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Contests
National and Regional Contests
Hungary Contests
Durer Math Competition
2021 Durer Math Competition (First Round)
2021 Durer Math Competition (First Round)
Part of
Durer Math Competition
Subcontests
(5)
5
2
Hide problems
21 bandits each bandit has 240 bullets, duels with all of his enemies
21
21
21
bandits live in the city of Warmridge, each of them having some enemies among the others. Initially each bandit has
240
240
240
bullets, and duels with all of his enemies. Every bandit distributes his bullets evenly between his enemies, this means that he takes the same number of bullets to each of his duels, and uses each of his bullets in only one duel. In case the number of his bullets is not divisible by the number of his enemies, he takes as many bullets to each duel as possible, but takes the same number of bullets to every duel, so it is possible that in the end the bandit will have some remaining bullets. Shooting is banned in the city, therefore a duel consists only of comparing the number of bullets in the guns of the opponents, and the winner is whoever has more bullets. After the duel the sheriff takes the bullets of the winner and as an act of protest the loser shoots all of his bullets into the air. What is the largest possible number of bullets the sheriff can have after all of the duels have ended?Being someones enemy is mutual. If two opponents have the same number of bullets in their guns during a duel, then the sheriff takes the bullets of the bandit who has the wider hat among them.Example: If a bandit has
13
13
13
enemies then he takes
18
18
18
bullets with himself to each duel, and they will have
6
6
6
leftover bullets after finishing all their duels.
max no of planes determined by these n lines
There are
n
n
n
distinct lines in three-dimensional space such that no two lines are parallel and no three lines meet at one point. What is the maximal possible number of planes determined by these
n
n
n
lines?We say that a plane is determined if it contains at least two of the lines.
4
2
Hide problems
[a, b] + [a, c] + [b, c] = [a, b, c] + (a, b, c)
Determine all triples of positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
that satisfya)
[
a
,
b
]
+
[
a
,
c
]
+
[
b
,
c
]
=
[
a
,
b
,
c
]
[a, b] + [a, c] + [b, c] = [a, b, c]
[
a
,
b
]
+
[
a
,
c
]
+
[
b
,
c
]
=
[
a
,
b
,
c
]
.b)
[
a
,
b
]
+
[
a
,
c
]
+
[
b
,
c
]
=
[
a
,
b
,
c
]
+
(
a
,
b
,
c
)
[a, b] + [a, c] + [b, c] = [a, b, c] + (a, b, c)
[
a
,
b
]
+
[
a
,
c
]
+
[
b
,
c
]
=
[
a
,
b
,
c
]
+
(
a
,
b
,
c
)
.Remark: Here
[
x
,
y
[x, y
[
x
,
y
] denotes the least common multiple of positive integers
x
x
x
and
y
y
y
, and
(
x
,
y
)
(x, y)
(
x
,
y
)
denotes their greatest common divisor.
p(x^2) + q(x^2) = p(x)q(x) for integer polynomials
Find all pairs of polynomials
(
p
,
q
)
(p, q)
(
p
,
q
)
with integer coefficients that satisfy the equation
p
(
x
2
)
+
q
(
x
2
)
=
p
(
x
)
q
(
x
)
p(x^2) + q(x^2) = p(x)q(x)
p
(
x
2
)
+
q
(
x
2
)
=
p
(
x
)
q
(
x
)
such that
p
p
p
is of degree
n
n
n
and has
n
n
n
nonnegative real roots (with multiplicity).
3
2
Hide problems
can from any point of the AB section one see the whole museum?
The floor plan of a contemporary art museum is a (not necessarily convex) polygon and its walls are solid. The security guard guarding the museum has two favourite spots (points
A
A
A
and
B
B
B
) because one can see the whole area of the museum standing at either point. Is it true that from any point of the
A
B
AB
A
B
section one can see the whole museum?
concyclic circumcenters wanted, starting with 2 ext. tangent circles
Let
k
1
k_1
k
1
and
k
2
k_2
k
2
be two circles that are externally tangent at point
C
C
C
. We have a point
A
A
A
on
k
1
k_1
k
1
and a point
B
B
B
on
k
2
k_2
k
2
such that
C
C
C
is an interior point of segment
A
B
AB
A
B
. Let
k
3
k_3
k
3
be a circle that passes through points
A
A
A
and
B
B
B
and intersects circles
k
1
k_1
k
1
and
k
2
k_2
k
2
another time at points
M
M
M
and
N
N
N
respectively. Let
k
4
k_4
k
4
be the circumscribed circle of triangle
C
M
N
CMN
CMN
. Prove that the centres of circles
k
1
,
k
2
,
k
3
k_1, k_2, k_3
k
1
,
k
2
,
k
3
and
k
4
k_4
k
4
all lie on the same circle.
2
1
Hide problems
18x 36 rectangle-shaped cake cut into convex pieces
The best part of grandma’s
18
18
18
cm
×
36
\times 36
×
36
cm rectangle-shaped cake is the chocolate covering on the edges. Her three grandchildren would like to split the cake between each other so that everyone gets the same amount (of the area) of the cake, and they all get the same amount of the delicious perimeter too. a) Can they cut the cake into three convex pieces like that? b) The next time grandma baked this cake, the whole family wanted to try it so they had to cut the cake into six convex pieces this way. Is this possible? c) Soon the entire neighbourhood has heard of the delicious cake. Can the cake be cut into
12
12
12
convex pieces with the same conditions?
1
1
Hide problems
Albrecht wants calculate the speed of the car
Albrecht is travelling in his car on the motorway at a constant speed. The journey is very long so Marvin who is sitting next to Albrecht gets bored and decides to calculate the speed of the car. He was a bit careless but he noted that at noon they passed milestone
X
Y
XY
X
Y
(where
X
X
X
and
Y
Y
Y
are digits), at
12
:
42
12:42
12
:
42
milestone
Y
X
YX
Y
X
and at
1
1
1
pm they arrived at milestone
X
0
Y
X0Y
X
0
Y
. What did Marvin deduce, what is the speed of the car?