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Problems
Contests
National and Regional Contests
Moldova Contests
Moldova National Olympiad
2006 Moldova National Olympiad
2006 Moldova National Olympiad
Part of
Moldova National Olympiad
Subcontests
(19)
12.2
1
Hide problems
Integral with floor function
Let
a
,
b
,
n
∈
N
a, b, n \in \mathbb{N}
a
,
b
,
n
∈
N
, with
a
,
b
≥
2.
a, b \geq 2.
a
,
b
≥
2.
Also, let
I
1
(
n
)
=
∫
0
1
⌊
a
n
x
⌋
d
x
I_{1}(n)=\int_{0}^{1} \left \lfloor{a^n x} \right \rfloor dx
I
1
(
n
)
=
∫
0
1
⌊
a
n
x
⌋
d
x
and
I
2
(
n
)
=
∫
0
1
⌊
b
n
x
⌋
d
x
.
I_{2} (n) = \int_{0}^{1} \left \lfloor{b^n x} \right \rfloor dx.
I
2
(
n
)
=
∫
0
1
⌊
b
n
x
⌋
d
x
.
Find
lim
n
→
∞
I
1
(
n
)
I
2
(
n
)
.
\lim_{n \to \infty} \dfrac{I_1(n)}{I_{2}(n)}.
lim
n
→
∞
I
2
(
n
)
I
1
(
n
)
.
9.1
1
Hide problems
Moldova MO 2006, 9.1 (tricky ineq.)
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
+
b
+
c
=
2005
a+b+c=2005
a
+
b
+
c
=
2005
. Find the minimum value of the expression:
E
=
a
2006
+
b
2006
+
c
2006
+
(
a
b
)
2004
+
(
b
c
)
2004
+
(
c
a
)
2004
(
a
b
c
)
2004
E=a^{2006}+b^{2006}+c^{2006}+\frac{(ab)^{2004}+(bc)^{2004}+(ca)^{2004}}{(abc)^{2004}}
E
=
a
2006
+
b
2006
+
c
2006
+
(
ab
c
)
2004
(
ab
)
2004
+
(
b
c
)
2004
+
(
c
a
)
2004
12.5
1
Hide problems
maybe easy
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_{1},a_{2},...,a_{n}
a
1
,
a
2
,
...
,
a
n
be real positive numbers and
k
>
m
,
k
,
m
k>m, k,m
k
>
m
,
k
,
m
natural numbers. Prove that
(
n
−
1
)
(
a
1
m
+
a
2
m
+
.
.
.
+
a
n
m
)
≤
a
2
k
+
a
3
k
+
.
.
.
+
a
n
k
a
1
k
−
m
+
a
1
k
+
a
3
k
+
.
.
.
+
a
n
k
a
2
k
−
m
+
.
.
.
+
a
1
k
+
a
2
k
+
.
.
.
+
a
n
−
1
k
a
n
k
−
m
(n-1)(a_{1}^m +a_{2}^m+...+a_{n}^m)\leq\frac{a_{2}^k+a_{3}^k+...+a_{n}^k}{a_{1}^{k-m}}+\frac{a_{1}^k+a_{3}^k+...+a_{n}^k}{a_2^{k-m}}+...+\frac{a_{1}^k+a_{2}^k+...+a_{n-1}^k}{a_{n}^{k-m}}
(
n
−
1
)
(
a
1
m
+
a
2
m
+
...
+
a
n
m
)
≤
a
1
k
−
m
a
2
k
+
a
3
k
+
...
+
a
n
k
+
a
2
k
−
m
a
1
k
+
a
3
k
+
...
+
a
n
k
+
...
+
a
n
k
−
m
a
1
k
+
a
2
k
+
...
+
a
n
−
1
k
8.4
1
Hide problems
100 numbers
Sum of
100
100
100
natural distinct numbers is
9999
9999
9999
. Prove that
2006
2006
2006
divide their product.
10.2
1
Hide problems
Moldova mo 2006, 10th grade
Let
n
n
n
be a positive integer,
n
≥
2
n\geq 2
n
≥
2
. Let
M
=
{
0
,
1
,
2
,
…
n
−
1
}
M=\{0,1,2,\ldots n-1\}
M
=
{
0
,
1
,
2
,
…
n
−
1
}
. For an integer nonzero number
a
a
a
we define the function
f
a
:
M
⟶
M
f_{a}: M\longrightarrow M
f
a
:
M
⟶
M
, such that
f
a
(
x
)
f_{a}(x)
f
a
(
x
)
is the remainder when dividing
a
x
ax
a
x
at
n
n
n
. Find a necessary and sufficient condition such that
f
a
f_{a}
f
a
is bijective. And if
f
a
f_{a}
f
a
is bijective and
n
n
n
is a prime number, prove that
a
n
(
n
−
1
)
−
1
a^{n(n-1)}-1
a
n
(
n
−
1
)
−
1
is divisible by
n
2
n^{2}
n
2
.
10.1
1
Hide problems
Moldova mo 2006, 10th grade
Let
a
,
b
a,b
a
,
b
be the smaller sides of a right triangle. Let
c
c
c
be the hypothenuse and
h
h
h
be the altitude from the right angle. Fint the maximal value of
c
+
h
a
+
b
\frac{c+h}{a+b}
a
+
b
c
+
h
.
10.4
1
Hide problems
Moldova mo 2006, 10th grade
Find all real values of the real parameter
a
a
a
such that the equation
2
x
2
−
6
a
x
+
4
a
2
−
2
a
−
2
+
log
2
(
2
x
2
+
2
x
−
6
a
x
+
4
a
2
)
=
2x^{2}-6ax+4a^{2}-2a-2+\log_{2}(2x^{2}+2x-6ax+4a^{2})=
2
x
2
−
6
a
x
+
4
a
2
−
2
a
−
2
+
lo
g
2
(
2
x
2
+
2
x
−
6
a
x
+
4
a
2
)
=
=
log
2
(
x
2
+
2
x
−
3
a
x
+
2
a
2
+
a
+
1
)
.
=\log_{2}(x^{2}+2x-3ax+2a^{2}+a+1).
=
lo
g
2
(
x
2
+
2
x
−
3
a
x
+
2
a
2
+
a
+
1
)
.
has a unique solution.
10.5
1
Hide problems
Moldova mo 2006, 10th grade
Let
x
1
x_{1}
x
1
,
x
2
x_{2}
x
2
,
…
\ldots
…
,
x
n
x_{n}
x
n
be
n
n
n
real numbers in
(
1
4
,
2
3
)
\left(\frac{1}{4},\frac{2}{3}\right)
(
4
1
,
3
2
)
. Find the minimal value of the expression:
log
3
2
x
1
(
1
2
−
1
36
x
2
2
)
+
log
3
2
x
2
(
1
2
−
1
36
x
3
2
)
+
⋯
+
log
3
2
x
n
(
1
2
−
1
36
x
1
2
)
.
\log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right).
lo
g
2
3
x
1
(
2
1
−
36
x
2
2
1
)
+
lo
g
2
3
x
2
(
2
1
−
36
x
3
2
1
)
+
⋯
+
lo
g
2
3
x
n
(
2
1
−
36
x
1
2
1
)
.
10.3
1
Hide problems
Moldova mo 2006, 10th grade
A convex quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle. The tangents to the circle through
A
A
A
and
C
C
C
intersect at a point
P
P
P
, such that this point
P
P
P
does not lie on
B
D
BD
B
D
, and such that
P
A
2
=
P
B
⋅
P
D
PA^{2}=PB\cdot PD
P
A
2
=
PB
⋅
P
D
. Prove that the line
B
D
BD
B
D
passes through the midpoint of
A
C
AC
A
C
.
10.7
1
Hide problems
Moldova mo 2006, 10th grade
Consider an octogon with equal angles and rational side lengths. Prove that it has a symmetry center.
11.7
1
Hide problems
2n+3 points
Let
n
∈
N
∗
n\in\mathbb{N}^*
n
∈
N
∗
.
2
n
+
3
2n+3
2
n
+
3
points on the plane are given so that no 3 lie on a line and no 4 lie on a circle. Is it possible to find 3 points so that the interior of the circle passing through them would contain exactly
n
n
n
of the remaining points.
11.3
1
Hide problems
Pyramid
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a right quadrangular pyramid with vertex
E
E
E
and height
E
O
EO
EO
. Point
S
S
S
divides this height in the ratio
E
S
:
S
O
=
m
ES: SO=m
ES
:
SO
=
m
. In which ratio does the plane
(
A
B
C
)
(ABC)
(
A
BC
)
divide the lateral area of the pyramid.
11.8
1
Hide problems
Words of maximal length
Given an alfabet of
n
n
n
letters. A sequence of letters such that between any 2 identical letters there are no 2 identical letters is called a word. a) Find the maximal possible length of a word. b) Find the number of the words of maximal length.
11.6
1
Hide problems
2 sequences
Sequences
(
x
n
)
n
≥
1
(x_n)_{n\ge1}
(
x
n
)
n
≥
1
,
(
y
n
)
n
≥
1
(y_n)_{n\ge1}
(
y
n
)
n
≥
1
satisfy the relations
x
n
=
4
x
n
−
1
+
3
y
n
−
1
x_n=4x_{n-1}+3y_{n-1}
x
n
=
4
x
n
−
1
+
3
y
n
−
1
and
y
n
=
2
x
n
−
1
+
3
y
n
−
1
y_n=2x_{n-1}+3y_{n-1}
y
n
=
2
x
n
−
1
+
3
y
n
−
1
for
n
≥
1
n\ge1
n
≥
1
. If
x
1
=
y
1
=
5
x_1=y_1=5
x
1
=
y
1
=
5
find
x
n
x_n
x
n
and
y
n
y_n
y
n
. Calculate
lim
n
→
∞
x
n
y
n
\lim_{n\rightarrow\infty}\frac{x_n}{y_n}
lim
n
→
∞
y
n
x
n
.
11.5
1
Hide problems
Trigonometric equation
Let
n
∈
N
∗
n\in\mathbb{N}^*
n
∈
N
∗
. Solve the equation
∑
k
=
0
n
C
n
k
cos
2
k
x
=
cos
n
x
\sum_{k=0}^n C_n^k\cos2kx=\cos nx
∑
k
=
0
n
C
n
k
cos
2
k
x
=
cos
n
x
in
R
\mathbb{R}
R
.
11.2
1
Hide problems
Exists c, f'(c)=...
Function
f
:
[
a
,
b
]
→
R
f: [a,b]\to\mathbb{R}
f
:
[
a
,
b
]
→
R
,
0
<
a
<
b
0<a<b
0
<
a
<
b
is continuous on
[
a
,
b
]
[a,b]
[
a
,
b
]
and differentiable on
(
a
,
b
)
(a,b)
(
a
,
b
)
. Prove that there exists
c
∈
(
a
,
b
)
c\in(a,b)
c
∈
(
a
,
b
)
such that
f
′
(
c
)
=
1
a
−
c
+
1
b
−
c
+
1
a
+
b
.
f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}.
f
′
(
c
)
=
a
−
c
1
+
b
−
c
1
+
a
+
b
1
.
11.1
1
Hide problems
Trigonometric limit
Let
n
∈
N
∗
n\in\mathbb{N}^*
n
∈
N
∗
. Prove that
lim
x
→
0
(
1
+
x
2
)
n
+
1
−
∏
k
=
1
n
cos
k
x
x
∑
k
=
1
n
sin
k
x
=
2
n
2
+
n
+
12
6
n
.
\lim_{x\to 0}\frac{ \displaystyle (1+x^2)^{n+1}-\prod_{k=1}^n\cos kx}{ \displaystyle x\sum_{k=1}^n\sin kx}=\frac{2n^2+n+12}{6n}.
x
→
0
lim
x
k
=
1
∑
n
sin
k
x
(
1
+
x
2
)
n
+
1
−
k
=
1
∏
n
cos
k
x
=
6
n
2
n
2
+
n
+
12
.
10.8
1
Hide problems
M={x^2+x}
Let
M
=
{
x
2
+
x
∣
x
∈
N
⋆
}
M=\{x^2+x \mid x\in \mathbb N^{\star} \}
M
=
{
x
2
+
x
∣
x
∈
N
⋆
}
. Prove that for every integer
k
≥
2
k\geq 2
k
≥
2
there exist elements
a
1
,
a
2
,
…
,
a
k
,
b
k
a_{1}, a_{2}, \ldots, a_{k},b_{k}
a
1
,
a
2
,
…
,
a
k
,
b
k
from
M
M
M
, such that
a
1
+
a
2
+
⋯
+
a
k
=
b
k
a_{1}+a_{2}+\cdots+a_{k}=b_{k}
a
1
+
a
2
+
⋯
+
a
k
=
b
k
.
12.4
1
Hide problems
P(-1) is real
Let
P
(
x
)
=
x
n
+
a
1
x
n
−
1
+
.
.
.
+
a
n
−
1
x
+
(
−
1
)
n
P(x)= x^n+a_{1}x^{n-1}+...+a_{n-1}x+(-1)^{n}
P
(
x
)
=
x
n
+
a
1
x
n
−
1
+
...
+
a
n
−
1
x
+
(
−
1
)
n
,
a
i
∈
C
a_{i} \in C
a
i
∈
C
,
n
≥
2
n\geq 2
n
≥
2
with all roots having same modulo. Prove that
P
(
−
1
)
∈
R
P(-1) \in R
P
(
−
1
)
∈
R