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Problems
Contests
National and Regional Contests
Serbia Contests
Serbia National Math Olympiad
2024 Serbia National Math Olympiad
2024 Serbia National Math Olympiad
Part of
Serbia National Math Olympiad
Subcontests
(6)
6
1
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NT with iteration of polynomial and perfect squares
Find all non-constant polynomials
P
(
x
)
P(x)
P
(
x
)
with integer coefficients and positive leading coefficient, such that
P
2
m
n
(
m
2
)
+
n
2
P^{2mn}(m^2)+n^2
P
2
mn
(
m
2
)
+
n
2
is a perfect square for all positive integers
m
,
n
m, n
m
,
n
.
5
1
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Sum of powers of cos is constant
Let
n
≥
3
n \geq 3
n
≥
3
be a positive integer. Find all positive integers
k
k
k
, such that the function
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
defined by
f
(
x
)
=
cos
k
(
x
)
+
cos
k
(
x
+
2
π
n
)
+
…
+
cos
k
(
x
+
2
(
n
−
1
)
π
n
)
f(x)=\cos^k(x)+\cos^k(x+\frac{2\pi}{n})+\ldots +\cos^k(x+\frac{2(n-1)\pi}{n})
f
(
x
)
=
cos
k
(
x
)
+
cos
k
(
x
+
n
2
π
)
+
…
+
cos
k
(
x
+
n
2
(
n
−
1
)
π
)
is constant.
4
1
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Incircle-excircle config geo
Let
A
B
C
ABC
A
BC
be a triangle with incenter and
A
A
A
-excenter
I
,
I
a
I, I_a
I
,
I
a
, whose incircle touches
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D, E, F
D
,
E
,
F
. The line
E
F
EF
EF
meets
B
C
BC
BC
at
P
P
P
and
X
X
X
is the midpoint of
P
D
PD
P
D
. Show that
X
I
⊥
D
I
a
XI \perp DI_a
X
I
⊥
D
I
a
.
3
1
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Existence of alpha and beta satisfying min condition
Let
n
n
n
be a positive integer and let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
and
b
1
,
b
2
,
…
,
b
n
b_1, b_2, \ldots, b_n
b
1
,
b
2
,
…
,
b
n
be reals. Show that for any positive integer
1
≤
m
≤
n
1 \leq m \leq n
1
≤
m
≤
n
, there exist two distinct reals
α
,
β
\alpha, \beta
α
,
β
,
α
2
+
β
2
>
0
\alpha^2+\beta^2>0
α
2
+
β
2
>
0
, such that
p
m
=
min
{
p
1
,
p
2
,
…
,
p
n
}
p_m=\min\{p_1, p_2, \ldots, p_n\}
p
m
=
min
{
p
1
,
p
2
,
…
,
p
n
}
, where
p
j
=
∑
i
=
1
n
∣
α
(
a
i
−
a
j
)
+
β
(
b
i
−
b
j
)
∣
p_j=\sum_{i=1}^n|\alpha(a_i-a_j)+\beta(b_i-b_j)|
p
j
=
i
=
1
∑
n
∣
α
(
a
i
−
a
j
)
+
β
(
b
i
−
b
j
)
∣
for
1
≤
j
≤
n
1\leq j \leq n
1
≤
j
≤
n
.
2
1
Hide problems
Tournaments of order n
A tournament of order
n
n
n
,
n
∈
N
n \in \mathbb{N}
n
∈
N
, consists of
2
n
2^n
2
n
players, which are numbered with
1
,
2
,
…
,
2
n
1, 2, \ldots, 2^n
1
,
2
,
…
,
2
n
, and has
n
n
n
rounds. In each round, the remaining players paired with each other to play a match and the winner from each match advances to the next round. The winner of the
n
n
n
-th round is considered the winner of the tournament. Two tournaments are considered different if there is a match that took place in the
k
k
k
-th round of one tournament, but not in the
k
k
k
-th round of the other, or if the tournaments have different winners. Determine how many different tournaments of order
n
n
n
there are with the property that in each round, the sum of the numbers of the players in each match is the same (but not necessarily the same for all rounds).
1
1
Hide problems
Differences of divisors form geometric progression
Find all positive integers
n
n
n
, such that if their divisors are
1
=
d
1
<
d
2
<
…
<
d
k
=
n
1=d_1<d_2<\ldots<d_k=n
1
=
d
1
<
d
2
<
…
<
d
k
=
n
for
k
≥
4
k \geq 4
k
≥
4
, then the numbers
d
2
−
d
1
,
d
3
−
d
2
,
…
,
d
k
−
d
k
−
1
d_2-d_1, d_3-d_2, \ldots, d_k-d_{k-1}
d
2
−
d
1
,
d
3
−
d
2
,
…
,
d
k
−
d
k
−
1
form a geometric progression in some order.