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Problems
Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan Junior National Olympiad
2016 Azerbaijan Junior Mathematical Olympiad
2016 Azerbaijan Junior Mathematical Olympiad
Part of
Azerbaijan Junior National Olympiad
Subcontests
(7)
4
1
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easy geometry
Let
O
O
O
be the circumcenter of
△
A
B
C
.
\triangle ABC.
△
A
BC
.
The circle
k
k
k
passing through
A
A
A
and
B
B
B
cuts
A
C
AC
A
C
and
B
C
BC
BC
at
P
P
P
and
Q
,
Q,
Q
,
respectively. Prove that
P
Q
PQ
PQ
and
O
C
OC
OC
are perpendicular.
7
1
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geometric proportion
In
△
A
B
C
\triangle ABC
△
A
BC
the median
A
M
AM
A
M
is drawn. The foot of perpendicular from
B
B
B
to the angle bisector of
∠
B
M
A
\angle BMA
∠
BM
A
is
B
1
B_1
B
1
and the foot of perpendicular from
C
C
C
to the angle bisector of
∠
A
M
C
\angle AMC
∠
A
MC
is
C
1
.
C_1.
C
1
.
Let
M
A
MA
M
A
and
B
1
C
1
B_1C_1
B
1
C
1
intersect at
A
1
.
A_1.
A
1
.
Find
B
1
A
1
A
1
C
1
.
\frac{B_1A_1}{A_1C_1}.
A
1
C
1
B
1
A
1
.
5
1
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number theory
Positive integers
(
p
,
a
,
b
,
c
)
(p,a,b,c)
(
p
,
a
,
b
,
c
)
called good quadruple if a)
p
p
p
is odd prime,b)
a
,
b
,
c
a,b,c
a
,
b
,
c
are distinct ,c)
a
b
+
1
,
b
c
+
1
ab+1,bc+1
ab
+
1
,
b
c
+
1
and
c
a
+
1
ca+1
c
a
+
1
are divisible by
p
p
p
.Prove that for all good quadruple
p
+
2
≤
a
+
b
+
c
3
p+2\le \frac {a+b+c}{3}
p
+
2
≤
3
a
+
b
+
c
, and show the equality case.
6
1
Hide problems
Inequality.
For all reals
x
,
y
,
z
x,y,z
x
,
y
,
z
prove that
x
2
+
1
y
2
+
y
2
+
1
z
2
+
z
2
+
1
x
2
≥
3
2
.
\sqrt {x^2+\frac {1}{y^2}}+ \sqrt {y^2+\frac {1}{z^2}}+ \sqrt {z^2+\frac {1}{x^2}}\geq 3\sqrt {2}.
x
2
+
y
2
1
+
y
2
+
z
2
1
+
z
2
+
x
2
1
≥
3
2
.
2
1
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Algebra.
Prove that if for a real number
a
a
a
,
a
+
1
a
a+\frac {1}{a}
a
+
a
1
is integer then
a
n
+
1
a
n
a^n+\frac {1}{a^n}
a
n
+
a
n
1
is also integer where
n
n
n
is positive integer.
1
1
Hide problems
Decimak Representation
In decimal representation
34!=295232799039a041408476186096435b0000000
.
\text {34!=295232799039a041408476186096435b0000000}.
34!=295232799039a041408476186096435b0000000
.
Find the numbers
a
a
a
and
b
b
b
.
3
1
Hide problems
combinatorics problem
65
65
65
distinct natural numbers not exceeding
2016
2016
2016
are given. Prove that among these numbers we can find four
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
such that
a
+
b
−
c
−
d
a+b-c-d
a
+
b
−
c
−
d
is divisible by
2016.
2016.
2016.