MathDB
Problems
Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan Junior National Olympiad
2017 Azerbaijan Junior National Olympiad
2017 Azerbaijan Junior National Olympiad
Part of
Azerbaijan Junior National Olympiad
Subcontests
(5)
P5
1
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getting 20^17+2016 from x=3
A student firstly wrote
x
=
3
x=3
x
=
3
on the board. For each procces, the stutent deletes the number x and replaces it with either
(
2
x
+
4
)
(2x+4)
(
2
x
+
4
)
or
(
3
x
+
8
)
(3x+8)
(
3
x
+
8
)
or
(
x
2
+
5
x
)
(x^2+5x)
(
x
2
+
5
x
)
. Is this possible to make the number
(
2
0
17
+
2016
)
(20^{17}+2016)
(
2
0
17
+
2016
)
on the board? \\ (Explain your answer) \\ This type of the question is well known but I am going to make a collection so, :blush:
P4
1
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Comparing Acute angles of Rhombus and Trapezoid
A Rhombus and an Isosceles trapezoid that has same area is drawn in the same circle's outside. Compare their acute angles \\ (explain your answer)
P2
1
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finding the number of 2015 and 2016 in an infinite sequence
For all
n
>
1
n>1
n
>
1
let
f
(
n
)
f(n)
f
(
n
)
be the sum of the smallest factor of
n
n
n
that is not 1 and
n
n
n
. The computer prints
f
(
2
)
,
f
(
3
)
,
f
(
4
)
,
.
.
.
f(2),f(3),f(4),...
f
(
2
)
,
f
(
3
)
,
f
(
4
)
,
...
with order:
4
,
6
,
6
,
.
.
.
4,6,6,...
4
,
6
,
6
,
...
( Because
f
(
2
)
=
2
+
2
=
4
,
f
(
3
)
=
3
+
3
=
6
,
f
(
4
)
=
4
+
2
=
6
f(2)=2+2=4,f(3)=3+3=6,f(4)=4+2=6
f
(
2
)
=
2
+
2
=
4
,
f
(
3
)
=
3
+
3
=
6
,
f
(
4
)
=
4
+
2
=
6
etc.). In this infinite sequence, how many times will be
2015
2015
2015
and
2016
2016
2016
written? (Explain your answer)
P1
1
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A system of equation with squares/roots for real numbers
Solve the system of equation for
(
x
,
y
)
∈
R
(x,y) \in \mathbb{R}
(
x
,
y
)
∈
R
{
x
2
+
y
2
+
(
x
−
4
)
2
+
(
y
−
3
)
2
=
5
3
x
2
+
4
x
y
=
24
\left\{\begin{matrix} \sqrt{x^2+y^2}+\sqrt{(x-4)^2+(y-3)^2}=5\\ 3x^2+4xy=24 \end{matrix}\right.
{
x
2
+
y
2
+
(
x
−
4
)
2
+
(
y
−
3
)
2
=
5
3
x
2
+
4
x
y
=
24
\\ Explain your answer
P3
1
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ASU 543 All Soviet Union MO 1991 (x+y+z)^2/3>=x\sqrt{yz}+y\sqrt{zx}+z\sqrt{xy}
Show that
(
x
+
y
+
z
)
2
3
≥
x
y
z
+
y
z
x
+
z
x
y
\frac{(x + y + z)^2}{3} \ge x\sqrt{yz} + y\sqrt{zx} + z\sqrt{xy}
3
(
x
+
y
+
z
)
2
≥
x
yz
+
y
z
x
+
z
x
y
for all non-negative reals
x
,
y
,
z
x, y, z
x
,
y
,
z
.