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National and Regional Contests
Romania Contests
Romania National Olympiad
1999 Romania National Olympiad
1999 Romania National Olympiad
Part of
Romania National Olympiad
Subcontests
(6)
2b
1
Hide problems
a^3+b^3+c^3 >= a+b+c if ab +be + ba <= 3abc 1999 Romania NMO VIII p2b
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers such that
a
b
+
b
e
+
b
a
≤
3
a
b
c
ab +be + ba \le 3abc
ab
+
b
e
+
ba
≤
3
ab
c
. Prove that
a
3
+
b
3
+
c
3
≥
a
+
b
+
c
.
a^3+b^3+c^3 \ge a+b+c.
a
3
+
b
3
+
c
3
≥
a
+
b
+
c
.
4
3
Hide problems
EF/BC+EG / AD=1 1999 Romania NMO VII p4
In the triangle
A
B
C
ABC
A
BC
, let
D
∈
(
B
C
)
D \in (BC)
D
∈
(
BC
)
,
E
∈
(
A
B
)
E \in (AB)
E
∈
(
A
B
)
,
E
F
∥
B
C
EF \parallel BC
EF
∥
BC
,
F
∈
(
A
C
)
F \in (AC)
F
∈
(
A
C
)
,
E
G
∥
A
D
EG\parallel AD
EG
∥
A
D
,
G
∈
(
B
C
)
G\in (BC)
G
∈
(
BC
)
and
M
,
N
M,N
M
,
N
be the midpoints of
(
A
D
)
(AD)
(
A
D
)
and
(
B
C
)
(BC)
(
BC
)
, respectively. Prove that:a)
E
F
B
C
+
E
G
A
D
=
1
\frac{EF}{BC}+\frac{EG}{AD}=1
BC
EF
+
A
D
EG
=
1
b) the midpoint of
[
F
G
]
[FG]
[
FG
]
lies on the line
M
N
MN
MN
.
_|_ planes wanted, regular pyramid 1999 Romania NMO VIII p4
Let
S
A
B
C
SABC
S
A
BC
be a regular pyramid,
O
O
O
the center of basis
A
B
C
ABC
A
BC
, and
M
M
M
the midpoint of
[
B
C
]
[BC]
[
BC
]
. If
N
∈
[
S
A
]
N \in [SA]
N
∈
[
S
A
]
such that
S
A
=
25
⋅
N
S
SA = 25 \cdot NS
S
A
=
25
⋅
NS
and
S
O
∩
M
N
=
{
P
}
SO \cap MN=\{P\}
SO
∩
MN
=
{
P
}
,
A
M
=
2
⋅
S
O
AM=2\cdot SO
A
M
=
2
⋅
SO
, prove that the planes
(
A
B
P
)
(ABP)
(
A
BP
)
and
(
S
B
C
)
(SBC)
(
SBC
)
are perpendicular.
f(m x+(1- m)y) < m f{x)+(1- m)f(y), parallelogram on graph
a) Let
a
,
b
∈
R
a,b\in R
a
,
b
∈
R
,
a
<
b
a <b
a
<
b
. Prove that
x
∈
(
a
,
b
)
x \in (a,b)
x
∈
(
a
,
b
)
if and only if there exists
λ
∈
(
0
,
1
)
\lambda \in (0,1)
λ
∈
(
0
,
1
)
such that
x
=
λ
a
+
(
1
−
λ
)
b
x=\lambda a +(1-\lambda)b
x
=
λa
+
(
1
−
λ
)
b
.b) If the function
f
:
R
→
R
f: R \to R
f
:
R
→
R
has the property:
f
(
λ
x
+
(
1
−
λ
)
y
)
<
λ
f
(
x
)
+
(
1
−
λ
)
f
(
y
)
,
∀
x
,
y
∈
R
,
x
≠
y
,
∀
λ
∈
(
0
,
1
)
,
f (\lambda x+(1-\lambda) y) < \lambda f(x) + (1-\lambda)f(y), \forall x,y \in R, x\ne y, \forall \lambda \in (0,1),
f
(
λ
x
+
(
1
−
λ
)
y
)
<
λ
f
(
x
)
+
(
1
−
λ
)
f
(
y
)
,
∀
x
,
y
∈
R
,
x
=
y
,
∀
λ
∈
(
0
,
1
)
,
prove that one cannot find four points on the function’s graph that are the vertices of a parallelogram
3
3
Hide problems
AB x DE = BC x CE, AC^2 < /2 (AD x AF + AB x AE), 1999 Romania NMO VII p3
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
∠
B
A
C
=
∠
C
A
D
\angle BAC = \angle CAD
∠
B
A
C
=
∠
C
A
D
,
∠
A
B
C
=
∠
A
C
D
\angle ABC =\angle ACD
∠
A
BC
=
∠
A
C
D
,
(
A
D
∩
(
B
C
=
{
E
}
(AD \cap (BC =\{E\}
(
A
D
∩
(
BC
=
{
E
}
,
(
A
B
∩
(
D
C
=
{
F
}
(AB \cap (DC = \{F\}
(
A
B
∩
(
D
C
=
{
F
}
. Prove that:a)
A
B
⋅
D
E
=
B
C
⋅
C
E
AB\cdot DE = BC \cdot CE
A
B
⋅
D
E
=
BC
⋅
CE
b)
A
C
2
<
1
2
(
A
D
⋅
A
F
+
A
B
⋅
A
E
)
.
AC^2 < \frac12 (AD \cdot AF + AB \cdot AE).
A
C
2
<
2
1
(
A
D
⋅
A
F
+
A
B
⋅
A
E
)
.
similar triangles on parallel planes , parallelepiped 1999 Romania NMO VIII p3
Let
A
B
C
D
A
′
B
′
C
′
D
′
ABCDA'B'C'D'
A
BC
D
A
′
B
′
C
′
D
′
be a right parallelepiped,
E
E
E
and
F
F
F
the projections of
A
A
A
on the lines
A
′
D
A'D
A
′
D
,
A
′
C
A'C
A
′
C
, respectively, and
P
,
Q
P, Q
P
,
Q
the projections of
B
′
B'
B
′
on the lines
A
′
C
′
A'C'
A
′
C
′
and
A
′
C
A'C
A
′
C
Prove thata) the planes
(
A
E
F
)
(AEF)
(
A
EF
)
and
(
B
′
P
Q
)
(B'PQ)
(
B
′
PQ
)
are parallelb) the triangles
A
E
F
AEF
A
EF
and
B
′
P
Q
B'PQ
B
′
PQ
are similar.
circumscriptible quad areas criterion S[AIB] + S[CID] =S[AID]+S[BIC]
In the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, the bisectors of angles
A
A
A
and
C
C
C
intersect in
I
I
I
. Prove that
A
B
C
D
ABCD
A
BC
D
is circumscriptible if and only if
S
[
A
I
B
]
+
S
[
C
I
D
]
=
S
[
A
I
D
]
+
S
[
B
I
C
]
S[AIB] + S[CID] =S[AID]+S[BIC]
S
[
A
I
B
]
+
S
[
C
I
D
]
=
S
[
A
I
D
]
+
S
[
B
I
C
]
(
S
[
X
Y
Z
]
S[XYZ]
S
[
X
Y
Z
]
denotes the area of the triangle
X
Y
Z
XYZ
X
Y
Z
)
2
3
Hide problems
a^2 +b^2 +c^2 never prime 1999 Romania NMO VII p3
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be non zero integers,
a
≠
c
a\ne c
a
=
c
such that
a
c
=
a
2
+
b
2
c
2
+
b
2
\frac{a}{c}=\frac{a^2+b^2}{c^2+b^2}
c
a
=
c
2
+
b
2
a
2
+
b
2
Prove that
a
2
+
b
2
+
c
2
a^2 +b^2 +c^2
a
2
+
b
2
+
c
2
cannot be a prime number.
min pos. root of (a+b)x^2-2(ab-1)x-(a+b) = 0
For
a
,
b
>
0
a, b > 0
a
,
b
>
0
, denote by
t
(
a
,
b
)
t(a,b)
t
(
a
,
b
)
the positive root of the equation
(
a
+
b
)
x
2
−
2
(
a
b
−
1
)
x
−
(
a
+
b
)
=
0.
(a+b)x^2-2(ab-1)x-(a+b) = 0.
(
a
+
b
)
x
2
−
2
(
ab
−
1
)
x
−
(
a
+
b
)
=
0.
Let
M
=
{
(
a
.
b
)
∣
a
≠
b
a
n
d
t
(
a
,
b
)
≤
a
b
}
M = \{ (a.b) | \, a \ne b \,\,\, and \,\,\,t(a,b) \le \sqrt{ab} \}
M
=
{(
a
.
b
)
∣
a
=
b
an
d
t
(
a
,
b
)
≤
ab
}
Determine, for
(
a
,
b
)
∈
M
(a, b)\in M
(
a
,
b
)
∈
M
, the mmimum value of
t
(
a
,
b
)
t(a,b)
t
(
a
,
b
)
.
MN x NP x PQ x QM >= AM x BN x CP xDQ , regular tetrahedron
On the sides
(
A
B
)
(AB)
(
A
B
)
,
(
B
C
)
(BC)
(
BC
)
,
(
C
D
)
(CD)
(
C
D
)
and
(
D
A
)
(DA)
(
D
A
)
of the regular tetrahedron
A
B
C
D
ABCD
A
BC
D
, one considers the points
M
M
M
,
N
N
N
,
P
P
P
,
Q
Q
Q
, respectively Prove that
M
N
⋅
N
P
⋅
P
Q
⋅
Q
M
≥
A
M
⋅
B
N
⋅
C
P
⋅
D
Q
.
MN \cdot NP \cdot PQ \cdot QM \ge AM \cdot BN \cdot CP \cdot DQ.
MN
⋅
NP
⋅
PQ
⋅
QM
≥
A
M
⋅
BN
⋅
CP
⋅
D
Q
.
1
3
Hide problems
a triangle
Source: Romania 1999 7.1 Determine the side lengths of a right trianlge if they are intgers and the product of the leg lengths is equal to three times the perimeter.
natural sets related to P(x) = 2x^3-3x^2+2 1999 Romania NMO VIII p2
Let
P
(
x
)
=
2
x
3
−
3
x
2
+
2
P(x) = 2x^3-3x^2+2
P
(
x
)
=
2
x
3
−
3
x
2
+
2
, and the sets:
A
=
{
P
(
n
)
∣
n
∈
N
,
n
≤
1999
}
,
B
=
{
p
2
+
1
∣
p
∈
N
}
,
C
=
{
q
2
+
2
∣
q
∈
N
}
A =\{ P(n) | n \in N, n \le 1999\}, B=\{p^2+1 |p \in N\}, C=\{ q^2+2 | q \in N\}
A
=
{
P
(
n
)
∣
n
∈
N
,
n
≤
1999
}
,
B
=
{
p
2
+
1∣
p
∈
N
}
,
C
=
{
q
2
+
2∣
q
∈
N
}
Prove that the sets
A
∩
B
A \cap B
A
∩
B
and
A
∩
C
A\cap C
A
∩
C
have the same number of elements
AD^3 = AB x AC x AP, equal sngles
Let
A
D
AD
A
D
be the bisector of angle
A
A
A
of the triangle
A
B
C
ABC
A
BC
. One considers the points M, N on the half-lines
(
A
B
(AB
(
A
B
and
(
A
C
(AC
(
A
C
, respectively, such that
∠
M
D
A
=
∠
B
\angle MDA = \angle B
∠
M
D
A
=
∠
B
and
∠
N
D
A
=
∠
C
\angle NDA = \angle C
∠
N
D
A
=
∠
C
. Let
A
D
∩
M
N
=
{
P
}
AD \cap MN=\{P\}
A
D
∩
MN
=
{
P
}
. Prove that:
A
D
3
=
A
B
⋅
A
C
⋅
A
P
AD^3 = AB \cdot AC\cdot AP
A
D
3
=
A
B
⋅
A
C
⋅
A
P
2a
1
Hide problems
making the product to the fraction and reversing the sign
let
x
i
,
y
i
1
≤
i
≤
n
x_i,y_i 1 \le i \le n
x
i
,
y
i
1
≤
i
≤
n
be positive numbers such that :
∑
i
=
1
n
x
i
≥
∑
i
=
1
n
x
i
y
i
\displaystyle \sum_{i=1}^n x_i \ge \sum_{i=1}^n x_iy_i
i
=
1
∑
n
x
i
≥
i
=
1
∑
n
x
i
y
i
Prove :
∑
i
=
1
n
x
i
≤
∑
i
=
1
n
x
i
y
i
\displaystyle \sum_{i=1}^n x_i \le \sum _{i=1}^n \frac{x_i}{y_i}
i
=
1
∑
n
x
i
≤
i
=
1
∑
n
y
i
x
i