MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgarian Spring Mathematical Competition
2022 Bulgarian Spring Math Competition
2022 Bulgarian Spring Math Competition
Part of
Bulgarian Spring Mathematical Competition
Subcontests
(20)
Problem 12.4
1
Hide problems
Sum of elements of sets give equal residue modulo a prime
Let
m
m
m
and
n
n
n
be positive integers and
p
p
p
be a prime number. Find the greatest positive integer
s
s
s
(as a function of
m
,
n
m,n
m
,
n
and
p
p
p
) such that from a random set of
m
n
p
mnp
mn
p
positive integers we can choose
s
n
p
snp
s
n
p
numbers, such that they can be partitioned into
s
s
s
sets of
n
p
np
n
p
numbers, such that the sum of the numbers in every group gives the same remainder when divided by
p
p
p
.
Problem 12.3
1
Hide problems
Polynomial equation implies real root
Let
P
,
Q
∈
R
[
x
]
P,Q\in\mathbb{R}[x]
P
,
Q
∈
R
[
x
]
, such that
Q
Q
Q
is a
2021
2021
2021
-degree polynomial and let
a
1
,
a
2
,
…
,
a
2022
,
b
1
,
b
2
,
…
,
b
2022
a_{1}, a_{2}, \ldots , a_{2022}, b_{1}, b_{2}, \ldots , b_{2022}
a
1
,
a
2
,
…
,
a
2022
,
b
1
,
b
2
,
…
,
b
2022
be real numbers such that
a
1
a
2
…
a
2022
≠
0
a_{1}a_{2}\ldots a_{2022}\neq 0
a
1
a
2
…
a
2022
=
0
. If for all real
x
x
x
P
(
a
1
Q
(
x
)
+
b
1
)
+
…
+
P
(
a
2021
Q
(
x
)
+
b
2021
)
=
P
(
a
2022
Q
(
x
)
+
b
2022
)
P(a_{1}Q(x) + b_{1}) + \ldots + P(a_{2021}Q(x) + b_{2021}) = P(a_{2022}Q(x) + b_{2022})
P
(
a
1
Q
(
x
)
+
b
1
)
+
…
+
P
(
a
2021
Q
(
x
)
+
b
2021
)
=
P
(
a
2022
Q
(
x
)
+
b
2022
)
prove that
P
(
x
)
P(x)
P
(
x
)
has a real root.
Problem 12.2
1
Hide problems
Ratios in a regular quadrangular pyramid
Let
A
B
C
D
V
ABCDV
A
BC
D
V
be a regular quadrangular pyramid with
V
V
V
as the apex. The plane
λ
\lambda
λ
intersects the
V
A
VA
V
A
,
V
B
VB
V
B
,
V
C
VC
V
C
and
V
D
VD
V
D
at
M
M
M
,
N
N
N
,
P
P
P
,
Q
Q
Q
respectively. Find
V
Q
:
Q
D
VQ : QD
V
Q
:
Q
D
, if
V
M
:
M
A
=
2
:
1
VM : MA = 2 : 1
V
M
:
M
A
=
2
:
1
,
V
N
:
N
B
=
1
:
1
VN : NB = 1 : 1
V
N
:
NB
=
1
:
1
and
V
P
:
P
C
=
1
:
2
VP : PC = 1 : 2
V
P
:
PC
=
1
:
2
.
Problem 12.1
1
Hide problems
Areas in circumscribed quadrilateral
A
B
C
D
ABCD
A
BC
D
is circumscribed in a circle
k
k
k
, such that
[
A
C
B
]
=
s
[ACB]=s
[
A
CB
]
=
s
,
[
A
C
D
]
=
t
[ACD]=t
[
A
C
D
]
=
t
,
s
<
t
s<t
s
<
t
. Determine the smallest value of
4
s
2
+
t
2
5
s
t
\frac{4s^2+t^2}{5st}
5
s
t
4
s
2
+
t
2
and when this minimum is achieved.
Problem 11.4
1
Hide problems
Find a big sum-free subset
Let
n
≥
2
n \geq 2
n
≥
2
be a positive integer. The set
M
M
M
consists of
2
n
2
−
3
n
+
2
2n^2-3n+2
2
n
2
−
3
n
+
2
positive rational numbers. Prove that there exists a subset
A
A
A
of
M
M
M
with
n
n
n
elements with the following property:
∀
\forall
∀
2
≤
k
≤
n
2 \leq k \leq n
2
≤
k
≤
n
the sum of any
k
k
k
(not necessarily distinct) numbers from
A
A
A
is not in
A
A
A
.
Problem 11.3
1
Hide problems
Letters in rows and columns overload inequality
In every cell of a table with
n
n
n
rows and
m
m
m
columns is written one of the letters
a
a
a
,
b
b
b
,
c
c
c
. Every two rows of the table have the same letter in at most
k
≥
0
k\geq 0
k
≥
0
positions and every two columns coincide at most
k
k
k
positions. Find
m
m
m
,
n
n
n
,
k
k
k
if
2
m
n
+
6
k
3
(
m
+
n
)
≥
k
+
1
\frac{2mn+6k}{3(m+n)}\geq k+1
3
(
m
+
n
)
2
mn
+
6
k
≥
k
+
1
Problem 11.2
1
Hide problems
Find the angles of triangle ABC
A circle through the vertices
A
A
A
and
B
B
B
of
△
A
B
C
\triangle ABC
△
A
BC
intersects segments
A
C
AC
A
C
and
B
C
BC
BC
at points
P
P
P
and
Q
Q
Q
respectively. If
A
Q
=
A
C
AQ=AC
A
Q
=
A
C
,
∠
B
A
Q
=
∠
C
B
P
\angle BAQ=\angle CBP
∠
B
A
Q
=
∠
CBP
and
A
B
=
(
3
+
1
)
P
Q
AB=(\sqrt{3}+1)PQ
A
B
=
(
3
+
1
)
PQ
, find the measures of the angles of
△
A
B
C
\triangle ABC
△
A
BC
.
Problem 11.1
1
Hide problems
Equation with logs
Solve the equation
(
x
+
1
)
log
3
2
x
+
4
x
log
3
x
−
16
=
0
(x+1)\log^2_{3}x+4x\log_{3}x-16=0
(
x
+
1
)
lo
g
3
2
x
+
4
x
lo
g
3
x
−
16
=
0
Problem 10.4
1
Hide problems
System of equations involving a prime
Find the smallest odd prime
p
p
p
, such that there exist coprime positive integers
k
k
k
and
ℓ
\ell
ℓ
which satisfy 4k-3\ell=12 \text{ and } \ell^2+\ell k +k^2\equiv 3\text{ }(\text{mod }p)
Problem 10.3
1
Hide problems
Number of good permutations
A permutation
σ
\sigma
σ
of the numbers
1
,
2
,
…
,
10
1,2,\ldots , 10
1
,
2
,
…
,
10
is called
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
b
a
d
<
/
s
p
a
n
>
<span class='latex-italic'>bad</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
ba
d
<
/
s
p
an
>
if there exist integers
i
,
j
,
k
i, j, k
i
,
j
,
k
which satisfy 1 \leq i < j < k \leq 10 \text{ and } \sigma(j) < \sigma(k) < \sigma(i) and
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
otherwise. Find the number of
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
permutations.
Problem 10.2
1
Hide problems
Incenter related geometry with specific ratio
Let
△
A
B
C
\triangle ABC
△
A
BC
have incenter
I
I
I
. The line
C
I
CI
C
I
intersects the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
for the second time at
L
L
L
, and
C
I
=
2
I
L
CI=2IL
C
I
=
2
I
L
. Points
M
M
M
and
N
N
N
lie on the segment
A
B
AB
A
B
, such that
∠
A
I
M
=
∠
B
I
N
=
9
0
∘
\angle AIM =\angle BIN = 90^{\circ}
∠
A
I
M
=
∠
B
I
N
=
9
0
∘
. Prove that
A
B
=
2
M
N
AB=2MN
A
B
=
2
MN
.
Problem 10.1
1
Hide problems
System of equations implies greatest value of an expression
If
x
,
y
,
z
∈
R
x, y, z \in \mathbb{R}
x
,
y
,
z
∈
R
are solutions to the system of equations
{
x
−
y
+
z
−
1
=
0
x
y
+
2
z
2
−
6
z
+
1
=
0
\begin{cases} x - y + z - 1 = 0\\ xy + 2z^2 - 6z + 1 = 0\\ \end{cases}
{
x
−
y
+
z
−
1
=
0
x
y
+
2
z
2
−
6
z
+
1
=
0
what is the greatest value of
(
x
−
1
)
2
+
(
y
+
1
)
2
(x - 1)^2 + (y + 1)^2
(
x
−
1
)
2
+
(
y
+
1
)
2
?
Problem 9.4
1
Hide problems
Favourite numbers at IMO training camp
14 students attend the IMO training camp. Every student has at least
k
k
k
favourite numbers. The organisers want to give each student a shirt with one of the student's favourite numbers on the back. Determine the least
k
k
k
, such that this is always possible if:
a
)
a)
a
)
The students can be arranged in a circle such that every two students sitting next to one another have different numbers.
b
)
b)
b
)
7
7
7
of the students are boys, the rest are girls, and there isn't a boy and a girl with the same number.
Problem 9.3
1
Hide problems
System of equations involving squares and primes
Find all primes
p
p
p
, such that there exist positive integers
x
x
x
,
y
y
y
which satisfy
{
p
+
49
=
2
x
2
p
2
+
49
=
2
y
2
\begin{cases} p + 49 = 2x^2\\ p^2 + 49 = 2y^2\\ \end{cases}
{
p
+
49
=
2
x
2
p
2
+
49
=
2
y
2
Problem 9.2
1
Hide problems
Asymmetric geometry involving... NT?
Let
△
A
B
C
\triangle ABC
△
A
BC
have median
C
M
CM
CM
(
M
∈
A
B
M\in AB
M
∈
A
B
) and circumcenter
O
O
O
. The circumcircle of
△
A
M
O
\triangle AMO
△
A
MO
bisects
C
M
CM
CM
. Determine the least possible perimeter of
△
A
B
C
\triangle ABC
△
A
BC
if it has integer side lengths.
Problem 9.1
1
Hide problems
Quadratic function is pinned down
Let
f
(
x
)
f(x)
f
(
x
)
be a quadratic function with integer coefficients. If we know that
f
(
0
)
f(0)
f
(
0
)
,
f
(
3
)
f(3)
f
(
3
)
and
f
(
4
)
f(4)
f
(
4
)
are all different and elements of the set
{
2
,
20
,
202
,
2022
}
\{2, 20, 202, 2022\}
{
2
,
20
,
202
,
2022
}
, determine all possible values of
f
(
1
)
f(1)
f
(
1
)
.
Problem 8.4
1
Hide problems
Optimistic permutations of a set
Let
p
=
(
a
1
,
a
2
,
…
,
a
12
)
p = (a_{1}, a_{2}, \ldots , a_{12})
p
=
(
a
1
,
a
2
,
…
,
a
12
)
be a permutation of
1
,
2
,
…
,
12
1, 2, \ldots, 12
1
,
2
,
…
,
12
. We will denote
S
p
=
∣
a
1
−
a
2
∣
+
∣
a
2
−
a
3
∣
+
…
+
∣
a
11
−
a
12
∣
S_{p} = |a_{1}-a_{2}|+|a_{2}-a_{3}|+\ldots+|a_{11}-a_{12}|
S
p
=
∣
a
1
−
a
2
∣
+
∣
a
2
−
a
3
∣
+
…
+
∣
a
11
−
a
12
∣
We'll call
p
p
p
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
o
p
t
i
m
i
s
t
i
c
<
/
s
p
a
n
>
<span class='latex-italic'>optimistic</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
o
pt
imi
s
t
i
c
<
/
s
p
an
>
if
a
i
>
min
(
a
i
−
1
,
a
i
+
1
)
a_{i} > \min(a_{i-1}, a_{i+1})
a
i
>
min
(
a
i
−
1
,
a
i
+
1
)
∀
i
=
2
,
…
,
11
\forall i = 2, \ldots, 11
∀
i
=
2
,
…
,
11
.
a
)
a)
a
)
What is the maximum possible value of
S
p
S_{p}
S
p
. How many permutations
p
p
p
achieve this maximum?
\newline
b
)
b)
b
)
What is the number of
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
o
p
t
i
m
i
s
t
i
c
<
/
s
p
a
n
>
<span class='latex-italic'>optimistic</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
o
pt
imi
s
t
i
c
<
/
s
p
an
>
permtations
p
p
p
?
c
)
c)
c
)
What is the maximum possible value of
S
p
S_{p}
S
p
for an
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
o
p
t
i
m
i
s
t
i
c
<
/
s
p
a
n
>
<span class='latex-italic'>optimistic</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
o
pt
imi
s
t
i
c
<
/
s
p
an
>
p
p
p
? How many
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
o
p
t
i
m
i
s
t
i
c
<
/
s
p
a
n
>
<span class='latex-italic'>optimistic</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
o
pt
imi
s
t
i
c
<
/
s
p
an
>
permutations
p
p
p
achieve this maximum?
Problem 8.3
1
Hide problems
Prove or disprove inequalities
Given the inequalities:
a
)
a)
a
)
(
2
a
b
+
c
)
2
+
(
2
b
a
+
c
)
2
+
(
2
c
a
+
b
)
2
≥
a
c
+
b
a
+
c
b
\left(\frac{2a}{b+c}\right)^2+\left(\frac{2b}{a+c}\right)^2+\left(\frac{2c}{a+b}\right)^2\geq \frac{a}{c}+\frac{b}{a}+\frac{c}{b}
(
b
+
c
2
a
)
2
+
(
a
+
c
2
b
)
2
+
(
a
+
b
2
c
)
2
≥
c
a
+
a
b
+
b
c
b
)
b)
b
)
(
a
+
b
c
)
2
+
(
b
+
c
a
)
2
+
(
c
+
a
b
)
2
≥
a
b
+
b
c
+
c
a
+
9
\left(\frac{a+b}{c}\right)^2+\left(\frac{b+c}{a}\right)^2+\left(\frac{c+a}{b}\right)^2\geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+9
(
c
a
+
b
)
2
+
(
a
b
+
c
)
2
+
(
b
c
+
a
)
2
≥
b
a
+
c
b
+
a
c
+
9
For each of them either prove that it holds for all positive real numbers
a
a
a
,
b
b
b
,
c
c
c
or present a counterexample
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
which doesn't satisfy the inequality.
Problem 8.2
1
Hide problems
Fixed construction geometry problem
Let
△
A
B
C
\triangle ABC
△
A
BC
have
A
B
=
1
AB = 1
A
B
=
1
cm,
B
C
=
2
BC = 2
BC
=
2
cm and
A
C
=
3
AC = \sqrt{3}
A
C
=
3
cm. Points
D
D
D
,
E
E
E
and
F
F
F
lie on segments
A
B
AB
A
B
,
A
C
AC
A
C
and
B
C
BC
BC
respectively are such that
A
E
=
B
D
AE = BD
A
E
=
B
D
and
B
F
=
A
D
BF = AD
BF
=
A
D
. The angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
intersects the circumcircle of
△
A
D
E
\triangle ADE
△
A
D
E
for the second time at
M
M
M
and the angle bisector of
∠
A
B
C
\angle ABC
∠
A
BC
intersects the circumcircle of
△
B
D
F
\triangle BDF
△
B
D
F
at
N
N
N
. Determine the length of
M
N
MN
MN
.
Problem 8.1
1
Hide problems
Cute and easy polynomial problem
Let
P
=
(
x
4
−
40
x
2
+
144
)
(
x
3
−
16
x
)
P=(x^4-40x^2+144)(x^3-16x)
P
=
(
x
4
−
40
x
2
+
144
)
(
x
3
−
16
x
)
.
a
)
a)
a
)
Factor
P
P
P
as a product of irreducible polynomials.
b
)
b)
b
)
We write down the values of
P
(
10
)
P(10)
P
(
10
)
and
P
(
91
)
P(91)
P
(
91
)
. What is the greatest common divisor of the two numbers?