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Contests
National and Regional Contests
Hungary Contests
Durer Math Competition
2023 Durer Math Competition Finals
2023 Durer Math Competition Finals
Part of
Durer Math Competition
Subcontests
(16)
14
1
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Zeus lightnings
Zeus’s lightning is made of a copper rod of length
60
60
60
by bending it
4
4
4
times in alternating directions so that the angle between two adjacent parts is always
6
0
o
60^o
6
0
o
. What is the minimum value of the square of the distance between the two endpoints of the lightning? All five segments of the lightning lie in the same plane. https://cdn.artofproblemsolving.com/attachments/5/1/a18206df4fde561421022c0f2b4332f5ac44a2.png
16
2
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8 plugs into 8 holes
For the Dürer final results announcement, four loudspeakers are used to provide sound in the hall. However, there are only two sockets in the wall from which the power comes. To solve the problem, Ádám got two extension cords and two power strips. One plug can be plugged into an extension cord, and two plugs can be plugged into a power strip. Gábor, in his haste before the announcement of the results, quickly plugs the
8
8
8
plugs into the
8
8
8
holes. Every possible way of plugging has the same probability, and it is also possible for Gábor to plug something into itself. What is the probability that all
4
4
4
speakers will have sound at the results announcement? For the solution, give the sum of the numerator and the denominator in the simplified form of the probability. A speaker sounds when it is plugged directly or indirectly into the wall.
2025^.... mod 2023
What is the remainder of
2025
∧
(
2024
∧
(
2022
∧
(
2021
∧
(
2020
∧
.
.
.
∧
(
2
∧
1
)
.
.
.
)
)
)
)
)
2025\wedge (2024\wedge (2022\wedge (2021\wedge (2020\wedge ...\wedge (2\wedge 1) . . .)))))
2025
∧
(
2024
∧
(
2022
∧
(
2021
∧
(
2020
∧
...
∧
(
2
∧
1
)
...
)))))
when it is divided by
2023
2023
2023
?Here
∧
\wedge
∧
is the exponential symbol, for example
2
∧
(
3
∧
2
)
=
2
∧
9
=
512
2\wedge (3\wedge 2) = 2\wedge 9 = 512
2
∧
(
3
∧
2
)
=
2
∧
9
=
512
. The power tower contains the integers from
2025
2025
2025
to
1
1
1
exactly once, except that the number
2023
2023
2023
is missing.
15
2
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12-sided convex polygon-shaped pizza.
Csongi bought a
12
12
12
-sided convex polygon-shaped pizza. The pizza has no interior point with three or more distinct diagonals passing through it. Áron wants to cut the pizza along
3
3
3
diagonals so that exactly
6
6
6
pieces of pizza are created. In how many ways can he do this? Two ways of slicing are different if one of them has a cut line that the other does not have.
biggest positive integer which divides p^4 - q^4
What is the biggest positive integer which divides
p
4
−
q
4
p^4 - q^4
p
4
−
q
4
for all primes
p
p
p
and
q
q
q
greater than
10
10
10
?
13
1
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fligths among 2023 cities
A country has
2023
2023
2023
cities and there are flights between these cities. Each flight connects two cities in both directions. We know that you can get from any city to any other using these flights, and from each city there are flights to at most
4
4
4
other cities. What is the maximum possible number of cities in the country from which there is a flight to only one city?
12
1
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pancakes
Marvin really likes pancakes, so he asked his grandma to make pancakes for him. Every time Grandma sends pancakes, she sends a package of
32
32
32
. When Marvin is in the mood for pancakes, he eats half of the pancakes he has. Marvin ate
157
157
157
pancakes for lunch today. At least how many times has Grandma sent pancakes to Marvin so far? Marvin does not necessarily eat an integer number of pancakes at once, and he is in the mood for pancakes at most once a day.
11
1
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binary sudoku
The binary sudoku is a puzzle in which a table should be filled with digits
0
0
0
and
1
1
1
such that in each row and column, the number of 0s is equal to the number of
1
1
1
s. Furthermore, there cannot exist three adjacent cells in a row or in a column such that they have the same digit written in them. Solving the given binary sudoku, what is the sum of the numbers in the two diagonals? https://cdn.artofproblemsolving.com/attachments/a/8/be7de94ce02a90b3cabf1b9795b94ec7ec677f.png
10
1
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four-digit numbers coloured gold
One day Mnemosyne decided to colour all natural numbers in increasing order. She coloured
0
0
0
,
1
1
1
and
2
2
2
in brown, and her favourite number,
3
3
3
, in gold. From then on, for any number whose sum of digits (in the decimal system) was a golden number less than the number itself, she coloured it gold, but coloured the rest of the numbers brown. How many four-digit numbers were coloured gold by Mnemosyne? The set of natural numbers includes
0
0
0
.
9
1
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4 circles and 1 square
Archimedes drew a square with side length
36
36
36
cm into the sand and he also drew a circle of radius
36
36
36
cm around each vertex of the square. If the total area of the grey parts is
n
⋅
π
n \cdot \pi
n
⋅
π
cm
2
^2
2
, what is the value of
n
n
n
? Do not disturb my circles! https://cdn.artofproblemsolving.com/attachments/e/7/a755007990625c74fc2e59b999f0a3eddb2371.png
8
1
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1-4 in 4x4 gird
Zoli wants to fill the given
4
×
4
4 \times 4
4
×
4
table with the digits
1
1
1
,
2
2
2
,
3
3
3
and
4
4
4
, such that in every row and column, and also in the diagonal going from the top left cell to the bottom right, each digit appears exactly once. What is the highest possible value of the sum of the digits in the six grey cells? https://cdn.artofproblemsolving.com/attachments/7/0/498e652cd7ce556d8a638f3d51b65b13154ee5.png
7
1
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square in a circle
The area of a rectangle is
64
64
64
cm
2
^2
2
, and the radius of its circumscribed circle is
7
7
7
cm. What is the perimeter of the rectangle in centimetres?
3
3
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4-digit number
Which is the largest four-digit number that has all four of its digits among its divisors and its digits are all different?
4 merchants want to travel from Athens to Rome by cart.
a) Four merchants want to travel from Athens to Rome by cart. On the same day, but different times they leave Athens and arrive on another day to Rome, but in reverse order. Every day, when the evening comes, each merchant enters the next inn on the way. When some merchants sleep in the same inn at night, then on the following day at dawn they leave in reverse order of arrival, because they can only park this way on the narrow streets next to the inns. They cannot overtake each other, their order only changes after a night spent together in the same inn. Eventually each merchant arrives in Rome while they sleep with every other merchant in the same inn exactly once. Is it possible, that the number of the inns they sleep in is even every night?b) Is it possible if there are
8
8
8
merchants instead of
4
4
4
and every other condition is the same?
y = 1000/(x^2+100)
Hapi, the god of the annual flooding of the Nile is preparing for this year’s flooding. The shape of the channel of the Nile can be described by the function
y
=
−
1000
x
2
+
100
y = \frac{-1000}{ x^2+100}
y
=
x
2
+
100
−
1000
where the
x
x
x
and
y
y
y
coordinates are in metres. The depth of the river is
5
5
5
metres now. Hapi plans to increase the water level by
3
3
3
metres. How many metres wide will the river be after the flooding? The depth of the river is always measured at its deepest point. https://cdn.artofproblemsolving.com/attachments/8/3/4e1d277e5cacf64bf82c110d521747592b928e.png
6
2
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Nim-like game on four piles
Two players play a game on four piles of pebbles labeled with the numbers
1
,
2
,
3
,
4
1,2,3,4
1
,
2
,
3
,
4
respectively. The players take turns in an alternating fashion. On his or her turn, a player selects integers
m
m
m
and
n
n
n
with
1
≤
m
<
n
≤
4
1\leq m<n\leq 4
1
≤
m
<
n
≤
4
, removes
m
m
m
pebbles from pile
n
n
n
, and places one pebble in each of the piles
n
−
1
,
n
−
2
,
…
,
n
−
m
n-1,n-2,\dots,n-m
n
−
1
,
n
−
2
,
…
,
n
−
m
. A player loses the game if he or she cannot make a legal move. For each starting position, determine the player with a winning strategy.
many days of the year can be Adel&rsquo;s birthday?
In Eldorado a year has
20
20
20
months, and each month has
20
20
20
days. One day Brigi asked Adél who lives in Eldorado what day her birthday is. Adél answered that she is only going to tell her the product of the month and the day in her birthday. (For example, if she was born on the
19
19
19
th day of the
4
4
4
th month, she would say
4
⋅
19
=
76
4 \cdot 19 = 76
4
⋅
19
=
76
.) From this, Brigi was able to tell Adél’s birthday. Based on this information, how many days of the year can be Adél’s birthday?
5
4
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4
3
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1 piece of 1 , 2 pieces of 1 , 3 pieces of 3 , . . . , 50 piecies of 50
Benedek wrote down the following numbers:
1
1
1
piece of one,
2
2
2
pieces of twos,
3
3
3
pieces of threes,
.
.
.
...
...
,
50
50
50
piecies of fifties. How many digits did Benedek write down?
infinite number of n-sided polygonal numbers = sum of 2 other polygonal numbers
Prove that for all
n
≥
3
n \ge 3
n
≥
3
there are an infinite number of
n
n
n
-sided polygonal numbers which are also the sum of two other (not necessarily different)
n
n
n
-sided polygonal numbers!The first
n
n
n
-sided polygonal number is
1
1
1
. The kth n-sided polygonal number for
k
≥
2
k \ge 2
k
≥
2
is the number of different points in a figure that consists of all of the regular
n
n
n
-sided polygons which have one common vertex, are oriented in the same direction from that vertex and their sides are
ℓ
\ell
ℓ
cm long where
1
≤
ℓ
≤
k
−
1
1 \le \ell \le k - 1
1
≤
ℓ
≤
k
−
1
cm and
ℓ
\ell
ℓ
is an integer.In this figure, what we call points are the vertices of the polygons and the points that break up the sides of the polygons into exactly
1
1
1
cm long segments. For example, the first four pentagonal numbers are 1,5,12, and 22, like it is shown in the figure. https://cdn.artofproblemsolving.com/attachments/1/4/290745d4be1888813678127e6d63b331adaa3d.png
Stable pyramid
For a given integer
n
≥
2
n\geq2
n
≥
2
, a pyramid of height
n
n
n
if defined as a collection of
1
2
+
2
2
+
⋯
+
n
2
1^2+2^2+\dots+n^2
1
2
+
2
2
+
⋯
+
n
2
stone cubes of equal size stacked in
n
n
n
layers such that the cubes in the
k
k
k
-th layer form a square with sidelength
n
+
1
−
k
n+1-k
n
+
1
−
k
and every cube (except for the ones in the bottom layer) rests on four cubes in the layer below. Some of the cubes are made of sandstone, some are made of granite. The top cube is made of granite, and to ensure the stability of the piramid, for each granite cube (except for the ones in the bottom layer), at least three out of four of the cubes supporting it have to be granite. What is the minimum possible number of granite cubes in such an arrangement?
2
3
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kn + k + n -> 9999999 or 99999999 or 48999999
When Andris entered the room, there were the numbers
3
3
3
and
24
24
24
on the board. In one step, if there are the (not necessarily different) numbers
k
k
k
and
n
n
n
on the board already, then Andris can write the number
k
n
+
k
+
n
kn + k + n
kn
+
k
+
n
on the board, too. a) Can Andris write the number
9999999
9999999
9999999
on the board after a few moves? b) What if he wants to get
99999999
99999999
99999999
? c) And what about
48999999
48999999
48999999
?
day date using only the digits 0,1,2
Timi was born in
1999
1999
1999
. Ever since her birth how many times has it happened that you could write that day’s date using only the digits
0
0
0
,
1
1
1
and
2
2
2
? For example,
2022.02.21
2022.02.21
2022.02.21
. is such a date.
Diophantine equation in primes
a) Find all solutions of the equation
p
2
+
q
2
+
r
2
=
p
q
r
p^2+q^2+r^2=pqr
p
2
+
q
2
+
r
2
=
pq
r
, where
p
,
q
,
r
p,q,r
p
,
q
,
r
are positive primes.\\ b) Show that for every positive integer
N
N
N
, there exist three integers
a
,
b
,
c
≥
N
a,b,c\geq N
a
,
b
,
c
≥
N
with
a
2
+
b
2
+
c
2
=
a
b
c
a^2+b^2+c^2=abc
a
2
+
b
2
+
c
2
=
ab
c
.
1
4
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