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Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2010 Argentina National Olympiad
2010 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(6)
6
1
Hide problems
+ or x amoing 1-201-
In a row the numbers
1
,
2
,
.
.
.
,
2010
1,2,...,2010
1
,
2
,
...
,
2010
have been written. Two players, taking turns, write
+
+
+
or
×
\times
×
between two consecutive numbers whenever possible. The first player wins if the algebraic sum obtained is divisible by
3
3
3
; otherwise, the second player wins. Find a winning strategy for one of the players.
5
1
Hide problems
v= 2uw/(u+w)
21
21
21
numbers are written in a row.
u
,
v
,
w
u,v,w
u
,
v
,
w
are three consecutive numbers so
v
=
2
u
w
u
+
w
v=\frac{2uw}{u+w}
v
=
u
+
w
2
u
w
. The first number is
1
100
\frac{1}{100}
100
1
, the last one is
1
101
\frac{1}{101}
101
1
. Find the
15
15
15
th number.
4
1
Hide problems
sum of all products a_1a_2...a_{50}
Find the sum of all products
a
1
a
2
.
.
.
a
50
a_1a_2...a_{50}
a
1
a
2
...
a
50
, where
a
1
,
a
2
,
.
.
.
,
a
50
a_1,a_2,...,a_{50}
a
1
,
a
2
,
...
,
a
50
are distinct positive integers, less than or equal to
101
101
101
, and such that no two of them add up to
101
101
101
.
3
1
Hide problems
min max of a+b+c if a^2+b^2=c^2+99^2 for integers 0<a,b,c <99
The positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
are less than
99
99
99
and satisfy
a
2
+
b
2
=
c
2
+
9
9
2
a^2+b^2=c^2+99^2
a
2
+
b
2
=
c
2
+
9
9
2
. . Find the minimum and maximum value of
a
+
b
+
c
a+b+c
a
+
b
+
c
.
1
1
Hide problems
replace two of them by their non-negative difference
Given several integers, the allowed operation is to replace two of them by their non-negative difference. The operation is repeated until only one number remains. If the initial numbers are
1
,
2
,
…
,
2010
1, 2, … , 2010
1
,
2
,
…
,
2010
, what can be the last remaining number?
2
1
Hide problems
computational with incircle of a right triangle, AP = QM
Let
A
B
C
ABC
A
BC
be a triangle with
∠
C
=
9
0
o
\angle C = 90^o
∠
C
=
9
0
o
and
A
C
=
1
AC = 1
A
C
=
1
. The median
A
M
AM
A
M
intersects the incircle at the points
P
P
P
and
Q
Q
Q
, with
P
P
P
between
A
A
A
and
Q
Q
Q
, such that
A
P
=
Q
M
AP = QM
A
P
=
QM
. Find the length of
P
Q
PQ
PQ
.