MathDB
Problems
Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1974 Yugoslav Team Selection Test
1974 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
Problem 2
1
Hide problems
triangle, transformation as composition of three
Given two directly congruent triangles
A
B
C
ABC
A
BC
and
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
in a plane, assume that the circles with centers
C
C
C
and
C
′
C'
C
′
and radii
C
A
CA
C
A
and
C
′
A
′
C'A'
C
′
A
′
intersect. Denote by
M
\mathcal M
M
the transformation that maps
△
A
B
C
\triangle ABC
△
A
BC
to
△
A
′
B
′
C
′
\triangle A'B'C'
△
A
′
B
′
C
′
. Prove that
M
\mathcal M
M
can be expressed as a composition of at most three rotations in the following way: The first rotation has the center in one of
A
,
B
,
C
A,B,C
A
,
B
,
C
and maps
△
A
B
C
\triangle ABC
△
A
BC
to
△
A
1
B
1
C
1
\triangle A_1B_1C_1
△
A
1
B
1
C
1
; The second rotation has the center in one of
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
, and maps
△
A
1
B
1
C
1
\triangle A_1B_1C_1
△
A
1
B
1
C
1
to
△
A
2
B
2
C
2
\triangle A_2B_2C_2
△
A
2
B
2
C
2
; The third rotation has the center in one of
A
2
,
B
2
,
C
2
A_2,B_2,C_2
A
2
,
B
2
,
C
2
and maps
△
A
2
B
2
C
2
\triangle A_2B_2C_2
△
A
2
B
2
C
2
to
△
A
′
B
′
C
′
\triangle A'B'C'
△
A
′
B
′
C
′
.
Problem 3
1
Hide problems
combinatorial geometry
Let
S
S
S
be a set of
n
n
n
points
P
1
,
P
2
,
…
,
P
n
P_1,P_2,\ldots,P_n
P
1
,
P
2
,
…
,
P
n
in a plane such that no three of the points are collinear. Let
α
\alpha
α
be the smallest of the angles
∠
P
i
P
j
P
k
\angle P_iP_jP_k
∠
P
i
P
j
P
k
(
i
≠
j
≠
k
≠
i
,
i
,
j
,
k
∈
{
1
,
2
,
…
,
n
}
i\ne j\ne k\ne i,i,j,k\in\{1,2,\ldots,n\}
i
=
j
=
k
=
i
,
i
,
j
,
k
∈
{
1
,
2
,
…
,
n
}
). Find
max
S
α
\max_S\alpha
max
S
α
and determine those sets
S
S
S
for which this maximal value is attained.
Problem 1
1
Hide problems
rational NT involving floor function
Assume that
a
a
a
is a given irrational number.(a) Prove that for each positive real number
ϵ
\epsilon
ϵ
there exists at least one integer
q
≥
0
q\ge0
q
≥
0
such that
a
q
−
⌊
a
q
⌋
<
ϵ
aq-\lfloor aq\rfloor<\epsilon
a
q
−
⌊
a
q
⌋
<
ϵ
. (b) Prove that for given
ϵ
>
0
\epsilon>0
ϵ
>
0
there exist infinitely many rational numbers
p
q
\frac pq
q
p
such that
q
>
0
q>0
q
>
0
and
∣
a
−
p
q
∣
<
ϵ
q
\left|a-\frac pq\right|<\frac\epsilon q
a
−
q
p
<
q
ϵ
.