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Contests
National and Regional Contests
Bulgaria Contests
Bulgaria JBMO Team Selection Test
2017 Bulgaria JBMO TST
2017 Bulgaria JBMO TST
Part of
Bulgaria JBMO Team Selection Test
Subcontests
(4)
4
2
Hide problems
Checkers on a board
Given is a board
n
×
n
n \times n
n
×
n
and in every square there is a checker. In one move, every checker simultaneously goes to an adjacent square (two squares are adjacent if they share a common side). In one square there can be multiple checkers. Find the minimum and the maximum number of covered cells for
n
=
5
,
6
,
7
n=5, 6, 7
n
=
5
,
6
,
7
.
Numbers with fixed number of divisors and sum
Find all positive integers such that they have
6
6
6
divisors (without
1
1
1
and the number itself) and the sum of the divisors is
14133
14133
14133
.
3
2
Hide problems
Sheets and boxes
Given are sheets and the numbers
00
,
01
,
…
,
99
00, 01, \ldots, 99
00
,
01
,
…
,
99
are written on them. We must put them in boxes
000
,
001
,
…
,
999
000, 001, \ldots, 999
000
,
001
,
…
,
999
so that the number on the sheet is the number on the box with one digit erased. What is the minimum number of boxes we need in order to put all the sheets?
4-variable inequality
Prove that for all positive real
m
,
n
,
p
,
q
m, n, p, q
m
,
n
,
p
,
q
and
t
=
m
+
n
+
p
+
q
2
t=\frac{m+n+p+q}{2}
t
=
2
m
+
n
+
p
+
q
,
m
t
+
n
+
p
+
q
+
n
t
+
m
+
p
+
q
+
p
t
+
m
+
n
+
q
+
q
t
+
m
+
n
+
p
≥
4
5
.
\frac{m}{t+n+p+q} +\frac{n}{t+m+p+q} +\frac{p} {t+m+n+q}+\frac{q}{t+m+n+p} \geq \frac{4}{5}.
t
+
n
+
p
+
q
m
+
t
+
m
+
p
+
q
n
+
t
+
m
+
n
+
q
p
+
t
+
m
+
n
+
p
q
≥
5
4
.
2
2
Hide problems
NT equation with two variables
Solve the following equation over the integers
25
x
2
y
2
+
10
x
2
y
+
25
x
y
2
+
x
2
+
30
x
y
+
2
y
2
+
5
x
+
7
y
+
6
=
0.
25x^2y^2+10x^2y+25xy^2+x^2+30xy+2y^2+5x+7y+6= 0.
25
x
2
y
2
+
10
x
2
y
+
25
x
y
2
+
x
2
+
30
x
y
+
2
y
2
+
5
x
+
7
y
+
6
=
0.
Incircle geo
Let
k
k
k
be the incircle of triangle
A
B
C
ABC
A
BC
. It touches
A
B
=
c
,
B
C
=
a
,
A
C
=
b
AB=c, BC=a, AC=b
A
B
=
c
,
BC
=
a
,
A
C
=
b
at
C
1
,
A
1
,
B
1
C_1, A_1, B_1
C
1
,
A
1
,
B
1
, respectively. Suppose that
K
C
1
KC_1
K
C
1
is a diameter of the incircle. Let
C
1
A
1
C_1A_1
C
1
A
1
intersect
K
B
1
KB_1
K
B
1
at
N
N
N
and
C
1
B
1
C_1B_1
C
1
B
1
intersect
K
A
1
KA_1
K
A
1
at
M
M
M
. Find the length of
M
N
MN
MN
.
1
2
Hide problems
Geometry with fixed angles
Given is a triangle
A
B
C
ABC
A
BC
and let
A
A
1
AA_1
A
A
1
,
B
B
1
BB_1
B
B
1
be angle bisectors. It turned out that
∠
A
A
1
B
=
2
4
∘
\angle AA_1B=24^{\circ}
∠
A
A
1
B
=
2
4
∘
and
∠
B
B
1
A
=
1
8
∘
\angle BB_1A=18^{\circ}
∠
B
B
1
A
=
1
8
∘
. Find the ratio
∠
B
A
C
:
∠
A
C
B
:
∠
A
B
C
\angle BAC:\angle ACB:\angle ABC
∠
B
A
C
:
∠
A
CB
:
∠
A
BC
.
Classical NT equation
Find all positive integers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
so that
a
2
+
b
2
+
c
2
+
d
2
=
13
⋅
4
n
a^2+b^2+c^2+d^2=13 \cdot 4^n
a
2
+
b
2
+
c
2
+
d
2
=
13
⋅
4
n