MathDB
Problems
Contests
National and Regional Contests
Argentina Contests
Argentina National Olympiad
2011 Argentina National Olympiad
2011 Argentina National Olympiad
Part of
Argentina National Olympiad
Subcontests
(5)
2
1
Hide problems
winning strategy for 2 pleayers tema up aginaast third on, piles of stones
Three players
A
,
B
A,B
A
,
B
and
C
C
C
take turns removing stones from a pile of
N
N
N
stones. They move in the order
A
,
B
,
C
,
A
,
B
,
C
,
…
A
A,B,C,A,B,C,…A
A
,
B
,
C
,
A
,
B
,
C
,
…
A
. The game begins, and the one who takes out the last stone loses the game. The players
A
A
A
and
C
C
C
team up against
B
B
B
, they agree on a joint strategy.
B
B
B
can take in each play
1
,
2
,
3
,
4
1,2,3,4
1
,
2
,
3
,
4
or
5
5
5
stones, while
A
A
A
and
C
C
C
, they can each get
1
,
2
1,2
1
,
2
or
3
3
3
stones each turn. Determine for what values of
N
N
N
have winning strategy
A
A
A
and
C
C
C
, and for what values the winning strategy is from
B
B
B
. .
1
1
Hide problems
S_1+S_1^2+S_2^2+\cdots +S_{2011}^2 if S_k = sum 1/i
For
k
=
1
,
2
,
…
,
2011
k=1,2,\ldots ,2011
k
=
1
,
2
,
…
,
2011
we denote
S
k
=
1
k
+
1
k
+
1
+
⋯
+
1
2011
S_k=\frac{1}{k}+\frac{1}{k+1}+\cdots +\frac{1}{2011}
S
k
=
k
1
+
k
+
1
1
+
⋯
+
2011
1
. Compute the sum
S
1
+
S
1
2
+
S
2
2
+
⋯
+
S
2011
2
S_1+S_1^2+S_2^2+\cdots +S_{2011}^2
S
1
+
S
1
2
+
S
2
2
+
⋯
+
S
2011
2
.
5
1
Hide problems
n^3-1 is divisible by 10^6 n−1
Find all integers
n
n
n
such that
1
<
n
<
1
0
6
1<n<10^6
1
<
n
<
1
0
6
and
n
3
−
1
n^3-1
n
3
−
1
is divisible by
1
0
6
n
−
1
10^6 n-1
1
0
6
n
−
1
.
6
1
Hide problems
2-player game with a square, drawing segments of given segment
We have a square of side
1
1
1
and a number
ℓ
\ell
ℓ
such that
0
<
ℓ
<
2
0 <\ell <\sqrt2
0
<
ℓ
<
2
. Two players
A
A
A
and
B
B
B
, in turn, draw in the square an open segment (without its two ends) of length
ℓ
\ell
ℓ
, starts A. Each segment after the first cannot have points in common with the previously drawn segments. He loses the player who cannot make his play. Determine if either player has a winning strategy.
3
1
Hide problems
computational staring with w 90-57-15 triangle, <APB =<CPQ, <BQA=<CQP
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
=
9
0
o
,
∠
B
=
7
5
o
\angle A = 90^o, \angle B = 75^o
∠
A
=
9
0
o
,
∠
B
=
7
5
o
and
A
B
=
2
AB = 2
A
B
=
2
. The points
P
P
P
and
Q
Q
Q
on the sides
A
C
AC
A
C
and
B
C
BC
BC
respectively are such that
∠
A
P
B
=
∠
C
P
Q
\angle APB = \angle CPQ
∠
A
PB
=
∠
CPQ
and
∠
B
Q
A
=
∠
C
Q
P
\angle BQA = \angle CQP
∠
BQ
A
=
∠
CQP
. Calculate the measurement of the segment
Q
A
QA
Q
A
.