MathDB
Problems
Contests
National and Regional Contests
Kazakhstan Contests
Kazakhstan National Olympiad
2021 Kazakhstan National Olympiad
2021 Kazakhstan National Olympiad
Part of
Kazakhstan National Olympiad
Subcontests
(4)
4
1
Hide problems
The line connecting centers bisection
Given acute triangle
A
B
C
ABC
A
BC
with circumcircle
Γ
\Gamma
Γ
and altitudes
A
D
,
B
E
,
C
F
AD, BE, CF
A
D
,
BE
,
CF
, line
A
D
AD
A
D
cuts
Γ
\Gamma
Γ
again at
P
P
P
and
P
F
,
P
E
PF, PE
PF
,
PE
meet
Γ
\Gamma
Γ
again at
R
,
Q
R, Q
R
,
Q
. Let
O
1
,
O
2
O_1, O_2
O
1
,
O
2
be the circumcenters of
△
B
F
R
\triangle BFR
△
BFR
and
△
C
E
Q
\triangle CEQ
△
CEQ
respectively. Prove that
O
1
O
2
O_{1}O_{2}
O
1
O
2
bisects
E
F
‾
\overline{EF}
EF
.
3
1
Hide problems
Two sequences
Let
(
a
n
)
(a_n)
(
a
n
)
and
(
b
n
)
(b_n)
(
b
n
)
be sequences of real numbers, such that
a
1
=
b
1
=
1
a_1 = b_1 = 1
a
1
=
b
1
=
1
,
a
n
+
1
=
a
n
+
a
n
a_{n+1} = a_n + \sqrt{a_n}
a
n
+
1
=
a
n
+
a
n
,
b
n
+
1
=
b
n
+
b
n
3
b_{n+1} = b_n + \sqrt[3]{b_n}
b
n
+
1
=
b
n
+
3
b
n
for all positive integers
n
n
n
. Prove that there is a positive integer
n
n
n
for which the inequality
a
n
≤
b
k
<
a
n
+
1
a_n \leq b_k < a_{n+1}
a
n
≤
b
k
<
a
n
+
1
holds for exactly 2021 values of
k
k
k
.
5
2
Hide problems
nice fe from r+ to r+
Find all functions
f
:
R
+
→
R
+
f : \mathbb{R^{+}}\to \mathbb{R^{+}}
f
:
R
+
→
R
+
such that
f
(
x
)
2
=
f
(
x
y
)
+
f
(
x
+
f
(
y
)
)
−
1
f(x)^2=f(xy)+f(x+f(y))-1
f
(
x
)
2
=
f
(
x
y
)
+
f
(
x
+
f
(
y
))
−
1
for all
x
,
y
∈
R
+
x, y\in \mathbb{R^{+}}
x
,
y
∈
R
+
Cubic diophantine equation solving algorithm
Let
a
a
a
be a positive integer. Prove that for any pair
(
x
,
y
)
(x,y)
(
x
,
y
)
of integer solutions of equation
x
(
y
2
−
2
x
2
)
+
x
+
y
+
a
=
0
x(y^2-2x^2)+x+y+a=0
x
(
y
2
−
2
x
2
)
+
x
+
y
+
a
=
0
we have:
∣
x
∣
⩽
a
+
2
a
2
+
2
|x| \leqslant a+\sqrt{2a^2+2}
∣
x
∣
⩽
a
+
2
a
2
+
2
1
1
Hide problems
İnequality
Given
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
such that
a
+
b
+
c
+
1
a
b
c
=
19
2
a+b+c+\frac{1}{abc}=\frac{19}{2}
a
+
b
+
c
+
ab
c
1
=
2
19
What is the greatest value for
a
a
a
?