MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
EGMO TST - Moldova
2021 Moldova EGMO TST
2021 Moldova EGMO TST
Part of
EGMO TST - Moldova
Subcontests
(11)
12
1
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\frac{\sqrt{x^2+4}}{\sqrt{x^2+1}+\sqrt{x^2+9}}
Find all real numbers
y
y
y
, for which there exists at least one real number
x
x
x
such that
y
=
x
2
+
4
x
2
+
1
+
x
2
+
9
.
y=\frac{\sqrt{x^2+4}}{\sqrt{x^2+1}+\sqrt{x^2+9}}.
y
=
x
2
+
1
+
x
2
+
9
x
2
+
4
.
10
1
Hide problems
Find the smallest positive integer $k$ with the property
Let
n
≥
3
n\geq3
n
≥
3
be an integer. Find the smallest positive integer
k
k
k
with the property that if in a group of
n
n
n
boys for each boy there are at least
k
k
k
other boys that are born in the same year with him, then all the boys are born in the same year.
9
1
Hide problems
Prove that $AH$ and $CK$ are parallel.
Let
A
B
C
D
ABCD
A
BC
D
be a square and
E
E
E
a on point diagonal
(
A
C
)
(AC)
(
A
C
)
, different from its midpoint.
H
H
H
and
K
K
K
are the orthoceneters of triangles
A
B
E
ABE
A
BE
and
A
D
E
ADE
A
D
E
. Prove that
A
H
AH
A
H
and
C
K
CK
C
K
are parallel.
8
1
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x(x+1)(x+2)(x+3)=1679^{p-1}+1680^{p-1}+1681^{p-1}
Find all pairs of nonnegative integers
(
x
,
p
)
(x, p)
(
x
,
p
)
, where
p
p
p
is prime, that verify
x
(
x
+
1
)
(
x
+
2
)
(
x
+
3
)
=
167
9
p
−
1
+
168
0
p
−
1
+
168
1
p
−
1
.
x(x+1)(x+2)(x+3)=1679^{p-1}+1680^{p-1}+1681^{p-1}.
x
(
x
+
1
)
(
x
+
2
)
(
x
+
3
)
=
167
9
p
−
1
+
168
0
p
−
1
+
168
1
p
−
1
.
7
1
Hide problems
Prove that the lines $AM, BN, CP$ and $OH$ are concurrent.
A triangle
A
B
C
ABC
A
BC
has the orthocenter
H
H
H
different from the vertexes and the circumcenter
O
O
O
. Let
M
,
N
M, N
M
,
N
and
P
P
P
be the circumcenters of triangles
H
B
C
,
H
C
A
HBC, HCA
H
BC
,
H
C
A
and
H
A
B
HAB
H
A
B
. Prove that the lines
A
M
,
B
N
,
C
P
AM, BN, CP
A
M
,
BN
,
CP
and
O
H
OH
O
H
are concurrent.
6
1
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How many $3$ digit integers are not divided by $5$ neither by $7$?
How many
3
3
3
digit positive integers are not divided by
5
5
5
neither by
7
7
7
?
5
1
Hide problems
2^{x^2-3y+z}+2^{y^2-3z+x}+2^{z^2-3x+y}=1,5
Find all triplets
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
of real numbers that satisfy the equation
2
x
2
−
3
y
+
z
+
2
y
2
−
3
z
+
x
+
2
z
2
−
3
x
+
y
=
1
,
5.
2^{x^2-3y+z}+2^{y^2-3z+x}+2^{z^2-3x+y}=1,5.
2
x
2
−
3
y
+
z
+
2
y
2
−
3
z
+
x
+
2
z
2
−
3
x
+
y
=
1
,
5.
4
1
Hide problems
On a board there are $4$ positive integers $a, b, c$ and $d$.
On a board there are
4
4
4
positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
. Dan chooses three of them and writes their product on a paper. Then he substracts
1
1
1
from the other number. He does this until
0
0
0
appears on the board. What are the possible values of the sum of the numbers written on the paper?
3
1
Hide problems
Prove that $9$ divides $A_n=16^n+4^n-2$ for every nonnegative integer $n$.
Prove that
9
9
9
divides
A
n
=
1
6
n
+
4
n
−
2
A_n=16^n+4^n-2
A
n
=
1
6
n
+
4
n
−
2
for every nonnegative integer
n
n
n
.
2
1
Hide problems
Prove that lines $MN$ and $AC$ are parallel.
In triangle
A
B
C
ABC
A
BC
point
M
M
M
is on side
A
B
AB
A
B
such that
A
M
:
A
B
=
3
:
4
AM:AB=3:4
A
M
:
A
B
=
3
:
4
and point
P
P
P
is on side
B
C
BC
BC
such that
C
P
:
C
B
=
3
:
8
CP:CB=3:8
CP
:
CB
=
3
:
8
. Point
N
N
N
is symmetric to
A
A
A
with respect to point
P
P
P
. Prove that lines
M
N
MN
MN
and
A
C
AC
A
C
are parallel.
1
1
Hide problems
\frac{a^3+a^2}{1+bc}+\frac{b^3+b^2}{1+ca}+\frac{c^3+c^2}{1+ab}\geq3.
Postive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy
a
b
c
=
1
abc=1
ab
c
=
1
. Show that
a
3
+
a
2
1
+
b
c
+
b
3
+
b
2
1
+
c
a
+
c
3
+
c
2
1
+
a
b
≥
3.
\frac{a^3+a^2}{1+bc}+\frac{b^3+b^2}{1+ca}+\frac{c^3+c^2}{1+ab}\geq3.
1
+
b
c
a
3
+
a
2
+
1
+
c
a
b
3
+
b
2
+
1
+
ab
c
3
+
c
2
≥
3.