MathDB
Problems
Contests
National and Regional Contests
Azerbaijan Contests
JBMO TST - Azerbaijan
2016 Azerbaijan JBMO TST
2016 Azerbaijan JBMO TST
Part of
JBMO TST - Azerbaijan
Subcontests
(4)
2
1
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Angle bisectors meets the circumcircle
Let the angle bisectors of
∠
B
A
C
,
\angle BAC,
∠
B
A
C
,
∠
C
B
A
,
\angle CBA,
∠
CB
A
,
and
∠
A
C
B
\angle ACB
∠
A
CB
meets the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at the points
M
,
N
,
M,N,
M
,
N
,
and
K
,
K,
K
,
respectively. Let the segments
A
B
AB
A
B
and
M
K
MK
M
K
intersects at the point
P
P
P
and the segments
A
C
AC
A
C
and
M
N
MN
MN
intersects at the point
Q
.
Q.
Q
.
Prove that
P
Q
∥
B
C
PQ\parallel BC
PQ
∥
BC
4
2
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three stack of tokens game , winning strategy
There are three stacks of tokens on the table: the first contains
a
,
a,
a
,
the second contains
b
,
b,
b
,
and the third contains
c
c
c
where
a
≥
b
≥
c
.
a \ge b \ge c.
a
≥
b
≥
c
.
Players
A
A
A
and
B
B
B
take turns playing a game of swapping tokens.
A
A
A
starts first. On each turn, the player who gets his turn chooses two stacks, then takes at least one token from the stack with the lowest number of tokens and places them on the stack with the highest number of tokens. If the number of tokens in the two piles he/she chooses is equal, then he/she takes at least one token from any of them and puts it in the other. If only one pile remains after a player's move, that player is considered a winner. At what values of
a
,
b
,
c
a, b, c
a
,
b
,
c
who has the winning strategy (
A
A
A
or
B
B
B
)?
Find all positive integers.....
Find all positive integers n such that
(
n
2
+
11
n
−
4
)
n
!
+
33.1
3
n
+
4
(n ^{ 2} + 11n - 4) n! + 33.13 ^ n + 4
(
n
2
+
11
n
−
4
)
n
!
+
33.1
3
n
+
4
is the perfect square
1
2
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inequalities
Let
a
,
b
,
c
≥
0
a,b,c \ge 0
a
,
b
,
c
≥
0
and
a
+
b
+
c
=
3
a+b+c=3
a
+
b
+
c
=
3
. Prove that:
2
(
a
b
+
b
c
+
c
a
)
−
3
a
b
c
≥
∑
c
y
c
a
b
2
+
c
2
2
2(ab+bc+ca)-3abc\ge \sum_{cyc}^{}a\sqrt{\frac{b^2+c^2}{2}}
2
(
ab
+
b
c
+
c
a
)
−
3
ab
c
≥
∑
cyc
a
2
b
2
+
c
2
inequality involving triangle
If
a
,
b
,
c
a,b,c
a
,
b
,
c
represent the lengths of the sides of a triangle, prove the inequality:
3
≤
∑
cyc
a
−
a
+
b
+
c
.
3\le\sum_{\text{cyc}}\sqrt{\frac{a}{-a+b+c}} .
3
≤
cyc
∑
−
a
+
b
+
c
a
.
3
2
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Equation
Find all the pime numbers
(
p
,
q
)
(p,q)
(
p
,
q
)
such that :
p
3
+
p
=
q
2
+
q
p^{3}+p=q^{2}+q
p
3
+
p
=
q
2
+
q
m*n table colored with white and black
All cells of the
m
×
n
m\times n
m
×
n
table are colored either white or black such that all corner cells of any rectangle containing the cells of this table with sides greater than one cell are not the same color. For values
m
=
2
,
3
,
4
,
m = 2, 3, 4,
m
=
2
,
3
,
4
,
find all
n
n
n
such that the mentioned coloring is possible.