MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
EGMO TST - Moldova
2018 Moldova EGMO TST
2018 Moldova EGMO TST
Part of
EGMO TST - Moldova
Subcontests
(8)
4
1
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Number Theory problem
Find all sets of positive integers
A
=
{
a
1
,
a
2
,
.
.
.
a
19
}
A=\big\{ a_1,a_2,...a_{19}\big\}
A
=
{
a
1
,
a
2
,
...
a
19
}
which satisfy the following:
1
)
a
1
+
a
2
+
.
.
.
+
a
19
=
2017
;
1\big) a_1+a_2+...+a_{19}=2017;
1
)
a
1
+
a
2
+
...
+
a
19
=
2017
;
2
)
S
(
a
1
)
=
S
(
a
2
)
=
.
.
.
=
S
(
a
19
)
2\big) S(a_1)=S(a_2)=...=S(a_{19})
2
)
S
(
a
1
)
=
S
(
a
2
)
=
...
=
S
(
a
19
)
where
S
(
n
)
S\big(n\big)
S
(
n
)
denotes digit sum of number
n
n
n
.
3
1
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Geometry problem
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute triangle.
O
O
O
denote its circumcenter.Points
D
D
D
,
E
E
E
,
F
F
F
are the midpoints of the sides
B
C
BC
BC
,
C
A
CA
C
A
,and
A
B
AB
A
B
.Let
M
M
M
be a point on the side
B
C
BC
BC
.
A
M
∩
E
F
=
{
N
}
AM \cap EF = \big\{ N \big\}
A
M
∩
EF
=
{
N
}
.
O
N
∩
(
O
D
M
)
=
{
P
}
ON \cap \big( ODM \big) = \big\{ P \big\}
ON
∩
(
O
D
M
)
=
{
P
}
Prove that
M
′
M'
M
′
lie on
(
D
E
F
)
\big(DEF\big)
(
D
EF
)
where
M
′
M'
M
′
is the symmetrical point of
M
M
M
thought the midpoint of
D
P
DP
D
P
.
2
1
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Interesting problem
Let
S
S
S
= {
x
1
x_1
x
1
,
x
2
x_2
x
2
} be the solutions of the equation
x
2
−
2
∗
a
∗
x
−
1
=
0
x^2-2*a*x -1 = 0
x
2
−
2
∗
a
∗
x
−
1
=
0
, where
a
a
a
is a positive integer.Prove that for any
n
∈
N
n \in\mathbb{N}
n
∈
N
the expression
E
=
1
8
E=\frac{1}{8}
E
=
8
1
(
x
1
2
n
−
x
2
2
n
x_1^{2n}-x_2^{2n}
x
1
2
n
−
x
2
2
n
)(
x
1
4
n
−
x
2
4
n
x_1^{4n}-x_2^{4n}
x
1
4
n
−
x
2
4
n
) is a product of consecutive numbers.
1
1
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Easy number theory problem
Find if there are solutions :
a
,
b
∈
N
a,b \in\mathbb{N}
a
,
b
∈
N
,
a
2
+
b
2
=
2018
a^2+b^2=2018
a
2
+
b
2
=
2018
,
7
∣
a
+
b
7|a+b
7∣
a
+
b
.
8
1
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Easy Geometry equality
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
c
AB=c
A
B
=
c
,
B
C
=
a
BC=a
BC
=
a
and
A
C
=
b
AC=b
A
C
=
b
. If
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
satisfy
1
x
+
1
y
+
z
=
1
a
\frac{1}{x} +\frac{1}{y+z} = \frac{1}{a}
x
1
+
y
+
z
1
=
a
1
,
1
y
+
1
x
+
z
=
1
b
\frac{1}{y} +\frac{1}{x+z} = \frac{1}{b}
y
1
+
x
+
z
1
=
b
1
,
1
z
+
1
y
+
x
=
1
c
\frac{1}{z} +\frac{1}{y+x} = \frac{1}{c}
z
1
+
y
+
x
1
=
c
1
. Prove that the following equality holds
x
(
p
−
a
)
+
y
(
p
−
b
)
+
z
(
p
−
c
)
=
3
r
2
+
12
R
∗
r
,
x(p-a) + y(p-b) + z(p-c) = 3r^2 + 12R*r ,
x
(
p
−
a
)
+
y
(
p
−
b
)
+
z
(
p
−
c
)
=
3
r
2
+
12
R
∗
r
,
Where
p
p
p
is semi-perimeter,
R
R
R
is the circumradius and
r
r
r
is the inradius.
7
1
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Geometry problem
Let
A
B
C
D
ABCD
A
BC
D
be a isosceles trapezoid with
A
B
∥
C
D
AB \| CD
A
B
∥
C
D
,
A
D
=
B
C
AD=BC
A
D
=
BC
,
A
C
∩
B
D
=
AC \cap BD =
A
C
∩
B
D
=
{
O
O
O
}.
M
M
M
is the midpoint of the side
A
D
AD
A
D
. The circumcircle of triangle
B
C
M
BCM
BCM
intersects again the side
A
D
AD
A
D
in
K
K
K
. Prove that
O
K
∥
A
B
OK \| AB
O
K
∥
A
B
.
6
1
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Trigonometry problem
Let
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
, and
x
,
y
∈
x,y \in
x
,
y
∈
(
0
,
π
2
)
\left(0,\frac{\pi}{2}\right)
(
0
,
2
π
)
, and
m
∈
(
2
,
+
∞
)
m \in \left(2,+\infty\right)
m
∈
(
2
,
+
∞
)
such that
tan
x
∗
tan
y
=
m
\tan x * \tan y = m
tan
x
∗
tan
y
=
m
. Find the minimum value of the expression
E
(
x
,
y
)
=
cos
x
+
cos
y
E(x,y) = \cos x + \cos y
E
(
x
,
y
)
=
cos
x
+
cos
y
.
5
1
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Interesting inequality
Let
a
a
a
and
b
b
b
be real numbers such that
a
+
b
=
1
a + b = 1
a
+
b
=
1
. Prove the inequality
1
+
5
a
2
+
5
2
+
b
2
≥
9.
\sqrt{1+5a^2} + 5\sqrt{2+b^2} \geq 9.
1
+
5
a
2
+
5
2
+
b
2
≥
9.
Proposed by Baasanjav Battsengel