MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
EGMO TST - Moldova
2019 Moldova EGMO TST
2019 Moldova EGMO TST
Part of
EGMO TST - Moldova
Subcontests
(9)
8
1
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a_{n+3}=7a_{n+2}-11a_{n+1}+5a_n-3\cdot2^n
The sequence
(
a
n
)
n
≥
1
(a_n)_{n\geq1}
(
a
n
)
n
≥
1
is defined as:
a
1
=
2
,
a
2
=
20
,
a
3
=
56
,
a
n
+
3
=
7
a
n
+
2
−
11
a
n
+
1
+
5
a
n
−
3
⋅
2
n
.
a_1=2, a_2=20, a_3=56, a_{n+3}=7a_{n+2}-11a_{n+1}+5a_n-3\cdot2^n.
a
1
=
2
,
a
2
=
20
,
a
3
=
56
,
a
n
+
3
=
7
a
n
+
2
−
11
a
n
+
1
+
5
a
n
−
3
⋅
2
n
.
Prove that
a
n
a_n
a
n
is positive for every positive integer
n
n{}
n
. Find the remainder of the divison of
a
673
a_{673}
a
673
to
673
673
673
.
7
1
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Prove that there are two elements in $A$ whose difference is $3$.
Let
A
A{}
A
be a subset formed of
16
16
16
elements of the set
B
=
{
1
,
2
,
3
,
…
,
105
,
106
}
B=\{1, 2, 3, \ldots, 105, 106\}
B
=
{
1
,
2
,
3
,
…
,
105
,
106
}
such that the difference between every two elements from
A
A
A
is different from
6
,
9
,
12
,
15
,
18
,
21
6, 9, 12, 15, 18, 21
6
,
9
,
12
,
15
,
18
,
21
. Prove that there are two elements in
A
A{}
A
whose difference is
3
3
3
.
6
1
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Prove that the lines $PQ$ and $AB$ are perpendicular.
There is a point
T
T
T
on a circle with the radius
R
R
R
. Points
A
A{}
A
and
B
B
B
are on the tangent to the circle that goes through
T
T
T
such that they are on the same side of
T
T
T
and
T
A
⋅
T
B
=
4
R
2
TA\cdot TB=4R^2
T
A
⋅
TB
=
4
R
2
. The point
S
S
S
is diametrically opposed to
T
T
T
. Lines
A
S
AS
A
S
and
B
S
BS
BS
intersect the circle again in
P
P{}
P
and
Q
Q{}
Q
. Prove that the lines
P
Q
PQ
PQ
and
A
B
AB{}
A
B
are perpendicular.
5
1
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Prove that the number $a=2019^{2020}+4^{2019}$ is a composite number.
Prove that the number
a
=
201
9
2020
+
4
2019
a=2019^{2020}+4^{2019}
a
=
201
9
2020
+
4
2019
is a composite number.
3
1
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weighing scale with two plates which shows the difference between the weights
There are
10
10{}
10
apples, each with a with a weight which is no more than
100
100{}
100
g. There is a weighing scale with two plates which shows the difference between the weights on the plates. Prove that 1) It is possible to put some (more than one) apples on the plates of the scale such that the difference between the weights on the plates will be less than
1
1
1
g. 2) It is possible to put an equal amount (more than one) of apples on each plate of the scale such that the difference between the weights on the plates will be less than
2
2
2
g.
2
1
Hide problems
Prove that $\frac{DB}{CE}=\frac{FB}{FC}=\left(\frac{AB}{AC}\right)^2$
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
A
C
AB<AC
A
B
<
A
C
. Point
M
M{}
M
from the side
(
B
C
)
(BC)
(
BC
)
is the foot of the bisector from the vertex
A
A{}
A
. The perpendicular bisector of the segment
[
A
M
]
[AM]
[
A
M
]
intersects the side
(
A
C
)
(AC)
(
A
C
)
in
E
E{}
E
, the side
(
A
B
)
(AB)
(
A
B
)
in
D
D
D
and the line
(
B
C
)
(BC)
(
BC
)
in
F
F{}
F
. Prove that
D
B
C
E
=
F
B
F
C
=
(
A
B
A
C
)
2
\frac{DB}{CE}=\frac{FB}{FC}=\left(\frac{AB}{AC}\right)^2
CE
D
B
=
FC
FB
=
(
A
C
A
B
)
2
.
1
1
Hide problems
Find the positive integer $n$ if
Find the positive integer
n
n{}
n
if
(
1
−
1
1
+
2
)
⋅
(
1
−
1
1
+
2
+
3
)
⋅
…
⋅
(
1
−
1
1
+
2
+
…
+
n
)
=
2021
6057
.
\left(1-\frac{1}{1+2}\right)\cdot\left(1-\frac{1}{1+2+3}\right)\cdot\ldots\cdot\left(1-\frac{1}{1+2+\ldots+n}\right)=\frac{2021}{6057}.
(
1
−
1
+
2
1
)
⋅
(
1
−
1
+
2
+
3
1
)
⋅
…
⋅
(
1
−
1
+
2
+
…
+
n
1
)
=
6057
2021
.
2
Hide problems
Problem about multiples with distinct digits
Find all
n
n
n
,in the range
10
,
11
…
2019
{10,11 \ldots 2019}
10
,
11
…
2019
such that every multiple of
n
n
n
has at least
2
2
2
distinct digits
Problem about a sequence
Given a sequence of positive real numbers such that
a
n
+
2
=
2
a
n
+
1
+
a
n
a_{n+2}=\frac{2}{a_{n+1}+a_{n}}
a
n
+
2
=
a
n
+
1
+
a
n
2
.Prove that there are two positive real numbers
s
,
t
s,t
s
,
t
such that
s
≤
a
n
≤
t
s \le a_n \le t
s
≤
a
n
≤
t
for all
n
n
n
4
1
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Easy inequality?
Let
x
,
y
>
0
x,y>0
x
,
y
>
0
be real numbers.Prove that:
1
x
2
+
y
2
+
1
x
2
+
1
y
2
≥
10
(
x
+
y
)
2
\frac{1}{x^2+y^2} +\frac{1}{x^2}+\frac{1}{y^2}\ge\frac{10}{(x+y)^2}
x
2
+
y
2
1
+
x
2
1
+
y
2
1
≥
(
x
+
y
)
2
10
I tried CBS, but it doesn't work... Can you give an idea, please?