MathDB
Problems
Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan Junior National Olympiad
2022 Azerbaijan Junior National Olympiad
2022 Azerbaijan Junior National Olympiad
Part of
Azerbaijan Junior National Olympiad
Subcontests
(5)
G5
1
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Prove that lines are colinear
Let
A
B
C
ABC
A
BC
be an acute triangle and
G
G
G
be the intersection of the meadians of triangle
A
B
C
ABC
A
BC
. Let
D
D
D
be the foot of the altitude drawn from
A
A
A
to
B
C
BC
BC
. Draw a parallel line such that it is parallel to
B
C
BC
BC
and one of the points of it is
A
A
A
.Donate the point
S
S
S
as the intersection of the parallel line and circumcircle
A
B
C
ABC
A
BC
. Prove that
S
,
G
,
D
S,G,D
S
,
G
,
D
are co-linear [asy] size(6cm); defaultpen(fontsize(10pt)); pair A = dir(50), S = dir(130), B = dir(200), C = dir(-20), G = (A+B+C)/3, D = foot(A, B, C);draw(A--B--C--cycle, black+linewidth(1)); draw(A--S^^A--D, magenta); draw(S--D, red+dashed); draw(circumcircle(A, B, C), heavymagenta);string[] names = {"
A
A
A
", "
B
B
B
", "
C
C
C
","
D
D
D
", "
G
G
G
","
S
S
S
"}; pair[] points = {A, B, C,D,G,S}; pair[] ll = {A, B, C,D, G,S}; int pt = names.length; for (int i=0; i
C4
1
Hide problems
8*8 board and the numbers 1,2,3,...,64
There is a
8
∗
8
8*8
8
∗
8
board and the numbers
1
,
2
,
3
,
4
,
.
.
.
,
63
,
64
1,2,3,4,...,63,64
1
,
2
,
3
,
4
,
...
,
63
,
64
. In all the unit squares of the board, these numbers are places such that only
1
1
1
numbers goes to only one unit square. Prove that there is atleast
4
4
4
2
∗
2
2*2
2
∗
2
squares such that the sum of the numbers in
2
∗
2
2*2
2
∗
2
is greater than
100
100
100
.
A3
1
Hide problems
Cyclic inequality where x^2+y^2+z^2=x+y+z
Let
x
,
y
,
z
∈
R
+
x,y,z \in \mathbb{R}^{+}
x
,
y
,
z
∈
R
+
and
x
2
+
y
2
+
z
2
=
x
+
y
+
z
x^2+y^2+z^2=x+y+z
x
2
+
y
2
+
z
2
=
x
+
y
+
z
. Prove that
x
+
y
+
z
+
3
≥
6
x
y
+
y
z
+
z
x
3
3
x+y+z+3 \ge 6 \sqrt[3]{\frac{xy+yz+zx}{3}}
x
+
y
+
z
+
3
≥
6
3
3
x
y
+
yz
+
z
x
N2
1
Hide problems
prove that 6|x,y,z
If
x
,
y
,
z
∈
N
x,y,z \in\mathbb{N}
x
,
y
,
z
∈
N
and
2
x
2
+
3
y
3
=
4
z
4
2x^2+3y^3=4z^4
2
x
2
+
3
y
3
=
4
z
4
, Prove that
6
∣
x
,
y
,
z
6|x,y,z
6∣
x
,
y
,
z
A1
1
Hide problems
Minimum value of a sequence
Find the minimum positive value of
1
∗
2
∗
3
∗
4
∗
.
.
.
∗
2020
∗
2021
∗
2022
1*2*3*4*...*2020*2021*2022
1
∗
2
∗
3
∗
4
∗
...
∗
2020
∗
2021
∗
2022
where you can replace
∗
*
∗
as
+
+
+
or
−
-
−