MathDB
Problems
Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan IZHO TST
2021 Azerbaijan IZhO TST
2021 Azerbaijan IZhO TST
Part of
Azerbaijan IZHO TST
Subcontests
(4)
4
1
Hide problems
Sharky-devil point with cicumcenter lying on BC
Let
A
B
C
ABC
A
BC
be a triangle with incircle touching
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
,
D, E, F,
D
,
E
,
F
,
respectively. Let
O
O
O
and
M
M
M
be its circumcenter and midpoint of
B
C
.
BC.
BC
.
Suppose that circumcircles of
A
E
F
AEF
A
EF
and
A
B
C
ABC
A
BC
intersect at
X
X
X
for the second time. Assume
Y
≠
X
Y \neq X
Y
=
X
is on the circumcircle of
A
B
C
ABC
A
BC
such that
O
M
X
Y
OMXY
OMX
Y
is cyclic. Prove that circumcenter of
D
X
Y
DXY
D
X
Y
lies on
B
C
.
BC.
BC
.
Proposed by tenplusten.
2
1
Hide problems
n*m board with white and black colors
Find the number of ways to color
n
×
m
n \times m
n
×
m
board with white and black colors such that any
2
×
2
2 \times 2
2
×
2
square contains the same number of black and white cells.
1
1
Hide problems
((a+b)^2 +1)/(c^2+2)+((b+c)^2 +1)/(a^2+2)+((c+a)^2 +1)/(b^2+2) >=3
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be real numbers with the property as
a
b
+
b
c
+
c
a
=
1
ab + bc + ca = 1
ab
+
b
c
+
c
a
=
1
. Show that:
(
a
+
b
)
2
+
1
c
2
+
2
+
(
b
+
c
)
2
+
1
a
2
+
2
+
(
c
+
a
)
2
+
1
b
2
+
2
≥
3
\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3
c
2
+
2
(
a
+
b
)
2
+
1
+
a
2
+
2
(
b
+
c
)
2
+
1
+
b
2
+
2
(
c
+
a
)
2
+
1
≥
3
.
3
1
Hide problems
good qstn
For each
n
∈
N
n \in N
n
∈
N
let
S
(
n
)
S(n)
S
(
n
)
be the sum of all numbers in the set {1,2,3,…,n} which are relatively prime to
n
n
n
. a. Show that
2
S
(
n
)
2S(n)
2
S
(
n
)
is not aperfect square for any
n
n
n
. b. Given positive integers
m
,
n
m,n
m
,
n
with odd n, show that the equation
2
S
(
x
)
=
y
n
2S(x)=y^n
2
S
(
x
)
=
y
n
has at least one solution
(
x
,
y
)
(x,y)
(
x
,
y
)
among positive integers such that
m
∣
x
m|x
m
∣
x
.