MathDB

2

area chasing inside a parallelogram

We choose a point on each side of a parallelogram

ABCD

, let these four points be

P, Q, R

and

S

. Then we divide the parallelogram into several regions using line segments as shown in the figure. The areas of the grey regions are given, except for one (see the figure). Find the area of the region marked with a question mark. https://cdn.artofproblemsolving.com/attachments/4/7/dbd009042dabdb2eafc8fc74960e9011038dae.png

reciprocals of every non-zero digit of integers from 1 to 10^n is integer

Find the smallest positive integer

n

with the following property: if we write down all positive integers from

1

to

10^n

and add together the reciprocals of every non-zero digit written down, we obtain an integer.

13

2

x_1 +k/x_2=x_2 +k/x_3=...=x_n +k/x_1

Let

k > 1

be a positive integer and

n \ge 2019

be an odd positive integer. The non-zero rational numbers

x_1, x_2,..., x_n

are not all equal, and satisfy the following chain of equalities:

x_1 +\frac{k}{x_2}= x_2 +\frac{k}{x_3}= x_3 +\frac{k}{x_4}= ... = x_{n-1} +\frac{k}{x_n}= x_n +\frac{k}{x_1}.

What is the smallest possible value of

k

?

12 numbered chairs arranged in a circle

There are

12

chairs arranged in a circle, numbered from

1

to

12

. How many ways are there to select some of the chairs in such a way that our selection includes

3

consecutive chairs somewhere?

16

2

paint the squares of a 6 x 6 board black or white

How many ways are there to paint the squares of a

6 \times 6

board black or white such that within each

2\times 2

square on the board, the number of black squares is odd?

circle ext. tangent to 3 circles, 13-14-15 triangle related

Triangle

ABC

has side lengths

13

,

14

and

15

. Let

k, k_A,k_B,k_C

be four circles of radius

r

inside the triangle such that

k_A

is tangent to sides

AB

and

AC

,

k_B

is tangent to sides

BA

and

BC

,

k_C

is tangent to sides

CA

and

CB

, and

k

is externally tangent to circles

k_A

,

k_B

and

k_C

. Let

r = m/n

where

m

and

n

are coprime. Find

m + n

.

15

2

<DAC=? AB = AC , <BAC = 96^o, <ACD = 48^o, AD=BC

ABC

is an isosceles triangle such that

AB = AC

and

\angle BAC = 96^o

.

D

is the point for which

\angle ACD = 48^o

,

AD = BC

and triangle

DAC

is obtuse-angled. Find

\angle DAC

.

m^{x_0} =\sum_{i=1}^{1001}m^{x_i}

The positive integer

m

and non-negative integers

x_0, x_1,..., x_{1001}

satisfy the following equation:

m^{x_0} =\sum_{i=1}^{1001}m^{x_i}.

How many possibilities are there for the value of

m

?

14

2

7 classmates are comparing their end-of-year grades in 12 subjects

Seven classmates are comparing their end-of-year grades in

12

subjects. They observe that for any two of them, there is some subject out of the

12

where the two students got different grades. It is possible to choose n subjects out of the

12

such that if the seven students only compare their grades in these

n

subjects, it will still be true that for any two, there is some subject out of the n where they got different grades. What is the smallest value of

n

for which such a selection is surely possible?
Note: In Hungarian high schools, students receive an integer grade from

1

to

5

in each subject at the end of the year.

last four digits in base 2 are the same as its last four digits in base 5

Let

S

be the set of all positive integers less than

10,000

whose last four digits in base

2

are the same as its last four digits in base

5

. What remainder do we get if we divide the sum of all elements of

S

by

10000

?

12

2

arrange the numbers 1, 2, 3, .. , 15 in some order

How many ways are there to arrange the numbers

1, 2, 3, .. , 15

in some order such that for any two numbers which are

2

or

3

positions apart, the one on the left is greater?

last four digits of Q(P(-1)) if P(Q(x)) = P(x)Q(x) and P(1) = P(-1) = 2019

P

and

Q

are two different non-constant polynomials such that

P(Q(x)) = P(x)Q(x)

and

P(1) = P(-1) = 2019

. What are the last four digits of

Q(P(-1))

?

11

2

min LCM of distinct a, b, c, d if a + b + c + d = 1000

What is the smallest possible value of the least common multiple of

a, b, c, d

if we know that these four numbers are distinct and

a + b + c + d = 1000

?

\sum_{k=1}^{N} k^{2018} is divisible by 2018?

What is the smallest

N

for which

\sum_{k=1}^{N} k^{2018}

is divisible by

2018

?

10

1

obtuse angle of isosceles triangle when 2 angle bisectors have ratio 2:1

In an isosceles, obtuse-angled triangle, the lengths of two internal angle bisectors are in a

2:1

ratio. Find the obtuse angle of the triangle.

9

1

cube divided into 27 equal-sized sub-cubes

A cube has been divided into

27

equal-sized sub-cubes. We take a line that passes through the interiors of as many sub-cubes as possible. How many does it pass through?

8

2

tour of a chess piece on a 8x8 chessboard

A chess piece is placed on one of the squares of an

8\times 8

chessboard where it begins a tour of the board: it moves from square to square, only moving horizontally or vertically. It visits every square precisely once, and ends up exactly where it started. What is the maximum number of times the piece can change direction along its tour?

N and N^2 both end in the same four digits

Let

N

be a positive integer such that

N

and

N^2

both end in the same four digits

\overline{abcd}

, where

a \ne 0

. What is the four-digit number

\overline{abcd}

?

6

2

min multiple of 81 that only contains the digit 1

Find the smallest multiple of

81

that only contains the digit

1

. How many

1

’s does it contain?

paper disks on table, into piles which contain various numbers of discs

(Game) At the beginning of the game, the organisers place paper disks on the table, grouped into piles which may contain various numbers of disks. The two players take turns. On a player’s turn, their opponent selects two piles (one if there is only one pile left), and the player must remove some number of disks from one of the piles selected. This means that at least one disk has to be removed, and removing all disks in the pile is also permitted. The player removing the last disk from the table wins. Defeat the organisers in this game twice in a row! A starting position will be given and then you can decide whether you want to go first or second.

5

4

3

sum of ratios wanted, similar isosceles triangles on sides of a triangle

Let

ABC

be an acute-angled triangle having angles

\alpha,\beta,\gamma

at vertices

A, B, C

respectively. Let isosceles triangles

BCA_1, CAB_1, ABC_1

be erected outwards on its sides, with apex angles

2\alpha ,2\beta ,2\gamma

respectively. Let

A_2

be the intersection point of lines

AA_1

and

B_1C_1

and let us define points

B_2

and

C_2

analogously. Find the exact value of the expression

\frac{AA_1}{A_2A_1}+\frac{BB_1}{B_2B_1}+\frac{CC_1}{C_2C_1}

2 tram lines in Miskolc

In Miskolc there are two tram lines: line

1

runs between Tiszai railway station and UpperMajláth, while line

2

runs between Tiszai railway station and the Ironworks. The timetable for trams leaving Tiszai railway station is as follows: tram

1

leaves at every minute ending in a

0

or

6

, and tram

2

leaves at every minute ending in a

3

. There are three types of passengers waiting for the trams: those who will take tram

1

only, those who will take tram

2

only and those who will take any tram. Every minute there is a constant number of passengers of each type arriving at the station. (This number is not necessarily the same for the different types.) Also, every tram departs with an equal number of passengers from Tiszai railway station. How many passengers are there on a departing tram, if we know that every minute there are

3

passengers arriving at the station who will take tram

2

only?

7 numbers are drawn out of 55 in the Intergalactic Lottery

In the Intergalactic Lottery,

7

numbers are drawn out of

55

. R2-D2 and C-3PO decide that they want to win this lottery, so they fill out lottery tickets separately such that for each possible draw one of them does have a winning ticket for that draw. Prove that one of them has

7

tickets with all different numbers.

3

p^q + p^r is a perfect square, 3 primes related

Determine all triples

(p, q, r)

of prime numbers for which

p^q + p^r

is a perfect square.

at most n true statements

On a piece of paper we have

2019

statements numbered from

1

to

2019

. The

n^{th}

statement is the following: "On this piece of paper there are at most

n

true statements". How many of the statements are true?

min perimeter of ABC, if 6 cevians divide the sides into 6 integer lengths

Let

P

be an interior point of triangle

ABC

. The lines

AP

,

BP

and

CP

divide each of the three sides into two segments. If the so-obtained six segments all have distinct integer lengths, what is the minimum possible perimeter of

ABC

?

2

3

fill in each cell of an 8x8 table with a 0 or 1

Albrecht fills in each cell of an

8 \times 8

table with a

0

or a

1

. Then at the end of each row and column he writes down the sum of the

8

digits in that row or column, and then he erases the original digits in the table. Afterwards, he claims to Berthold that given only the sums, it is possible to restore the

64

digits in the table uniquely. Show that the

8 \times 8

table contained either a row full of

0

’s or a column full of

1

’s

perfect squares after multiplying 2-digit numbers with 588

Anne multiplies each two-digit number by

588

in turn, and writes down the so-obtained products. How many perfect squares does she write down?

right triangle when sidelengths are integers and circumradius prime

Prove that if a triangle has integral side lengths and its circumradius is a prime number then the triangle is right-angled.

1

3

colouring positive rational numbers with blue and red

a) Is it possible to colour the positive rational numbers with blue and red in a way that there are numbers of both colours and the sum of any two numbers of the same colour is the same colour as them? b) Is it possible to colour the positive rational numbers with blue and red in a way that there are numbers of both colours and the product of any two numbers of the same colour is the same colour as them?
Note: When forming a sum or a product, it is allowed to pick the same number twice.

no of integer triangles whose sum of shorter sides is 19

Find the number of non-isosceles triangles (up to congruence) with integral side lengths, in which the sum of the two shorter sides is

19

.

(\sum a_i )^2 >= sum a_i^3 if a_n non-decreasing, a_0 = 0,a_j-a_i<=j -i for j>1

Let

a_o,a_1,a_2,..,a_ n

be a non-decreasing sequence of

n+1

real numbers where

a_0 = 0

and for every

j > i

we have

a_j - a_i \le j - i

. Show that

\left (\sum_{i=0}^n a_i \right )^2 \ge \sum_{i=0}^n a_i^3

2019 Durer Math Competition Finals

Subcontests

area chasing inside a parallelogram

reciprocals of every non-zero digit of integers from 1 to 10^n is integer

x_1 +k/x_2=x_2 +k/x_3=...=x_n +k/x_1

12 numbered chairs arranged in a circle

paint the squares of a 6 x 6 board black or white

circle ext. tangent to 3 circles, 13-14-15 triangle related

<DAC=? AB = AC , <BAC = 96^o, <ACD = 48^o, AD=BC

m^{x_0} =\sum_{i=1}^{1001}m^{x_i}

7 classmates are comparing their end-of-year grades in 12 subjects

last four digits in base 2 are the same as its last four digits in base 5

arrange the numbers 1, 2, 3, .. , 15 in some order

last four digits of Q(P(-1)) if P(Q(x)) = P(x)Q(x) and P(1) = P(-1) = 2019

min LCM of distinct a, b, c, d if a + b + c + d = 1000

\sum_{k=1}^{N} k^{2018} is divisible by 2018?

obtuse angle of isosceles triangle when 2 angle bisectors have ratio 2:1

cube divided into 27 equal-sized sub-cubes

tour of a chess piece on a 8x8 chessboard

N and N^2 both end in the same four digits

min multiple of 81 that only contains the digit 1

paper disks on table, into piles which contain various numbers of discs

sum of ratios wanted, similar isosceles triangles on sides of a triangle

2 tram lines in Miskolc

7 numbers are drawn out of 55 in the Intergalactic Lottery

p^q + p^r is a perfect square, 3 primes related

at most n true statements

min perimeter of ABC, if 6 cevians divide the sides into 6 integer lengths

fill in each cell of an 8x8 table with a 0 or 1

perfect squares after multiplying 2-digit numbers with 588

right triangle when sidelengths are integers and circumradius prime

colouring positive rational numbers with blue and red

no of integer triangles whose sum of shorter sides is 19

(\sum a_i )^2 >= sum a_i^3 if a_n non-decreasing, a_0 = 0,a_j-a_i<=j -i for j>1

2019 Durer Math Competition Finals

Subcontests

area chasing inside a parallelogram

reciprocals of every non-zero digit of integers from 1 to 10^n is integer

x_1 +k/x_2=x_2 +k/x_3=...=x_n +k/x_1

12 numbered chairs arranged in a circle

paint the squares of a 6 x 6 board black or white

circle ext. tangent to 3 circles, 13-14-15 triangle related

&lt;DAC=? AB = AC , &lt;BAC = 96^o, &lt;ACD = 48^o, AD=BC

m^{x_0} =\sum_{i=1}^{1001}m^{x_i}

7 classmates are comparing their end-of-year grades in 12 subjects

last four digits in base 2 are the same as its last four digits in base 5

arrange the numbers 1, 2, 3, .. , 15 in some order

last four digits of Q(P(-1)) if P(Q(x)) = P(x)Q(x) and P(1) = P(-1) = 2019

min LCM of distinct a, b, c, d if a + b + c + d = 1000

\sum_{k=1}^{N} k^{2018} is divisible by 2018?

obtuse angle of isosceles triangle when 2 angle bisectors have ratio 2:1

cube divided into 27 equal-sized sub-cubes

tour of a chess piece on a 8x8 chessboard

N and N^2 both end in the same four digits

min multiple of 81 that only contains the digit 1

paper disks on table, into piles which contain various numbers of discs

sum of ratios wanted, similar isosceles triangles on sides of a triangle

2 tram lines in Miskolc

7 numbers are drawn out of 55 in the Intergalactic Lottery

p^q + p^r is a perfect square, 3 primes related

at most n true statements

min perimeter of ABC, if 6 cevians divide the sides into 6 integer lengths

fill in each cell of an 8x8 table with a 0 or 1

perfect squares after multiplying 2-digit numbers with 588

right triangle when sidelengths are integers and circumradius prime

colouring positive rational numbers with blue and red

no of integer triangles whose sum of shorter sides is 19

(\sum a_i )^2 &gt;= sum a_i^3 if a_n non-decreasing, a_0 = 0,a_j-a_i&lt;=j -i for j&gt;1

<DAC=? AB = AC , <BAC = 96^o, <ACD = 48^o, AD=BC

(\sum a_i )^2 >= sum a_i^3 if a_n non-decreasing, a_0 = 0,a_j-a_i<=j -i for j>1