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Problems
Contests
National and Regional Contests
Hungary Contests
Durer Math Competition
2019 Dürer Math Competition (First Round)
2019 Dürer Math Competition (First Round)
Part of
Durer Math Competition
Subcontests
(5)
P5
2
Hide problems
geometry , collinear points wanted
Let
A
B
C
ABC
A
BC
be a non-right-angled triangle, with
A
C
≠
B
C
AC\ne BC
A
C
=
BC
. Let
F
F
F
be the midpoint of side
B
C
BC
BC
. Let
D
D
D
be a point on line
A
B
AB
A
B
satisfying
C
A
=
C
D
CA=CD
C
A
=
C
D
,and let
E
E
E
be a point on line
B
C
BC
BC
satisfying
E
B
=
E
D
EB = ED
EB
=
E
D
. The line passing through
A
A
A
and parallel to
E
D
ED
E
D
meets line
F
D
FD
F
D
at point
I
I
I
. Line
A
F
AF
A
F
meets line
E
D
ED
E
D
at point
J
J
J
. Prove that points
C
C
C
,
I
I
I
and
J
J
J
are collinear.
orthocenters coincide
Let
A
B
C
ABC
A
BC
and
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
be similar triangles with different orientation such that their orthocenters coincide. Show that lines $AA′, BB′, CC′ are concurrent or parallel.
P4
2
Hide problems
writing numbers on the points of the first quadrant
Albrecht writes numbers on the points of the first quadrant with integer coordinates in the following way: If at least one of the coordinates of a point is 0, he writes 0; in all other cases the number written on point
(
a
,
b
)
(a, b)
(
a
,
b
)
is one greater than the average of the numbers written on points
(
a
+
1
,
b
−
1
)
(a+1 , b-1)
(
a
+
1
,
b
−
1
)
and
(
a
−
1
,
b
+
1
)
(a-1,b+1)
(
a
−
1
,
b
+
1
)
. Which numbers could he write on point
(
121
,
212
)
(121, 212)
(
121
,
212
)
? Note: The elements of the first quadrant are points where both of the coordinates are non- negative.
N-tuples irrationality
An
n
n
n
-tuple
(
x
1
,
x
2
,
…
,
x
n
)
(x_1, x_2,\dots, x_n)
(
x
1
,
x
2
,
…
,
x
n
)
is called unearthly if
q
1
x
1
+
q
2
x
2
+
⋯
+
q
n
x
n
q_1x_1 +q_2x_2 +\dots+q_nx_n
q
1
x
1
+
q
2
x
2
+
⋯
+
q
n
x
n
is irrational for any non-negative rational coefficients
q
1
,
q
2
,
…
,
q
n
q_1, q_2, \dots, q_n
q
1
,
q
2
,
…
,
q
n
where
q
i
q_i
q
i
’s are not all zero. Prove that it is possible to select an unearthly
n
n
n
-tuple from any
2
n
−
1
2n-1
2
n
−
1
distinct irrational numbers.
P3
2
Hide problems
Game on 3X3 table
a) We are playing the following game on this table: In each move we select a row or a column of the table, reduce two neighboring numbers in that row or column by
1
1
1
and increase the third one by
1
1
1
. After some of these moves can we get to a table with all the same entries? b) This time we have the choice to arrange the integers from
1
1
1
to
9
9
9
in the
3
×
3
3 \times3
3
×
3
table. Still using the same moves now our aim is to create a table with all the same entries, maximising the value of the entries. What is the highest possible number we can achieve?
Anne and bob play a game on R^2
Anne has thought of a finite set
A
⊆
R
2
A \subseteq \mathbb{R}^2
A
⊆
R
2
. Bob does not know how many elements
A
A
A
has, but his goal is to completely determine
A
A
A
. To achieve this, Bob can chooseany point
b
∈
R
2
b \in \mathbb{R}^2
b
∈
R
2
and ask Anne how far it is from
A
A
A
. Anne replies with the distance, defined as
m
i
n
{
d
(
a
,
b
)
∣
a
∈
A
}
min \{d(a, b) | a \in A\}
min
{
d
(
a
,
b
)
∣
a
∈
A
}
. (Here
d
(
a
,
b
)
d(a, b)
d
(
a
,
b
)
denotes the distance between points
a
,
b
∈
R
a, b \in \mathbb{R}
a
,
b
∈
R
.) Bob can ask as many questions of this type as he wants, until he can determine A with certainty. a) Can Bob achieve his goal with finitely many questions? b) What if Anne tells Bob in advance that all points of A have both coordinates in the interval
[
0
,
1
]
\ [0, 1]\
[
0
,
1
]
? Note:
R
2
\mathbb{R}^2
R
2
is the set of points in the plane.
P2
2
Hide problems
kayakers with high-fives
a) 11 kayakers row on the Danube from Szentendre to Kopaszi-gát. They do not necessarily start at the same time, but we know that they all take the same route and that each kayaker rows with a constant speed. Whenever a kayaker passes another one, they do a high five. After they all arrive, everybody claims to have done precisely
10
10
10
high fives in total. Show that it is possible for the kayakers to have rowed in such a way that this is true.b) At a different occasion
13
13
13
kayakers rowed in the same manner; now after arrival everybody claims to have done precisely
6
6
6
high fives. Prove that at least one kayaker has miscounted.
sets of prime numbers
For a positive integer
n
n
n
let
P
(
n
)
P(n)
P
(
n
)
denote the set of primes
p
p
p
for which there exist positive integers
a
,
b
a, b
a
,
b
such that
n
=
a
p
+
b
p
n=a^p+b^p
n
=
a
p
+
b
p
. Is it true that for any finite set
H
H
H
consisting of primes, there is an n such that
P
(
n
)
=
H
P(n) = H
P
(
n
)
=
H
?
P1
1
Hide problems
Non-integers a,b,c,d
Non-negative integers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
satisfy the equation
a
+
b
+
c
+
d
=
100
a + b + c + d = 100
a
+
b
+
c
+
d
=
100
and there exists a non-negative integer n such that
a
+
n
=
b
−
n
=
c
×
n
=
d
n
a+ n =b- n= c \times n = \frac{d}{n}
a
+
n
=
b
−
n
=
c
×
n
=
n
d
Find all 5-tuples
(
a
,
b
,
c
,
d
,
n
)
(a, b, c, d, n)
(
a
,
b
,
c
,
d
,
n
)
satisfying all the conditions above.