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National and Regional Contests
Romania Contests
Romania EGMO Team Selection Test
2022 Romania EGMO TST
2022 Romania EGMO TST
Part of
Romania EGMO Team Selection Test
Subcontests
(4)
P4
2
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Romania EGMO TST 2022 Day 1 P4
For every positive integer
N
≥
2
N\geq 2
N
≥
2
with prime factorisation
N
=
p
1
a
1
p
2
a
2
⋯
p
k
a
k
N=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}
N
=
p
1
a
1
p
2
a
2
⋯
p
k
a
k
we define
f
(
N
)
:
=
1
+
p
1
a
1
+
p
2
a
2
+
⋯
+
p
k
a
k
.
f(N):=1+p_1a_1+p_2a_2+\cdots+p_ka_k.
f
(
N
)
:=
1
+
p
1
a
1
+
p
2
a
2
+
⋯
+
p
k
a
k
.
Let
x
0
≥
2
x_0\geq 2
x
0
≥
2
be a positive integer. We define the sequence
x
n
+
1
=
f
(
x
n
)
x_{n+1}=f(x_n)
x
n
+
1
=
f
(
x
n
)
for all
n
≥
0.
n\geq 0.
n
≥
0.
Prove that this sequence is eventually periodic and determine its fundamental period.
Romania EGMO TST 2022 Day 2 P4
Let
p
≥
3
p\geq 3
p
≥
3
be an odd positive integer. Show that
p
p
p
is prime if and only if however we choose
(
p
+
1
)
/
2
(p+1)/2
(
p
+
1
)
/2
pairwise distinct positive integers, we can find two of them,
a
a
a
and
b
b
b
, such that
(
a
+
b
)
/
gcd
(
a
,
b
)
≥
p
.
(a+b)/\gcd(a,b)\geq p.
(
a
+
b
)
/
g
cd
(
a
,
b
)
≥
p
.
P2
2
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Romania EGMO TST 2022 Day 1 P2
At first, on a board, the number
1
1
1
is written
100
100
100
times. Every minute, we pick a number
a
a
a
from the board, erase it, and write
a
/
3
a/3
a
/3
thrice instead. We say that a positive integer
n
n
n
is persistent if after any amount of time, regardless of the numbers we pick, we can find at least
n
n
n
equal numbers on the board. Find the greatest persistent number.
Romania EGMO TST 2022 Day 2 P2
On a board there is a regular polygon
A
1
A
2
…
A
99
.
A_1A_2\ldots A_{99}.
A
1
A
2
…
A
99
.
Ana and Barbu alternatively occupy empty vertices of the polygon and write down triangles on a list: Ana only writes obtuse triangles, while Barbu only writes acute ones.At the first turn, Ana chooses three vertices
X
,
Y
X,Y
X
,
Y
and
Z
Z
Z
and writes down
△
X
Y
Z
.
\triangle XYZ.
△
X
Y
Z
.
Then, Barbu chooses two of
X
,
Y
X,Y
X
,
Y
and
Z
,
Z,
Z
,
for example
X
X
X
and
Y
Y
Y
, and an unchosen vertex
T
T
T
, and writes down
△
X
Y
T
.
\triangle XYT.
△
X
Y
T
.
The game goes on and at each turn, the player must choose a new vertex
R
R
R
and write down
△
P
Q
R
\triangle PQR
△
PQR
, where
P
P
P
is the last vertex chosen by the other player, and
Q
Q
Q
is one of the other vertices of the last triangle written down by the other player.If one player cannot perform a move, then the other one wins. If both people play optimally, determine who has a winning strategy.
P1
2
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Romania EGMO TST 2022 Day 1 P1
A finite set
M
M
M
of real numbers has the following properties:
M
M
M
has at least
4
4
4
elements, and there exists a bijective function
f
:
M
→
M
f:M\to M
f
:
M
→
M
, different from the identity, such that
a
b
≤
f
(
a
)
f
(
b
)
ab\leq f(a)f(b)
ab
≤
f
(
a
)
f
(
b
)
for all
a
≠
b
∈
M
.
a\neq b\in M.
a
=
b
∈
M
.
Prove that the sum of the elements of
M
M
M
is
0.
0.
0.
Romania EGMO TST 2022 Day 2 P1
Determine all functions
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that all real numbers
x
x
x
and
y
y
y
satisfy
f
(
f
(
x
)
+
y
)
=
f
(
x
2
−
y
)
+
4
f
(
x
)
y
.
f(f(x)+y)=f(x^2-y)+4f(x)y.
f
(
f
(
x
)
+
y
)
=
f
(
x
2
−
y
)
+
4
f
(
x
)
y
.
P3
2
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Romania EGMO TST 2022 Day 1 P3
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral and let
O
O
O
be the intersection of its diagonals. Let
P
,
Q
,
R
,
P,Q,R,
P
,
Q
,
R
,
and
S
S
S
be the projections of
O
O
O
on
A
B
,
B
C
,
C
D
,
AB,BC,CD,
A
B
,
BC
,
C
D
,
and
D
A
DA
D
A
respectively. Prove that
2
(
O
P
+
O
Q
+
O
R
+
O
S
)
≤
A
B
+
B
C
+
C
D
+
D
A
.
2(OP+OQ+OR+OS)\leq AB+BC+CD+DA.
2
(
OP
+
OQ
+
OR
+
OS
)
≤
A
B
+
BC
+
C
D
+
D
A
.
Equal angles
Let be given a parallelogram
A
B
C
D
ABCD
A
BC
D
and two points
A
1
A_1
A
1
,
C
1
C_1
C
1
on its sides
A
B
AB
A
B
,
B
C
BC
BC
, respectively. Lines
A
C
1
AC_1
A
C
1
and
C
A
1
CA_1
C
A
1
meet at
P
P
P
. Assume that the circumcircles of triangles
A
A
1
P
AA_1P
A
A
1
P
and
C
C
1
P
CC_1P
C
C
1
P
intersect at the second point
Q
Q
Q
inside triangle
A
C
D
ACD
A
C
D
. Prove that \angle PDA \equal{} \angle QBA.