MathDB
Problems
Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1981 Yugoslav Team Selection Test
1981 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
Problem 3
1
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existence, x+2y+3z+7t=a and y+2z+5t=b
Let
a
,
b
a,b
a
,
b
be nonnegative integers. Prove that
5
a
>
7
b
5a>7b
5
a
>
7
b
if and only if there exist nonnegative integers
x
,
y
,
z
,
t
x,y,z,t
x
,
y
,
z
,
t
such that \begin{align*} x+2y+3z+7t&=a,\\ y+2z+5t&=b. \end{align*}
Problem 2
1
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point divides quadrilateral into four equal triangles
Suppose that there is a point
S
S
S
inside a quadrilateral
A
B
C
D
ABCD
A
BC
D
such that segments
S
A
,
S
B
,
S
C
,
S
D
SA,SB,SC,SD
S
A
,
SB
,
SC
,
S
D
divide the quadrilateral into four triangles of equal areas. Prove that one of the diagonals of the quadrilateral bisects the other one.
Problem 1
1
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increasing arithmetic sequences in set
Let
n
≥
3
n\ge3
n
≥
3
be a natural number. For a set
S
S
S
of
n
n
n
real numbers,
A
(
S
)
A(S)
A
(
S
)
denotes the set of all strictly increasing arithmetic sequences of three terms in
S
S
S
. At most, how many elements can the set
A
(
S
)
A(S)
A
(
S
)
have?