MathDB
Problems
Contests
National and Regional Contests
Kazakhstan Contests
Kazakhstan National Olympiad
2023 Kazakhstan National Olympiad
2023 Kazakhstan National Olympiad
Part of
Kazakhstan National Olympiad
Subcontests
(6)
6
2
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Beautiful geometry problem, HM line
The altitudes of an acute triangle
A
B
C
ABC
A
BC
intersect at
H
H
H
. The tangent line at
H
H
H
to the circumcircle of triangle
B
H
C
BHC
B
H
C
intersects the lines
A
B
AB
A
B
and
A
C
AC
A
C
at points
Q
Q
Q
and
P
P
P
respectively. The circumcircles of triangles
A
B
C
ABC
A
BC
and
A
P
Q
APQ
A
PQ
intersect at point
K
K
K
(
K
≠
A
K\neq A
K
=
A
). The tangent lines at the points
A
A
A
and
K
K
K
to the circumcircle of triangle
A
P
Q
APQ
A
PQ
intersect at
T
T
T
. Prove that
T
H
TH
T
H
passes through the midpoint of segment
B
C
BC
BC
.
1.061, intersection of two rhombuses
Inside an equilateral triangle with side
3
3
3
there are two rhombuses with sides
1
,
061
1,061
1
,
061
and acute angles
6
0
∘
60^\circ
6
0
∘
. Prove that these two rhombuses intersect each other. (The vertices of the rhombus are strictly inside the triangle.)
4
1
Hide problems
Find minimum value
Given
x
,
y
>
0
x,y>0
x
,
y
>
0
such that
x
2
y
2
+
2
x
3
y
=
1
x^2y^2+2x^3y=1
x
2
y
2
+
2
x
3
y
=
1
. Find the minimum value of sum
x
+
y
x+y
x
+
y
5
2
Hide problems
Equation in primes
Solve the given equation in prime numbers
p
3
+
q
3
+
r
3
=
p
2
q
r
p^3+q^3+r^3=p^2qr
p
3
+
q
3
+
r
3
=
p
2
q
r
Two weird numbers are coprime
Given are positive integers
a
,
b
,
m
,
k
a, b, m, k
a
,
b
,
m
,
k
with
k
≥
2
k \geq 2
k
≥
2
. Prove that there exist infinitely many
n
n
n
, such that
gcd
(
φ
m
(
n
)
,
⌊
a
n
+
b
k
⌋
)
=
1
\gcd (\varphi_m(n), \lfloor \sqrt[k] {an+b} \rfloor)=1
g
cd
(
φ
m
(
n
)
,
⌊
k
an
+
b
⌋)
=
1
, where
φ
m
(
n
)
\varphi_m(n)
φ
m
(
n
)
is the
m
m
m
-th iteration of
φ
(
n
)
\varphi(n)
φ
(
n
)
.
1
2
Hide problems
Excircle geo with equal segments
The
C
C
C
-excircle of a triangle
A
B
C
ABC
A
BC
touches
A
B
,
A
C
,
B
C
AB, AC, BC
A
B
,
A
C
,
BC
at
M
,
N
,
K
M, N, K
M
,
N
,
K
. The points
P
,
Q
P, Q
P
,
Q
lie on
N
K
NK
N
K
so that
A
N
=
A
P
,
B
K
=
B
Q
AN=AP, BK=BQ
A
N
=
A
P
,
B
K
=
BQ
. Prove that the circumradius of
△
M
P
Q
\triangle MPQ
△
MPQ
is equal to the inradius of
△
A
B
C
\triangle ABC
△
A
BC
.
Geo with obtuse triangle
A triangle
A
B
C
ABC
A
BC
with obtuse angle
C
C
C
and
A
C
>
B
C
AC>BC
A
C
>
BC
has center
O
O
O
of its circumcircle
ω
\omega
ω
. The tangent at
C
C
C
to
ω
\omega
ω
meets
A
B
AB
A
B
at
D
D
D
. Let
Ω
\Omega
Ω
be the circumcircle of
A
O
B
AOB
A
OB
. Let
O
D
,
A
C
OD, AC
O
D
,
A
C
meet
Ω
\Omega
Ω
at
E
,
F
E, F
E
,
F
and let
O
F
∩
C
E
=
T
OF \cap CE=T
OF
∩
CE
=
T
,
O
D
∩
B
C
=
K
OD \cap BC=K
O
D
∩
BC
=
K
. Prove that
O
T
B
K
OTBK
OTB
K
is cyclic.
3
1
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Inequality with some condition
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive real numbers such that
max
{
a
(
b
+
c
)
a
2
+
b
c
,
b
(
c
+
a
)
b
2
+
c
a
,
c
(
a
+
b
)
c
2
+
a
b
}
≤
5
2
\max\{\frac{a(b+c)}{a^2+bc},\frac{b(c+a)}{b^2+ca},\frac{c(a+b)}{c^2+ab}\}\le \frac{5}{2}
max
{
a
2
+
b
c
a
(
b
+
c
)
,
b
2
+
c
a
b
(
c
+
a
)
,
c
2
+
ab
c
(
a
+
b
)
}
≤
2
5
. Prove inequality
a
(
b
+
c
)
a
2
+
b
c
+
b
(
c
+
a
)
b
2
+
c
a
+
c
(
a
+
b
)
c
2
+
a
b
≤
3
\frac{a(b+c)}{a^2+bc}+\frac{b(c+a)}{b^2+ca}+\frac{c(a+b)}{c^2+ab}\le 3
a
2
+
b
c
a
(
b
+
c
)
+
b
2
+
c
a
b
(
c
+
a
)
+
c
2
+
ab
c
(
a
+
b
)
≤
3
2
2
Hide problems
Inequality x^2+y^2+z^2=2xyz+1
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive real numbers such that
a
+
b
+
c
≥
3
a+b+c\ge 3
a
+
b
+
c
≥
3
and
a
2
+
b
2
+
c
2
=
2
a
b
c
+
1
a^2+b^2+c^2=2abc+1
a
2
+
b
2
+
c
2
=
2
ab
c
+
1
. Prove that
a
+
b
+
c
≤
2
a
b
c
+
1
a+b+c\le 2\sqrt{abc}+1
a
+
b
+
c
≤
2
ab
c
+
1
Weird algebra with combinatorial flavour
Let
n
>
100
n>100
n
>
100
be an integer. The numbers
1
,
2
…
,
4
n
1,2 \ldots, 4n
1
,
2
…
,
4
n
are split into
n
n
n
groups of
4
4
4
. Prove that there are at least
(
n
−
6
)
2
2
\frac{(n-6)^2}{2}
2
(
n
−
6
)
2
quadruples
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
such that they are all in different groups,
a
<
b
<
c
<
d
a<b<c<d
a
<
b
<
c
<
d
and
c
−
b
≤
∣
a
d
−
b
c
∣
≤
d
−
a
c-b \leq |ad-bc|\leq d-a
c
−
b
≤
∣
a
d
−
b
c
∣
≤
d
−
a
.