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Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
1992 Romania Team Selection Test
1992 Romania Team Selection Test
Part of
Romania Team Selection Test
Subcontests
(11)
8
1
Hide problems
S_{ij} +S_{n-1-i,m-1-j} = \frac{2S}{mn}, equal segments
Let
m
,
n
≥
2
m,n \ge 2
m
,
n
≥
2
be integers. The sides
A
00
A
0
m
A_{00}A_{0m}
A
00
A
0
m
and
A
n
m
A
n
0
A_{nm}A_{n0}
A
nm
A
n
0
of a convex quadrilateral
A
00
A
0
m
A
n
m
A
n
0
A_{00}A_{0m}A_{nm}A_{n0}
A
00
A
0
m
A
nm
A
n
0
are divided into
m
m
m
equal segments by points
A
0
j
A_{0j}
A
0
j
and
A
n
j
A_{nj}
A
nj
respectively (
j
=
1
,
.
.
.
,
m
−
1
j = 1,...,m-1
j
=
1
,
...
,
m
−
1
). The other two sides are divided into
n
n
n
equal segments by points
A
i
0
A_{i0}
A
i
0
and
A
i
m
A_{im}
A
im
(
i
=
1
,
.
.
.
,
n
−
1
i = 1,...,n -1
i
=
1
,
...
,
n
−
1
). Denote by
A
i
j
A_{ij}
A
ij
the intersection of lines
A
0
j
A
n
j
A_{0j}A{nj}
A
0
j
A
nj
and
A
i
0
A
i
m
A_{i0}A_{im}
A
i
0
A
im
, by
S
i
j
S_{ij}
S
ij
the area of quadrilateral
A
i
j
A
i
,
j
+
1
A
i
+
1
,
j
+
1
A
i
+
1
,
j
A_{ij}A_{i, j+1}A_{i+1, j+1}A_{i+1, j}
A
ij
A
i
,
j
+
1
A
i
+
1
,
j
+
1
A
i
+
1
,
j
and by
S
S
S
the area of the big quadrilateral. Show that
S
i
j
+
S
n
−
1
−
i
,
m
−
1
−
j
=
2
S
m
n
S_{ij} +S_{n-1-i,m-1-j} = \frac{2S}{mn}
S
ij
+
S
n
−
1
−
i
,
m
−
1
−
j
=
mn
2
S
1
2
Hide problems
increasing f : N \to N , t f(f(n)) = 3n , f(1992) =?
Suppose that
f
:
N
→
N
f : N \to N
f
:
N
→
N
is an increasing function such that
f
(
f
(
n
)
)
=
3
n
f(f(n)) = 3n
f
(
f
(
n
))
=
3
n
for all
n
n
n
. Find
f
(
1992
)
f(1992)
f
(
1992
)
.
there exist k vertices of these rectangles which lie on a line
Let
S
>
1
S > 1
S
>
1
be a real number. The Cartesian plane is partitioned into rectangles whose sides are parallel to the axes of the coordinate system. and whose vertices have integer coordinates. Prove that if the area of each triangle if at most
S
S
S
, then for any positive integer
k
k
k
there exist
k
k
k
vertices of these rectangles which lie on a line.
4
2
Hide problems
1992−th term of the ordered set A of all ordered sequences (a_1,a_2,...,a_{11})
Let
A
A
A
be the set of all ordered sequences
(
a
1
,
a
2
,
.
.
.
,
a
11
)
(a_1,a_2,...,a_{11})
(
a
1
,
a
2
,
...
,
a
11
)
of zeros and ones. The elements of
A
A
A
are ordered as follows: The first element is
(
0
,
0
,
.
.
.
,
0
)
(0,0,...,0)
(
0
,
0
,
...
,
0
)
, and the
n
+
1
n + 1
n
+
1
−th is obtained from the
n
n
n
−th by changing the first component from the right such that the newly obtained sequence was not obtained before. Find the
1992
1992
1992
−th term of the ordered set
A
A
A
x_1^2 +x_2^2+...+x_n^2= 1, [x_1 +x_2 +...+x_n] = m, x_1 +x_2 +...+x_m \ge 1
Let
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
be real numbers with
1
≥
x
1
≥
x
2
≥
.
.
.
≥
x
n
≥
0
1 \ge x_1 \ge x_2\ge ... \ge x_n \ge 0
1
≥
x
1
≥
x
2
≥
...
≥
x
n
≥
0
and
x
1
2
+
x
2
2
+
.
.
.
+
x
n
2
=
1
x_1^2 +x_2^2+...+x_n^2= 1
x
1
2
+
x
2
2
+
...
+
x
n
2
=
1
. If
[
x
1
+
x
2
+
.
.
.
+
x
n
]
=
m
[x_1 +x_2 +...+x_n] = m
[
x
1
+
x
2
+
...
+
x
n
]
=
m
, prove that
x
1
+
x
2
+
.
.
.
+
x
m
≥
1
x_1 +x_2 +...+x_m \ge 1
x
1
+
x
2
+
...
+
x
m
≥
1
.
5
1
Hide problems
R = 2p, p prime, sidelengths wanted in triangle
Let
O
O
O
be the circumcenter of an acute triangle
A
B
C
ABC
A
BC
. Suppose that the circumradius of the triangle is
R
=
2
p
R = 2p
R
=
2
p
, where
p
p
p
is a prime number. The lines
A
O
,
B
O
,
C
O
AO,BO,CO
A
O
,
BO
,
CO
meet the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
, respectively. Given that the lengths of
O
A
1
,
O
B
1
,
O
C
1
OA_1,OB_1,OC_1
O
A
1
,
O
B
1
,
O
C
1
are positive integers, find the side lengths of the triangle.
6
1
Hide problems
(7^m + p \cdot 2^n)\(7^m - p \cdot 2^n) is integer then is prime
Let
m
,
n
m,n
m
,
n
be positive integers and
p
p
p
be a prime number. Show that if
7
m
+
p
⋅
2
n
7
m
−
p
⋅
2
n
\frac{7^m + p \cdot 2^n}{7^m - p \cdot 2^n}
7
m
−
p
⋅
2
n
7
m
+
p
⋅
2
n
is an integer, then it is a prime number.
10
1
Hide problems
\frac{AA'}{A'V}S_a +\frac{BB'}{B'V}S_b +\frac{CC'}{C'V}S_c = S_v, tetrahedron
In a tetrahedron
V
A
B
C
VABC
V
A
BC
, let
I
I
I
be the incenter and
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
be arbitrary points on the edges
A
V
,
B
V
,
C
V
AV,BV,CV
A
V
,
B
V
,
C
V
, and let
S
a
,
S
b
,
S
c
,
S
v
S_a,S_b,S_c,S_v
S
a
,
S
b
,
S
c
,
S
v
be the areas of triangles
V
B
C
,
V
A
C
,
V
A
B
,
A
B
C
VBC,VAC,VAB,ABC
V
BC
,
V
A
C
,
V
A
B
,
A
BC
, respectively. Show that points
A
′
,
B
′
,
C
′
,
I
A',B',C',I
A
′
,
B
′
,
C
′
,
I
are coplanar if and only if
A
A
′
A
′
V
S
a
+
B
B
′
B
′
V
S
b
+
C
C
′
C
′
V
S
c
=
S
v
\frac{AA'}{A'V}S_a +\frac{BB'}{B'V}S_b +\frac{CC'}{C'V}S_c = S_v
A
′
V
A
A
′
S
a
+
B
′
V
B
B
′
S
b
+
C
′
V
C
C
′
S
c
=
S
v
11
1
Hide problems
lattice polygon P with sides parallel to coordinate axes
In the Cartesian plane is given a polygon
P
P
P
whose vertices have integer coordinates and with sides parallel to the coordinate axes. Show that if the length of each edge of
P
P
P
is an odd integer, then the surface of P cannot be partitioned into
2
×
1
2\times 1
2
×
1
rectangles.
9
1
Hide problems
Romanian TST 1992
Let
x
,
y
x, y
x
,
y
be real numbers such that
1
≤
x
2
−
x
y
+
y
2
≤
2
1\le x^2-xy+y^2\le2
1
≤
x
2
−
x
y
+
y
2
≤
2
. Show that: a)
2
9
≤
x
4
+
y
4
≤
8
\dfrac{2}{9}\le x^4+y^4\le 8
9
2
≤
x
4
+
y
4
≤
8
; b)
x
2
n
+
y
2
n
≥
2
3
n
x^{2n}+y^{2n}\ge\dfrac{2}{3^n}
x
2
n
+
y
2
n
≥
3
n
2
, for all
n
≥
3
n\ge3
n
≥
3
.Laurențiu Panaitopol and Ioan Tomescu
2
2
Hide problems
x_{n+2 }= (a^4 +4a^2 +2)x_{n+1} -x_n -2a^2, perfext square, matrix
For a positive integer
a
a
a
, define the sequence (
x
n
x_n
x
n
) by
x
1
=
x
2
=
1
x_1 = x_2 = 1
x
1
=
x
2
=
1
and
x
n
+
2
=
(
a
4
+
4
a
2
+
2
)
x
n
+
1
−
x
n
−
2
a
2
x_{n+2 }= (a^4 +4a^2 +2)x_{n+1} -x_n -2a^2
x
n
+
2
=
(
a
4
+
4
a
2
+
2
)
x
n
+
1
−
x
n
−
2
a
2
, for n
≥
1
\ge 1
≥
1
. Show that
x
n
x_n
x
n
is a perfect square and that for
n
>
2
n > 2
n
>
2
its square root equals the first entry in the matrix
(
a
2
+
1
a
a
1
)
n
−
2
\begin{pmatrix} a^2+1 & a \\ a & 1 \end{pmatrix}^{n-2}
(
a
2
+
1
a
a
1
)
n
−
2
Romanian TST 1992
Let
a
1
,
a
2
,
.
.
.
,
a
k
a_1, a_2, ..., a_k
a
1
,
a
2
,
...
,
a
k
be distinct positive integers such that the
2
k
2^k
2
k
sums
∑
i
=
1
k
ϵ
i
a
i
\displaystyle\sum\limits_{i=1}^{k}{\epsilon_i a_i}
i
=
1
∑
k
ϵ
i
a
i
,
ϵ
i
∈
{
0
,
1
}
\epsilon_i\in\left\{0,1\right\}
ϵ
i
∈
{
0
,
1
}
are distinct. a) Show that
1
a
1
+
1
a
2
+
.
.
.
+
1
a
k
≤
2
(
1
−
2
−
k
)
\dfrac{1}{a_1}+\dfrac{1}{a_2}+...+\dfrac{1}{a_k}\le2(1-2^{-k})
a
1
1
+
a
2
1
+
...
+
a
k
1
≤
2
(
1
−
2
−
k
)
; b) Find the sequences
(
a
1
,
a
2
,
.
.
.
,
a
k
)
(a_1,a_2,...,a_k)
(
a
1
,
a
2
,
...
,
a
k
)
for which the equality holds.Șerban Buzețeanu
3
2
Hide problems
Tetrahedron ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a tetrahedron;
B
′
,
C
′
,
D
′
B', C', D'
B
′
,
C
′
,
D
′
be the midpoints of the edges
A
B
,
A
C
,
A
D
AB, AC, AD
A
B
,
A
C
,
A
D
;
G
A
,
G
B
,
G
C
,
G
D
G_A, G_B, G_C, G_D
G
A
,
G
B
,
G
C
,
G
D
be the barycentres of the triangles
B
C
D
,
A
C
D
,
A
B
D
,
A
B
C
BCD, ACD, ABD, ABC
BC
D
,
A
C
D
,
A
B
D
,
A
BC
, and
G
G
G
be the barycentre of the tetrahedron. Show that
A
,
G
,
G
B
,
G
C
,
G
D
A, G, G_B, G_C, G_D
A
,
G
,
G
B
,
G
C
,
G
D
are all on a sphere if and only if
A
,
G
,
B
′
,
C
′
,
D
′
A, G, B', C', D'
A
,
G
,
B
′
,
C
′
,
D
′
are also on a sphere.Dan Brânzei
f(\pi) is a square, f : \pi \to \pi, image of any triangle is a square
Let
π
\pi
π
be the set of points in a plane and
f
:
π
→
π
f : \pi \to \pi
f
:
π
→
π
be a mapping such that the image of any triangle (as its polygonal line) is a square. Show that
f
(
π
)
f(\pi)
f
(
π
)
is a square.
7
1
Hide problems
sequence infinitely
Let
(
a
n
)
n
≥
1
(a_{n})_{n\geq 1}
(
a
n
)
n
≥
1
and
(
b
n
)
n
≥
1
(b_{n})_{n\geq 1}
(
b
n
)
n
≥
1
be the sequence of positive integers defined by
a
n
+
1
=
n
a
n
+
1
a_{n+1}=na_{n}+1
a
n
+
1
=
n
a
n
+
1
and
b
n
+
1
=
n
b
n
−
1
b_{n+1}=nb_{n}-1
b
n
+
1
=
n
b
n
−
1
for
n
≥
1
n\geq 1
n
≥
1
. Show that the two sequence cannot have infinitely many common terms. Laurentiu Panaitopol