MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
1998 Moldova Team Selection Test
1998 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(10)
12
1
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Compute $\sum_{n=0}^{3^k} \frac{1}{b_n}$
Let
k
k{}
k
be a positive integer. For every positive integer
n
≤
3
k
n \leq 3^k
n
≤
3
k
, denote
b
n
b_n
b
n
the greatest power of
3
3
3
that divides
C
3
k
n
C_{3^k}^n
C
3
k
n
. Compute
∑
n
=
1
3
k
−
1
1
b
n
\sum_{n=1}^{3^k-1} \frac{1}{b_n}
∑
n
=
1
3
k
−
1
b
n
1
.
6
1
Hide problems
Find $r_1+r_2$
Two triangles
A
B
C
ABC
A
BC
and
B
D
E
BDE
B
D
E
have vertexes
C
C
C
and
E
E
E
on the same side of the line
A
B
AB{}
A
B
and
A
B
=
a
<
B
D
AB=a<BD
A
B
=
a
<
B
D
. Denote
{
P
}
=
C
E
∩
A
B
\{P\}=CE\cap AB
{
P
}
=
CE
∩
A
B
and
γ
=
m
(
∠
C
P
A
)
\gamma=m(\angle CPA)
γ
=
m
(
∠
CP
A
)
. Let
r
1
r_1
r
1
be the radius of the inscribed cricle of triangle
P
A
C
PAC
P
A
C
and
r
2
r_2
r
2
the radius of the excircle of triangle
P
D
E
PDE
P
D
E
, tangent to the side
D
E
DE
D
E
. Find
r
1
+
r
2
r_1+r_2
r
1
+
r
2
.
8
1
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arithmetic progression of maximum length l
Let
M
=
{
1
n
∣
n
∈
N
}
M=\{\frac{1}{n}|n\in\mathbb{N}\}
M
=
{
n
1
∣
n
∈
N
}
. Numbers
a
1
,
a
2
,
…
,
a
l
a_1,a_2,\ldots,a_l
a
1
,
a
2
,
…
,
a
l
from an arithmetic progression of maximum length
l
l
l
(
l
≥
3
)
(l\geq 3)
(
l
≥
3
)
if they verify the properties: a) numbers
a
1
,
a
2
,
…
,
a
l
a_1,a_2,\ldots,a_l
a
1
,
a
2
,
…
,
a
l
from a finite arithmetic progression; b) there is no number
b
∈
M
b\in M
b
∈
M
such that numbers
b
,
a
1
,
a
2
,
…
,
a
l
b,a_1,a_2,\ldots,a_l
b
,
a
1
,
a
2
,
…
,
a
l
or
a
1
,
a
2
,
…
,
a
l
,
b
a_1,a_2,\ldots,a_l, b
a
1
,
a
2
,
…
,
a
l
,
b
form a finite arithmetic progression. For example numbers
1
6
,
1
3
,
1
2
∈
M
\frac{1}{6},\frac{1}{3},\frac{1}{2}\in M
6
1
,
3
1
,
2
1
∈
M
form an arithmetic progression of maximum length
3
3
3
. a) FInd an arithmetic progression of maximum length
1998
1998
1998
. b) Prove that there exist maximum arithmetic progressions of any length
l
≥
3
l \geq 3
l
≥
3
.
7
1
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perimeter and area are equal
Find all triangles with integer sidelenghts such that their perimeter and area are equal.
5
1
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$$\max_{1\leq i\leq n} |a_i-f(a_i)| \geq \max_{1\leq i\leq n} |a_i-b_i|$$
Let
A
=
{
a
1
,
a
2
,
…
,
a
n
}
A=\{a_1,a_2,\ldots,a_n\}
A
=
{
a
1
,
a
2
,
…
,
a
n
}
be a set with
a
1
<
a
2
…
<
a
n
a_1<a_2\ldots<a_n
a
1
<
a
2
…
<
a
n
and
B
=
{
b
1
,
b
2
,
…
,
b
n
}
B=\{b_1,b_2,\ldots,b_n\}
B
=
{
b
1
,
b
2
,
…
,
b
n
}
be a set with
b
1
<
b
2
…
<
b
n
b_1<b_2\ldots<b_n
b
1
<
b
2
…
<
b
n
. Show that for every bijective function
f
:
A
→
B
f:A\rightarrow B
f
:
A
→
B
the following relation takes place
max
1
≤
i
≤
n
∣
a
i
−
f
(
a
i
)
∣
≥
max
1
≤
i
≤
n
∣
a
i
−
b
i
∣
.
\max_{1\leq i\leq n} |a_i-f(a_i)| \geq \max_{1\leq i\leq n} |a_i-b_i|.
1
≤
i
≤
n
max
∣
a
i
−
f
(
a
i
)
∣
≥
1
≤
i
≤
n
max
∣
a
i
−
b
i
∣.
4
1
Hide problems
\frac{a+z}{a+x}\cdot x+\frac{a+x}{a+y}\cdot y+\frac{a+y}{a+z}\cdot z \leq x+y+z
Show that for any positive real numbers
a
,
x
,
y
,
z
a, x, y, z
a
,
x
,
y
,
z
the following inequalities are true
a
+
z
a
+
x
⋅
x
+
a
+
x
a
+
y
⋅
y
+
a
+
y
a
+
z
⋅
z
≤
x
+
y
+
z
≤
a
+
y
a
+
z
⋅
x
+
a
+
z
a
+
x
⋅
y
+
a
+
x
a
+
y
⋅
z
.
\frac{a+z}{a+x}\cdot x+\frac{a+x}{a+y}\cdot y+\frac{a+y}{a+z}\cdot z \leq x+y+z \leq \frac{a+y}{a+z}\cdot x+\frac{a+z}{a+x}\cdot y+\frac{a+x}{a+y}\cdot z.
a
+
x
a
+
z
⋅
x
+
a
+
y
a
+
x
⋅
y
+
a
+
z
a
+
y
⋅
z
≤
x
+
y
+
z
≤
a
+
z
a
+
y
⋅
x
+
a
+
x
a
+
z
⋅
y
+
a
+
y
a
+
x
⋅
z
.
1
1
Hide problems
infinity of multiples of $1997$ that have $1998$ as first four digits and last
Prove that there exists and infinity of multiples of
1997
1997
1997
that have
1998
1998
1998
as first four digits and last four digits.
11
1
Hide problems
max AP/PE in lattice ZxZ, unique P lies interior to ABC
Let
A
,
B
,
C
A,B,C
A
,
B
,
C
be nodes of the lattice
Z
×
Z
Z\times Z
Z
×
Z
such that inside the triangle
A
B
C
ABC
A
BC
lies a unique node
P
P
P
of the lattice. Denote
E
=
A
P
∩
B
C
E = AP \cap BC
E
=
A
P
∩
BC
. Determine max
A
P
P
E
\frac{AP}{PE}
PE
A
P
, over all such configurations.
3
1
Hide problems
Properties of triangle
Prove that in a triangle
S
u
m
o
f
m
e
d
i
a
n
s
>
3
4
(
p
e
r
i
m
e
t
e
r
o
f
t
r
i
a
n
g
l
e
)
Sum of medians >\frac{3}{4}(perimeter of triangle )
S
u
m
o
f
m
e
d
ian
s
>
4
3
(
p
er
im
e
t
ero
f
t
r
ian
g
l
e
)
2
1
Hide problems
floor and square root
Determine the natural numbers that cannot be written as
⌊
n
+
n
+
1
2
⌋
\lfloor n + \sqrt{n} + \frac{1}{2} \rfloor
⌊
n
+
n
+
2
1
⌋
for any
n
∈
N
n \in \mathbb{N}
n
∈
N
.