MathDB

1

divisive graphs

G

is divisive, if we can write a positive integer on each of its vertices such that all the integers are distinct, and any two of these integers divide each other if and only if there is an edge running between them in

G

. Which Platonic solids form a divisive graph? https://cdn.artofproblemsolving.com/attachments/1/5/7c81439ee148ccda09c429556e0740865723e0.png

5

2

products of two different numbers from n different reals

Let

n \ge 3

be an integer. Timi thought of

n

different real numbers and then wrote down the numbers which she could produce as the product of two different numbers she had in mind. At most how many different positive prime numbers did she write down (depending on

n

)?

H_DB^2 + H_EC^2 + H_FA^2 = H_DC^2 + H_EA^2 + H_FB^2

Consider an acute triangle

ABC

. Let

D

,

E

and

F

be the feet of the altitudes through vertices

A

,

B

and

C

. Denote by

A'

,

B'

,

C'

the projections of

A

,

B

,

C

onto lines

EF

,

FD

,

DE

, respectively. Further, let

H_D

,

H_E

,

H_F

be the orthocenters of triangles

DB'C'

,

EC'A'

,

FA'B'

. Show that

H_DB^2 + H_EC^2 + H_FA^2 = H_DC^2 + H_EA^2 + H_FB^2.

4

2

5 points conyclic

Let

k

be a circle with diameter

AB

and centre

O

. Let C be an arbitrary point on the circle different from

A

and

B

. Let

D

be the point for which

O

,

B

,

D

and

C

(in this order) are the four vertices of a parallelogram. Let

E

be the intersection of the line

BD

and the circle

k

, and let

F

be the orthocenter of the triangle

OAC

. Prove that the points

O, D, E, C, F

lie on a circle.

ant travels inside a circular disc

We are given an angle

0^o < \phi \le 180^o

and a circular disc. An ant begins its journey from an interior point of the disc, travelling in a straight line in a certain direction. When it reaches the edge of the disc, it does the following: it turns clockwise by the angle

\phi

, and if its new direction does not point towards the interior of the disc, it turns by the angle

\phi

again, and repeats this until it faces the interior. Then it continues its journey in this new direction and turns as before every time when it reaches the edge. For what values of

\phi

is it true that for any starting point and initial direction the ant eventually returns to its starting position?

3

12 Olympians and 88 humans in a Greek village

In a Greek village of

100

inhabitants in the beginning there lived

12

Olympians and

88

humans, they were the first generation. The Olympians are

100\%

gods while humans are

0\%

gods. In each generation they formed

50

couples and each couple had

2

children, who form the next generation (also consisting of

100

members). From the second generation onwards, every person’s percentage of godness is the average of the percentages of his/her parents’ percentages. (For example the children of

25\%

and

12.5\%

gods are

18.75\%

gods.) a) Which is the earliest generation in which it is possible that there are equally many

100\%

gods as

0\%

gods? b) At most how many members of the fifth generation are at least 25% gods?

a positive integer next to points

Pythagoras drew some points in the plane and and connected some of these with segments. Now Tortillagoras wants to write a positive integer next to every point, such that one number divides another number if and only if these numbers are written next to points that Pythagoras has connected.Can Tortillagoras do this for the following drawings?
In part b), vertices in the same row or column but not adjacent are not connected. https://cdn.artofproblemsolving.com/attachments/1/e/7356e39e44e45e3263275292af3719595e2dd2.png

max different xy = prime. combo NT

Let

n \ge 3

be an integer and

A

be a subset of the real numbers of size n. Denote by

B

the set of real numbers that are of the form

x \cdot y

, where

x, y \in A

and

x\ne y

. At most how many distinct positive primes could

B

contain (depending on

n

)?

1

2

group of students counting from 00-99 taking turns

A group of students play the following game: they are counting one by one from

00

to

99

taking turns, but instead of every number they only say one of its digits. (The numbers in order are

00

,

01

,

02

,

...

., meaning that one-digit numbers are regarded as two-digit numbers with a first digit

0

.) One way of starting the counting could be for example

0

,

1

,

2

,

0

,

4

,

0

,

6

,

7

,

8

,

9,

1

,

1

,

2

,

1

,

1

,

5

,

6

,

1

,

8

,

1

,

0

,

2

etc. When they reach

99

, the counting restarts from

00

. At some point Csongor enters the room and after listening to the counting for a while, he discovers that he is able to tell what number the counting is at. How many digits has Csongor heard at least?

[ \sqrt{n} ] + [ n / [ \sqrt{n} ] ] > 2 \sqrt{n}

Find all positive integers

n

such that

\lfloor \sqrt{n} \rfloor + \left\lfloor \frac{n}{\lfloor \sqrt{n} \rfloor} \right \rfloor> 2\sqrt{n}.

If

k

is a real number, then

\lfloor k \rfloor

means the floor of

k

, this is the greatest integer less than or equal to

k

.

2023 Durer Math Competition (First Round)

Subcontests

divisive graphs

products of two different numbers from n different reals

H_DB^2 + H_EC^2 + H_FA^2 = H_DC^2 + H_EA^2 + H_FB^2

5 points conyclic

ant travels inside a circular disc

12 Olympians and 88 humans in a Greek village

a positive integer next to points

max different xy = prime. combo NT

group of students counting from 00-99 taking turns

[ \sqrt{n} ] + [ n / [ \sqrt{n} ] ] > 2 \sqrt{n}

2023 Durer Math Competition (First Round)

Subcontests

divisive graphs

products of two different numbers from n different reals

H_DB^2 + H_EC^2 + H_FA^2 = H_DC^2 + H_EA^2 + H_FB^2

5 points conyclic

ant travels inside a circular disc

12 Olympians and 88 humans in a Greek village

a positive integer next to points

max different xy = prime. combo NT

group of students counting from 00-99 taking turns

[ \sqrt{n} ] + [ n / [ \sqrt{n} ] ] &gt; 2 \sqrt{n}

[ \sqrt{n} ] + [ n / [ \sqrt{n} ] ] > 2 \sqrt{n}