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Problems
Contests
International Contests
Danube Competition in Mathematics
2019 Danube Mathematical Competition
2019 Danube Mathematical Competition
Part of
Danube Competition in Mathematics
Subcontests
(4)
4
2
Hide problems
Inscriptible quadrilateral; angle chasing
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral,
M
M
M
midpoint of
A
C
AC
A
C
and
N
N
N
midpoint of
B
D
.
BD.
B
D
.
If
∠
A
M
B
=
∠
A
M
D
,
\angle AMB =\angle AMD,
∠
A
MB
=
∠
A
M
D
,
prove that
∠
A
N
B
=
∠
B
N
C
.
\angle ANB =\angle BNC.
∠
A
NB
=
∠
BNC
.
Hard pure geometry
Let
A
P
D
APD
A
P
D
be an acute-angled triangle and let
B
,
C
B,C
B
,
C
be two points on the segments (excluding their endpoints)
A
P
,
P
D
,
AP,PD,
A
P
,
P
D
,
respectively. The diagonals of
A
B
C
D
ABCD
A
BC
D
meet at
Q
.
Q.
Q
.
Denote by
H
1
,
H
2
H_1,H_2
H
1
,
H
2
the orthocenters of
A
P
D
,
B
P
C
,
APD,BPC,
A
P
D
,
BPC
,
respectively. The circumcircles of
A
B
Q
ABQ
A
BQ
and
C
D
Q
CDQ
C
D
Q
intersect at
X
≠
Q
,
X\neq Q,
X
=
Q
,
and the circumcircles of
A
D
Q
,
B
C
Q
ADQ,BCQ
A
D
Q
,
BCQ
meet at
Y
≠
Q
.
Y\neq Q.
Y
=
Q
.
Prove that if the line
H
1
H
2
H_1H_2
H
1
H
2
passes through
X
,
X,
X
,
then it also passes through
Y
.
Y.
Y
.
3
2
Hide problems
A seqence of 51 terms
Let be a sequence of
51
51
51
natural numbers whose sum is
100.
100.
100.
Show that for any natural number
1
≤
k
<
100
1\le k<100
1
≤
k
<
100
there are some consecutive numbers from this sequence whose sum is
k
k
k
or
100
−
k
.
100-k.
100
−
k
.
99x99 colored square grid
We color some unit squares in a
99
×
99
99\times 99
99
×
99
square grid with one of
5
5
5
given distinct colors, such that each color appears the same number of times. On each row and on each column there are no differently colored unit squares. Find the maximum possible number of colored unit squares.
2
2
Hide problems
n irational numbers
Let be a natural number
n
,
n,
n
,
and
n
n
n
real numbers
a
1
,
a
2
,
…
,
a
n
.
a_1,a_2,\ldots ,a_n.
a
1
,
a
2
,
…
,
a
n
.
Prove that there exists a real number
a
a
a
such that
a
+
a
1
,
a
+
a
2
,
…
,
a
+
a
n
a+a_1,a+a_2,\ldots ,a+a_n
a
+
a
1
,
a
+
a
2
,
…
,
a
+
a
n
are all irrational.
f(f(x²)+y+f(y))=x²+2f(y)
Find all nondecreasing functions
f
:
R
⟶
R
f:\mathbb{R}\longrightarrow\mathbb{R}
f
:
R
⟶
R
that verify the relation
f
(
f
(
x
2
)
+
y
+
f
(
y
)
)
=
x
2
+
2
f
(
y
)
,
f\left( f\left( x^2 \right) +y+f(y) \right) =x^2+2f(y) ,
f
(
f
(
x
2
)
+
y
+
f
(
y
)
)
=
x
2
+
2
f
(
y
)
,
for any real numbers
x
,
y
.
x,y.
x
,
y
.
1
2
Hide problems
An equation in two variables from this year
Solve in
Z
2
\mathbb{Z}^2
Z
2
the equation:
x
2
(
1
+
x
2
)
=
−
1
+
2
1
y
.
x^2\left( 1+x^2 \right) =-1+21^y.
x
2
(
1
+
x
2
)
=
−
1
+
2
1
y
.
Lucian Petrescu
p^3-4p+9 is perfect square
Find all prime
p
p
p
numbers such that
p
3
−
4
p
+
9
p^3-4p+9
p
3
−
4
p
+
9
is perfect square.