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Problems
Contests
National and Regional Contests
Moldova Contests
EGMO TST - Moldova
2023 Moldova EGMO TST
2023 Moldova EGMO TST
Part of
EGMO TST - Moldova
Subcontests
(12)
12
1
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players can be arranged in a list: $P_1, P_2, P_3,\ldots,P_n$
Let there be an integer
n
≥
2
n\geq2
n
≥
2
. In a chess tournament
n
n
n
players play between each other one game. No game ended in a draw. Show that after the end of the tournament the players can be arranged in a list:
P
1
,
P
2
,
P
3
,
…
,
P
n
P_1, P_2, P_3,\ldots,P_n
P
1
,
P
2
,
P
3
,
…
,
P
n
such that for every
i
(
1
≤
i
≤
n
−
1
)
i (1\leq i\leq n-1)
i
(
1
≤
i
≤
n
−
1
)
the player
P
i
P_i
P
i
won against player
P
i
+
1
P_{i+1}
P
i
+
1
.
11
1
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after their greatest digit is switched to $1$ become multiples of $30$.
Find all three digit positive integers that have distinct digits and after their greatest digit is switched to
1
1
1
become multiples of
30
30
30
.
10
1
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FInd $\angle BEC$
Cirlce
Ω
\Omega
Ω
is inscribed in triangle
A
B
C
ABC
A
BC
with
∠
B
A
C
=
40
\angle BAC=40
∠
B
A
C
=
40
. Point
D
D
D
is inside the angle
B
A
C
BAC
B
A
C
and is the intersection of exterior bisectors of angles
B
B
B
and
C
C
C
with the common side
B
C
BC
BC
. Tangent form
D
D
D
touches
Ω
\Omega
Ω
in
E
E
E
. FInd
∠
B
E
C
\angle BEC
∠
BEC
.
9
1
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\left[\frac{x^2+1}{x}\right]-\left[\frac{x}{x^2+1}\right]=3
Solve the equation
[
x
2
+
1
x
]
−
[
x
x
2
+
1
]
=
3.
\left[\frac{x^2+1}{x}\right]-\left[\frac{x}{x^2+1}\right]=3.
[
x
x
2
+
1
]
−
[
x
2
+
1
x
]
=
3.
8
1
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Prove that the number $1$ can be written as a sum of $2023$ fractions
Prove that the number
1
1
1
can be written as a sum of
2023
2023
2023
fractions of the form
1
k
i
\frac{1}{k_i}
k
i
1
, where all nonnegative integers
k
i
(
1
≤
i
≤
2023
)
k_i (1\leq i\leq 2023)
k
i
(
1
≤
i
≤
2023
)
are distinct.
7
1
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$$|a+3|+b^2+4\cdot c^2-14\cdot b-12\cdot c+55=0.$$
Find all triplets of integers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
, that verify the equation
∣
a
+
3
∣
+
b
2
+
4
⋅
c
2
−
14
⋅
b
−
12
⋅
c
+
55
=
0.
|a+3|+b^2+4\cdot c^2-14\cdot b-12\cdot c+55=0.
∣
a
+
3∣
+
b
2
+
4
⋅
c
2
−
14
⋅
b
−
12
⋅
c
+
55
=
0.
6
1
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Prove that $EF$ and $DG$ are perpendicular.
Let there be a square
A
B
C
D
ABCD
A
BC
D
. Points
E
E
E
and
F
F
F
are on sides
(
B
C
)
(BC)
(
BC
)
and
(
A
B
)
(AB)
(
A
B
)
such that
B
F
=
C
E
BF=CE
BF
=
CE
. LInes
A
E
AE
A
E
and
C
F
CF
CF
intersect in point
G
G
G
. Prove that
E
F
EF
EF
and
D
G
DG
D
G
are perpendicular.
5
1
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6(1-x)^2=\frac{1}{y},\\6(1-y)^2=\frac{1}{x}
Find all pairs of real numbers
(
x
,
y
)
(x, y)
(
x
,
y
)
, that satisfy the system of equations:
{
6
(
1
−
x
)
2
=
1
y
6
(
1
−
y
)
2
=
1
x
.
\left\{\begin{matrix} 6(1-x)^2=\dfrac{1}{y} \\ \\6(1-y)^2=\dfrac{1}{x}.\end{matrix}\right.
⎩
⎨
⎧
6
(
1
−
x
)
2
=
y
1
6
(
1
−
y
)
2
=
x
1
.
4
1
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2m-n+13p=2072,\\3m+11n+13p=2961.
Find all triplets of prime numbers
(
m
,
n
,
p
)
(m, n, p)
(
m
,
n
,
p
)
, that satisfy the system of equations:
{
2
m
−
n
+
13
p
=
2072
,
3
m
+
11
n
+
13
p
=
2961.
\left\{\begin{matrix} 2m-n+13p=2072,\\3m+11n+13p=2961.\end{matrix}\right.
{
2
m
−
n
+
13
p
=
2072
,
3
m
+
11
n
+
13
p
=
2961.
3
1
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Find $\angle DBC$
Let there be a quadrilateral
A
B
C
D
ABCD
A
BC
D
such that
∠
C
A
D
=
45
,
∠
A
C
D
=
30
,
∠
B
A
C
=
∠
B
C
A
=
15
\angle CAD=45, \angle ACD=30, \angle BAC=\angle BCA=15
∠
C
A
D
=
45
,
∠
A
C
D
=
30
,
∠
B
A
C
=
∠
BC
A
=
15
. Find
∠
D
B
C
\angle DBC
∠
D
BC
.
2
1
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there are two distinct powers of $n$ such that their sum is greater than
Show that for every integer
n
≥
2
n\geq2
n
≥
2
there are two distinct powers of
n
n
n
such that their sum is greater than
1
0
2023
10^{2023}
1
0
2023
and their positive difference is divisible with
2023
2023
2023
.
1
1
Hide problems
n=(ab-cd)\cdot(bc-ad)\cdot(ca-bd)
Integers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
satisfy
a
+
b
+
c
+
d
=
0
a+b+c+d=0
a
+
b
+
c
+
d
=
0
. Show that
n
=
(
a
b
−
c
d
)
⋅
(
b
c
−
a
d
)
⋅
(
c
a
−
b
d
)
n=(ab-cd)\cdot(bc-ad)\cdot(ca-bd)
n
=
(
ab
−
c
d
)
⋅
(
b
c
−
a
d
)
⋅
(
c
a
−
b
d
)
is a perfect square.