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Problems
Contests
National and Regional Contests
Romania Contests
Romania Team Selection Test
1990 Romania Team Selection Test
1990 Romania Team Selection Test
Part of
Romania Team Selection Test
Subcontests
(11)
11
1
Hide problems
m(n-k -1) = k(k -l -1), in a group of n persons,
In a group of
n
n
n
persons, (i) each person is acquainted to exactly
k
k
k
others, (ii) any two acquainted persons have exactly
l
l
l
common acquaintances, (iii) any two non-acquainted persons have exactly
m
m
m
common acquaintances. Prove that
m
(
n
−
k
−
1
)
=
k
(
k
−
l
−
1
)
m(n-k -1) = k(k -l -1)
m
(
n
−
k
−
1
)
=
k
(
k
−
l
−
1
)
.
9
1
Hide problems
distance of two points is >= \sqrt{(5+\sqrt5)/2}
The distance between any two of six given points in the plane is at least
1
1
1
. Prove that the distance between some two points is at least
5
+
5
2
\sqrt{\frac{5+\sqrt5}{2}}
2
5
+
5
8
1
Hide problems
every coloring function f_k of S satisfies | f_k(S)| \le m
For a set
S
S
S
of
n
n
n
points, let
d
1
>
d
2
>
.
.
.
>
d
k
>
.
.
.
d_1 > d_2 >... > d_k > ...
d
1
>
d
2
>
...
>
d
k
>
...
be the distances between the points. A function
f
k
:
S
→
N
f_k: S \to N
f
k
:
S
→
N
is called a coloring function if, for any pair
M
,
N
M,N
M
,
N
of points in
S
S
S
with
M
N
≤
d
k
MN \le d_k
MN
≤
d
k
, it takes the value
f
k
(
M
)
+
f
k
(
N
)
f_k(M)+ f_k(N)
f
k
(
M
)
+
f
k
(
N
)
at some point. Prove that for each
m
∈
N
m \in N
m
∈
N
there are positive integers
n
,
k
n,k
n
,
k
and a set
S
S
S
of
n
n
n
points such that every coloring function
f
k
f_k
f
k
of
S
S
S
satisfies
∣
f
k
(
S
)
∣
≤
m
| f_k(S)| \le m
∣
f
k
(
S
)
∣
≤
m
6
1
Hide problems
partition of {1,2,...,3n} into 3 subsets such as a_i +b_i = c_i
Prove that there are infinitely many n’s for which there exists a partition of
{
1
,
2
,
.
.
.
,
3
n
}
\{1,2,...,3n\}
{
1
,
2
,
...
,
3
n
}
into subsets
{
a
1
,
.
.
.
,
a
n
}
,
{
b
1
,
.
.
.
,
b
n
}
,
{
c
1
,
.
.
.
,
c
n
}
\{a_1,...,a_n\}, \{b_1,...,b_n\}, \{c_1,...,c_n\}
{
a
1
,
...
,
a
n
}
,
{
b
1
,
...
,
b
n
}
,
{
c
1
,
...
,
c
n
}
such that
a
i
+
b
i
=
c
i
a_i +b_i = c_i
a
i
+
b
i
=
c
i
for all
i
i
i
, and prove that there are infinitely many
n
n
n
’s for which there is no such partition.
5
1
Hide problems
area set of points in the plane belonging to at least two of the disks >R^2/8
Let
O
O
O
be the circumcenter of an acute triangle
A
B
C
ABC
A
BC
and
R
R
R
be its circumcenter. Consider the disks having
O
A
,
O
B
,
O
C
OA,OB,OC
O
A
,
OB
,
OC
as diameters, and let
Δ
\Delta
Δ
be the set of points in the plane belonging to at least two of the disks. Prove that the area of
Δ
\Delta
Δ
is greater than
R
2
/
8
R^2/8
R
2
/8
.
4
2
Hide problems
6 faces of a hexahedron are quadrilaterals, 8 vertices on sphere
The six faces of a hexahedron are quadrilaterals. Prove that if seven its vertices lie on a sphere, then the eighth vertex also lies on the sphere.
6 planes are concurrent
Let
M
M
M
be a point on the edge
C
D
CD
C
D
of a tetrahedron
A
B
C
D
ABCD
A
BC
D
such that the tetrahedra
A
B
C
M
ABCM
A
BCM
and
A
B
D
M
ABDM
A
B
D
M
have the same total areas. We denote by
π
A
B
\pi_{AB}
π
A
B
the plane
A
B
M
ABM
A
BM
. Planes
π
A
C
,
.
.
.
,
π
C
D
\pi_{AC},...,\pi_{CD}
π
A
C
,
...
,
π
C
D
are analogously defined. Prove that the six planes
π
A
B
,
.
.
.
,
π
C
D
\pi_{AB},...,\pi_{CD}
π
A
B
,
...
,
π
C
D
are concurrent in a certain point
N
N
N
, and show that
N
N
N
is symmetric to the incenter
I
I
I
with respect to the barycenter
G
G
G
.
1
2
Hide problems
set $\{k | f(k) < k\}$ is finite, the set $\{k | g(f(k)) \le k\}$ is infinite
Let
f
:
N
→
N
f : N \to N
f
:
N
→
N
be a function such that the set
{
k
∣
f
(
k
)
<
k
}
\{k | f(k) < k\}
{
k
∣
f
(
k
)
<
k
}
is finite. Prove that the set
{
k
∣
g
(
f
(
k
)
)
≤
k
}
\{k | g(f(k)) \le k\}
{
k
∣
g
(
f
(
k
))
≤
k
}
is infinite for all functions
g
:
N
→
N
g : N \to N
g
:
N
→
N
.
[x/a]+[y/b]=[a^{n-1}/b]+[b^{n-1}/a]
Let a,b,n be positive integers such that
(
a
,
b
)
=
1
(a,b) = 1
(
a
,
b
)
=
1
. Prove that if
(
x
,
y
)
(x,y)
(
x
,
y
)
is a solution of the equation
a
x
+
b
y
=
a
n
+
b
n
ax+by = a^n + b^n
a
x
+
b
y
=
a
n
+
b
n
then
[
x
b
]
+
[
y
a
]
=
[
a
n
−
1
b
]
+
[
b
n
−
1
a
]
\left[\frac{x}{b}\right]+\left[\frac{y}{a}\right]=\left[\frac{a^{n-1}}{b}\right]+\left[\frac{b^{n-1}}{a}\right]
[
b
x
]
+
[
a
y
]
=
[
b
a
n
−
1
]
+
[
a
b
n
−
1
]
7
1
Hide problems
Sequence and floor function
The sequence
(
x
n
)
n
≥
1
(x_n)_{n \geq 1}
(
x
n
)
n
≥
1
is defined by: x_1\equal{}1 x_{n\plus{}1}\equal{}\frac{x_n}{n}\plus{}\frac{n}{x_n} Prove that
(
x
n
)
(x_n)
(
x
n
)
increases and [x_n^2]\equal{}n.
10
1
Hide problems
just q>p
Let
p
,
q
p,q
p
,
q
be positive prime numbers and suppose
q
>
5
q>5
q
>
5
. Prove that if
q
∣
2
p
+
3
p
q \mid 2^{p}+3^{p}
q
∣
2
p
+
3
p
, then
q
>
p
q>p
q
>
p
. Laurentiu Panaitopol
3
2
Hide problems
the lcm and prime
Prove that for any positive integer
n
n
n
, the least common multiple of the numbers
1
,
2
,
…
,
n
1,2,\ldots,n
1
,
2
,
…
,
n
and the least common multiple of the numbers:
(
n
1
)
,
(
n
2
)
,
…
,
(
n
n
)
\binom{n}{1},\binom{n}{2},\ldots,\binom{n}{n}
(
1
n
)
,
(
2
n
)
,
…
,
(
n
n
)
are equal if and only if
n
+
1
n+1
n
+
1
is a prime number. Laurentiu Panaitopol
2P(2x^2 -1) = P(x)^2 -1
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
such that
2
P
(
2
x
2
−
1
)
=
P
(
x
)
2
−
1
2P(2x^2 -1) = P(x)^2 -1
2
P
(
2
x
2
−
1
)
=
P
(
x
)
2
−
1
for all
x
x
x
.
2
2
Hide problems
geom ineq lp
Prove that in any triangle
A
B
C
ABC
A
BC
the following inequality holds:
a
2
b
+
c
−
a
+
b
2
a
+
c
−
b
+
c
2
a
+
b
−
c
≥
3
3
R
.
\frac{a^{2}}{b+c-a}+\frac{b^{2}}{a+c-b}+\frac{c^{2}}{a+b-c}\geq 3\sqrt{3}R.
b
+
c
−
a
a
2
+
a
+
c
−
b
b
2
+
a
+
b
−
c
c
2
≥
3
3
R
.
Laurentiu Panaitopol
binomial sums equality
Prove the following equality for all positive integers
m
,
n
m,n
m
,
n
:
∑
k
=
0
n
(
m
+
k
k
)
2
n
−
k
+
∑
k
=
0
m
(
n
+
k
k
)
2
m
−
k
=
2
m
+
n
+
1
\sum_{k=0}^{n} {m+k \choose k} 2^{n-k} +\sum_{k=0}^m {n+k \choose k}2^{m-k}= 2^{m+n+1}
k
=
0
∑
n
(
k
m
+
k
)
2
n
−
k
+
k
=
0
∑
m
(
k
n
+
k
)
2
m
−
k
=
2
m
+
n
+
1