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National and Regional Contests
Romania Contests
Romania Team Selection Test
1994 Romania TST for IMO
1994 Romania TST for IMO
Part of
Romania Team Selection Test
Subcontests
(4)
3:
3
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romania tst
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, . . ., a_n
a
1
,
a
2
,
...
,
a
n
be a finite sequence of
0
0
0
and
1
1
1
. Under any two consecutive terms of this sequence
0
0
0
is written if the digits are equal and
1
1
1
is written otherwise. This way a new sequence of length
n
−
1
n -1
n
−
1
is obtained. By repeating this procedure
n
−
1
n - 1
n
−
1
times one obtains a triangular table of
0
0
0
and
1
1
1
. Find the maximum possible number of ones that can appear on this table
easy old exercice
Prove that the sequence
a
n
=
3
n
−
2
n
a_n = 3^n- 2^n
a
n
=
3
n
−
2
n
contains no three numbers in geometric progression.
easy problem
Determine all integer solutions of the equation
x
n
+
y
n
=
1994
x^n+y^n=1994
x
n
+
y
n
=
1994
where
n
≥
2
n\geq 2
n
≥
2
4:
3
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old problem
Inscribe an equilateral triangle of minimum side in a given acute-angled triangle
A
B
C
ABC
A
BC
(one vertex on each side).
romania tsts
Let be given two concentric circles of radii
R
R
R
and
R
1
>
R
R_1 > R
R
1
>
R
. Let quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in the smaller circle and let the rays
C
D
,
D
A
,
A
B
,
B
C
CD, DA, AB, BC
C
D
,
D
A
,
A
B
,
BC
meet the larger circle at
A
1
,
B
1
,
C
1
,
D
1
A_1, B_1, C_1, D_1
A
1
,
B
1
,
C
1
,
D
1
respectively. Prove that
σ
(
A
1
B
1
C
1
D
1
)
σ
(
A
B
C
D
)
≥
R
1
2
R
2
\frac{\sigma(A_1B_1C_1D_1)}{\sigma(ABCD)} \geq \frac{R_1^2}{R^2}
σ
(
A
BC
D
)
σ
(
A
1
B
1
C
1
D
1
)
≥
R
2
R
1
2
where
σ
(
P
)
\sigma(P)
σ
(
P
)
denotes the area of a polygon
P
.
P.
P
.
Romania TST 1994, 3rd P4
Find a sequence of positive integer
f
(
n
)
f(n)
f
(
n
)
,
n
∈
N
n \in \mathbb{N}
n
∈
N
such that
(
1
)
(1)
(
1
)
f
(
n
)
≤
n
8
f(n) \leq n^8
f
(
n
)
≤
n
8
for any
n
≥
2
n \geq 2
n
≥
2
,
(
2
)
(2)
(
2
)
for any pairwisely distinct natural numbers
a
1
,
a
2
,
⋯
,
a
k
a_1,a_2,\cdots, a_k
a
1
,
a
2
,
⋯
,
a
k
and
n
n
n
, we have that
f
(
n
)
≠
f
(
a
1
)
+
f
(
a
2
)
+
⋯
+
f
(
a
k
)
f(n) \neq f(a_1)+f(a_2)+ \cdots + f(a_k)
f
(
n
)
=
f
(
a
1
)
+
f
(
a
2
)
+
⋯
+
f
(
a
k
)
2:
3
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Divisible by $((n-1)^n+1)^2$ (Romania TST 1994)
Let
n
n
n
be an odd positive integer. Prove that
(
(
n
−
1
)
n
+
1
)
2
((n-1)^n+1)^2
((
n
−
1
)
n
+
1
)
2
divides
n
(
n
−
1
)
(
n
−
1
)
n
+
1
+
n
n(n-1)^{(n-1)^n+1}+n
n
(
n
−
1
)
(
n
−
1
)
n
+
1
+
n
.
romania tsts
Let
S
1
,
S
2
,
S
3
S_1, S_2,S_3
S
1
,
S
2
,
S
3
be spheres of radii
a
,
b
,
c
a, b, c
a
,
b
,
c
respectively whose centers lie on a line
l
l
l
. Sphere
S
2
S_2
S
2
is externally tangent to
S
1
S_1
S
1
and
S
3
S_3
S
3
, whereas
S
1
S_1
S
1
and
S
3
S_3
S
3
have no common points. A straight line t touches each of the spheres, Find the sine of the angle between
l
l
l
and
t
t
t
romania tst
Let
n
n
n
be a positive integer. Find the number of polynomials
P
(
x
)
P(x)
P
(
x
)
with coefficients in
{
0
,
1
,
2
,
3
}
\{0, 1, 2, 3\}
{
0
,
1
,
2
,
3
}
for which
P
(
2
)
=
n
P(2) = n
P
(
2
)
=
n
.
1:
3
Hide problems
"odd" sets and "even" sets
Let X_n\equal{}\{1,2,...,n\},where
n
≥
3
n \geq 3
n
≥
3
. We define the measure
m
(
X
)
m(X)
m
(
X
)
of
X
⊂
X
n
X\subset X_n
X
⊂
X
n
as the sum of its elements.(If |X|\equal{}0,then m(X)\equal{}0). A set
X
⊂
X
n
X \subset X_n
X
⊂
X
n
is said to be even(resp. odd) if
m
(
X
)
m(X)
m
(
X
)
is even(resp. odd). (a)Show that the number of even sets equals the number of odd sets. (b)Show that the sum of the measures of the even sets equals the sum of the measures of the odd sets. (c)Compute the sum of the measures of the odd sets.
Find the smallest nomial
Find the smallest nomial of this sequence that
a
1
=
199
3
199
4
1995
a_1=1993^{1994^{1995}}
a
1
=
199
3
199
4
1995
and
a
n
+
1
=
{
a
n
2
if
n
is even
a
n
+
7
if
n
is odd.
a_{n+1}=\begin{cases}\frac{a_n}{2}&\text{if $n$ is even}\\a_n+7 &\text{if $n$ is odd.} \end{cases}
a
n
+
1
=
{
2
a
n
a
n
+
7
if
n
is even
if
n
is odd.
Romania tst
Let
p
p
p
be a (positive) prime number. Suppose that real numbers
a
1
,
a
2
,
.
.
.
,
a
p
+
1
a_1, a_2, . . ., a_{p+1}
a
1
,
a
2
,
...
,
a
p
+
1
have the property that, whenever one of the numbers is deleted, the remaining numbers can be partitioned into two classes with the same arithmetic mean. Show that these numbers must be equal.