MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
1999 Moldova Team Selection Test
1999 Moldova Team Selection Test
Part of
Moldova Team Selection Test
Subcontests
(15)
15
1
Hide problems
$(x-y)(y-z)(z-x)=x+y+z$
Distinct integers
x
,
y
,
z
x,y,z{}
x
,
y
,
z
verify the relation
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
=
x
+
y
+
z
(x-y)(y-z)(z-x)=x+y+z
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
=
x
+
y
+
z
. Find the smallest possibile value of
∣
x
+
y
+
z
∣
|x+y+z|
∣
x
+
y
+
z
∣
.
14
1
Hide problems
A square with sidelength $1$ is covered by $3$ congruent disks
A square with sidelength
1
1
1
is covered by
3
3
3
congruent disks. Find the smallest possible value of the radius of the disks.
12
1
Hide problems
$$x^2+y^2+1998=1997x-1999y.$$
Solve the equation in postive integers
x
2
+
y
2
+
1998
=
1997
x
−
1999
y
.
x^2+y^2+1998=1997x-1999y.
x
2
+
y
2
+
1998
=
1997
x
−
1999
y
.
11
1
Hide problems
is greater than $\frac{2}{3}$ of area of $ABC$
Let
A
B
C
ABC
A
BC
be a triangle. Show that there exists a lin
l
l
l
in the plane of
A
B
C
ABC
A
BC
such that the overlapping area of
A
B
C
ABC
A
BC
and
A
′
B
′
C
′
A^{'}B^{'}C^{'}
A
′
B
′
C
′
, the symmetric of
A
B
C
ABC
A
BC
with respect to
l
l
l
, is greater than
2
3
\frac{2}{3}
3
2
of area of
A
B
C
ABC
A
BC
.
10
1
Hide problems
noncongruent triangles with integer sidelengths and a perimeter of $2n$
Let
n
n{}
n
be a positive integer. Find the number of noncongruent triangles with integer sidelengths and a perimeter of
2
n
2n
2
n
.
9
1
Hide problems
Show that $P(X)$ is a first degree polynomial.
Let
P
(
X
)
P(X)
P
(
X
)
be a nonconstant polynomial with real coefficients such that for every rational number
q
q{}
q
the equation
P
(
X
)
=
q
P(X)=q
P
(
X
)
=
q
has no irrational solutions. Show that
P
(
X
)
P(X)
P
(
X
)
is a first degree polynomial.
7
1
Hide problems
Show that $S$ and $S^{'}$ are disjunctive.
Let
A
B
C
ABC
A
BC
be an equilateral triangle and
n
,
n
>
1
n{}, n>1
n
,
n
>
1
an integer. Let
S
S{}
S
be the set of the
n
−
1
n-1
n
−
1
lines parallel with
B
C
BC
BC
that cut
A
B
C
ABC
A
BC
in
n
n{}
n
figures with equal areas and
S
′
S^{'}
S
′
be the set of the
n
−
1
n-1
n
−
1
lines parallel with
B
C
BC
BC
that cut
A
B
C
ABC
A
BC
in
n
n{}
n
figures with equal perimeters. Show that
S
S{}
S
and
S
′
S^{'}
S
′
are disjunctive.
6
1
Hide problems
\frac{x_i}{\sqrt{1+x_0+x_1+\ldots+x_{i-1}}\cdot\sqrt{x_i+x_{i+1}+\ldots+x_n}}
Let
n
∈
N
,
x
0
=
0
n\in\mathbb{N}, x_0=0
n
∈
N
,
x
0
=
0
and
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
be postive real numbers such that
x
1
+
x
2
+
…
+
x
n
=
1
x_1+x_2+\ldots+x_n=1
x
1
+
x
2
+
…
+
x
n
=
1
. Show that
1
≤
∑
i
=
1
n
x
i
1
+
x
0
+
x
1
+
…
+
x
i
−
1
⋅
x
i
+
x
i
+
1
+
…
+
x
n
<
π
2
.
1\leq\sum_{i=1}^{n}\frac{x_i}{\sqrt{1+x_0+x_1+\ldots+x_{i-1}}\cdot\sqrt{x_i+x_{i+1}+\ldots+x_n}}<\frac{\pi}{2}.
1
≤
i
=
1
∑
n
1
+
x
0
+
x
1
+
…
+
x
i
−
1
⋅
x
i
+
x
i
+
1
+
…
+
x
n
x
i
<
2
π
.
5
1
Hide problems
$$\sqrt{x+a_1}+\sqrt{x+a_2}+\ldots+\sqrt{x+a_n}=n\sqrt{x}$$
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
be real numbers, but not all of them null. Show that the equation
x
+
a
1
+
x
+
a
2
+
…
+
x
+
a
n
=
n
x
\sqrt{x+a_1}+\sqrt{x+a_2}+\ldots+\sqrt{x+a_n}=n\sqrt{x}
x
+
a
1
+
x
+
a
2
+
…
+
x
+
a
n
=
n
x
has at most one real solution.
4
1
Hide problems
perpendiculars form $A, B$ and $C$ on $(EF), (FD)$ and $(DE)$ are concurrent
Outside the triangle
A
B
C
ABC
A
BC
the isosceles triangles
A
F
B
,
B
D
C
AFB, BDC
A
FB
,
B
D
C
and
C
E
A
CEA
CE
A
with the bases
A
B
,
B
C
AB, BC
A
B
,
BC
and
C
A
CA
C
A
respectively, are constructed. Show that the perpendiculars form
A
,
B
A, B
A
,
B
and
C
C
C
on
(
E
F
)
,
(
F
D
)
(EF), (FD)
(
EF
)
,
(
F
D
)
and
(
D
E
)
(DE)
(
D
E
)
, respectively, are concurrent.
3
1
Hide problems
$$f(x_1)+f(x_2)+\ldots+f(x_n)=nf(\sqrt[n]{x_1x_2\ldots x_n}),$$
The fuction
f
(
0
,
∞
)
→
R
f(0,\infty)\rightarrow\mathbb{R}
f
(
0
,
∞
)
→
R
verifies
f
(
x
)
+
f
(
y
)
=
2
f
(
x
y
)
,
∀
x
,
y
>
0
f(x)+f(y)=2f(\sqrt{xy}), \forall x,y>0
f
(
x
)
+
f
(
y
)
=
2
f
(
x
y
)
,
∀
x
,
y
>
0
. Show that for every positive integer
n
>
2
n>2
n
>
2
the following relation takes place
f
(
x
1
)
+
f
(
x
2
)
+
…
+
f
(
x
n
)
=
n
f
(
x
1
x
2
…
x
n
n
)
,
f(x_1)+f(x_2)+\ldots+f(x_n)=nf(\sqrt[n]{x_1x_2\ldots x_n}),
f
(
x
1
)
+
f
(
x
2
)
+
…
+
f
(
x
n
)
=
n
f
(
n
x
1
x
2
…
x
n
)
,
for every positive integers
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
.
1
1
Hide problems
$\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]}$
Let
a
,
b
,
c
,
d
,
e
a, b, c, d, e
a
,
b
,
c
,
d
,
e
(
a
<
b
<
c
<
d
<
e
)
(a < b < c < d < e)
(
a
<
b
<
c
<
d
<
e
)
be positive integers. FInd the greatest possible value of the expression
1
[
a
,
b
]
+
1
[
b
,
c
]
+
1
[
c
,
d
]
+
1
[
d
,
e
]
\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]}
[
a
,
b
]
1
+
[
b
,
c
]
1
+
[
c
,
d
]
1
+
[
d
,
e
]
1
, where
[
x
,
y
]
[x,y]
[
x
,
y
]
denotes the least common multiple of
x
x{}
x
and
y
y{}
y
.
16
1
Hide problems
functional equation
Define functions
f
,
g
:
R
→
R
f,g: \mathbb{R}\to \mathbb{R}
f
,
g
:
R
→
R
,
g
g
g
is injective, satisfy:
f
(
g
(
x
)
+
y
)
=
g
(
f
(
y
)
+
x
)
f(g(x)+y)=g(f(y)+x)
f
(
g
(
x
)
+
y
)
=
g
(
f
(
y
)
+
x
)
13
1
Hide problems
find the number of solutions of a flool function equation
Let
N
N
N
be a natural number. Find (with prove) the number of solutions in the segment
[
1
,
N
]
[1,N]
[
1
,
N
]
of the equation
x
2
−
[
x
2
]
=
(
x
−
[
x
]
)
2
x^2-[x^2]=(x-[x])^2
x
2
−
[
x
2
]
=
(
x
−
[
x
]
)
2
, where
[
x
]
[x]
[
x
]
means the floor function of
x
x
x
.
8
1
Hide problems
Find an f such that f(f(n))=n^2 [Romania 1978]
Find a function
f
:
N
→
N
f: \mathbb N \to \mathbb N
f
:
N
→
N
such that for all positive integers
n
n
n
, f(f(n))\equal{}n^2.