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Problems
Contests
National and Regional Contests
Romania Contests
Romania EGMO Team Selection Test
2021 Romania EGMO TST
2021 Romania EGMO TST
Part of
Romania EGMO Team Selection Test
Subcontests
(4)
P4
2
Hide problems
Hidden lattice points
Consider a coordinate system in the plane, with the origin
O
O
O
. We call a lattice point
A
A{}
A
hidden if the open segment
O
A
OA
O
A
contains at least one lattice point. Prove that for any positive integer
n
n
n
there exists a square of side-length
n
n
n
such that any lattice point lying in its interior or on its boundary is hidden.
Exponential NT
Determine all non-negative integers
n
n
n
for which there exist two relatively prime non-negative integers
x
x
x
and
y
y
y
and a positive integer
k
⩾
2
k\geqslant 2
k
⩾
2
such that
3
n
=
x
k
+
y
k
3^n=x^k+y^k
3
n
=
x
k
+
y
k
.
P3
2
Hide problems
Large set without inclusions
Let
X
X
X
be a finite set with
n
⩾
3
n\geqslant 3
n
⩾
3
elements and let
A
1
,
A
2
,
…
,
A
p
A_1,A_2,\ldots, A_p
A
1
,
A
2
,
…
,
A
p
be
3
3
3
-element subsets of
X
X
X
satisfying
∣
A
i
∩
A
j
∣
⩽
1
|A_i\cap A_j|\leqslant 1
∣
A
i
∩
A
j
∣
⩽
1
for all indices
i
,
j
i,j
i
,
j
. Show that there exists a subset
A
A{}
A
of
X
X
X
so that none of
A
1
,
A
2
,
…
,
A
p
A_1,A_2,\ldots, A_p
A
1
,
A
2
,
…
,
A
p
is included in
A
A{}
A
and
∣
A
∣
⩾
⌊
2
n
⌋
|A|\geqslant\lfloor\sqrt{2n}\rfloor
∣
A
∣
⩾
⌊
2
n
⌋
.
Tromino tilings
Determine all pairs of positive integers
(
m
,
n
)
(m,n)
(
m
,
n
)
for which an
m
×
n
m\times n
m
×
n
rectangle can be tiled with (possibly rotated) L-shaped trominos.
P2
2
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Just another geo
Through the midpoint
M
M
M
of the side
B
C
BC
BC
of the triangle
A
B
C
ABC
A
BC
passes a line which intersects the rays
A
B
AB
A
B
and
A
C
AC
A
C
at
D
D
D
and
E
E
E
, respectively, such that
A
D
=
A
E
AD=AE
A
D
=
A
E
. Let
F
F
F
be the foot of the perpendicular from
A
A
A
onto
B
C
BC
BC
and
P
P{}
P
the circumcenter of triangle
A
D
E
ADE
A
D
E
. Prove that
P
F
=
P
M
PF=PM
PF
=
PM
.
Two circles and midpoints of arcs
Two circles intersect at points
A
≠
B
A\neq B
A
=
B
. A line passing through
A
A{}
A
intersects the circles again at
C
C
C
and
D
D
D
. Let
E
E
E
and
F
F
F
be the midpoints of the arcs \overarc{BC} and \overarc{BD} which do not contain
A
A{}
A
and let
M
M
M
be the midpoint of the segment
C
D
CD
C
D
. Prove that
M
E
ME
ME
and
M
F
MF
MF
are perpendicular.
P1
2
Hide problems
An easy recurent sequence a_1+a_2+...+a_n<1
Let
(
a
n
)
n
≥
1
(a_n)_{n\geq 1}
(
a
n
)
n
≥
1
be a sequence for real numbers given by
a
1
=
1
/
2
a_1=1/2
a
1
=
1/2
and for each positive integer
n
n
n
a
n
+
1
=
a
n
2
a
n
2
−
a
n
+
1
.
a_{n+1}=\frac{a_n^2}{a_n^2-a_n+1}.
a
n
+
1
=
a
n
2
−
a
n
+
1
a
n
2
.
Prove that for every positive integer
n
n
n
we have
a
1
+
a
2
+
⋯
+
a
n
<
1
a_1+a_2+\cdots + a_n<1
a
1
+
a
2
+
⋯
+
a
n
<
1
.
Periodic Sequence
Let
x
>
1
x>1
x
>
1
be a real number which is not an integer. For each
n
∈
N
n\in\mathbb{N}
n
∈
N
, let
a
n
=
⌊
x
n
+
1
⌋
−
x
⌊
x
n
⌋
a_n=\lfloor x^{n+1}\rfloor - x\lfloor x^n\rfloor
a
n
=
⌊
x
n
+
1
⌋
−
x
⌊
x
n
⌋
. Prove that the sequence
(
a
n
)
(a_n)
(
a
n
)
is not periodic.