1
Hide problems
A funny inequality with (possibly negative) real numbers
Real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy
a
+
b
≠
0
a+b \ne 0
a
+
b
=
0
,
b
+
c
≠
0
b+c \ne 0
b
+
c
=
0
and
c
+
a
≠
0
c+a \ne 0
c
+
a
=
0
. Show that
(
a
2
c
a
+
b
+
b
2
a
b
+
c
+
c
2
b
c
+
a
)
⋅
(
b
2
c
a
+
b
+
c
2
a
b
+
c
+
a
2
b
c
+
a
)
≥
0.
\left(\frac{a^2c}{a+b}+\frac{b^2a}{b+c}+\frac{c^2b}{c+a}\right) \cdot \left(\frac{b^2c}{a+b}+\frac{c^2a}{b+c}+\frac{a^2b}{c+a}\right) \ge 0.
(
a
+
b
a
2
c
+
b
+
c
b
2
a
+
c
+
a
c
2
b
)
⋅
(
a
+
b
b
2
c
+
b
+
c
c
2
a
+
c
+
a
a
2
b
)
≥
0.
2
1
Hide problems
Cutting a nxn square in 1x1 and 2x2 pieces
Determine the smallest integer
n
≥
1
n \ge 1
n
≥
1
such that a
n
×
n
n \times n
n
×
n
square can be cut into square pieces of size
1
×
1
1 \times 1
1
×
1
and
2
×
2
2 \times 2
2
×
2
with both types occuring the same number of times.
1
1
Hide problems
A convex quadrilateral with an interior point
Can we find a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
with an interior point
P
P
P
satisfying AB=AP, BC=BP, CD=CP, \text{and} DA=DP ?