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Contests
National and Regional Contests
Serbia Contests
Serbia Team Selection Test
1998 Yugoslav Team Selection Test
1998 Yugoslav Team Selection Test
Part of
Serbia Team Selection Test
Subcontests
(3)
Problem 3
1
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arith. seq: nCk, nC(k+1), nC(k+2), nC(k+3)
Prove that there are no positive integers
n
n
n
and
k
≤
n
k\le n
k
≤
n
such that the numbers
(
n
k
)
,
(
n
k
+
1
)
,
(
n
k
+
2
)
,
(
n
k
+
3
)
\binom nk,\binom n{k+1},\binom n{k+2},\binom n{k+3}
(
k
n
)
,
(
k
+
1
n
)
,
(
k
+
2
n
)
,
(
k
+
3
n
)
in this order form an arithmetic progression.
Problem 2
1
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inradii inequality
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, the diagonal
A
C
AC
A
C
intersects the diagonal
B
D
BD
B
D
at its midpoint
S
S
S
. The radii of incircles of triangles
A
B
S
,
B
C
S
,
C
D
S
,
D
A
S
ABS,BCS,CDS,DAS
A
BS
,
BCS
,
C
D
S
,
D
A
S
are
r
1
,
r
2
,
r
3
,
r
4
r_1,r_2,r_3,r_4
r
1
,
r
2
,
r
3
,
r
4
, respectively. Prove that
∣
r
1
−
r
2
+
r
3
−
r
4
∣
≤
1
8
∣
A
B
−
B
C
+
C
D
−
D
A
∣
.
|r_1-r_2+r_3-r_4|\le\frac18|AB-BC+CD-DA|.
∣
r
1
−
r
2
+
r
3
−
r
4
∣
≤
8
1
∣
A
B
−
BC
+
C
D
−
D
A
∣.
Problem 1
1
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winning strategy in card game
From a deck of playing cards, four threes, four fours and four fives are selected and put down on a table with the main side up. Players
A
A
A
and
B
B
B
alternately take the cards one by one and put them on the pile. Player
A
A
A
begins. A player after whose move the sum of values of the cards on the pile is (a) greater than 34; (b) greater than 37; loses the game. Which player has a winning strategy?