MathDB

1

largest cardinality of a set with a given binary operation

A

is endowed with a binary operation

*

satisfying the following four conditions: (1) If

a, b, c

are elements of

A

, then

a * (b * c) = (a * b) * c

, (2) If

a, b, c

are elements of

A

such that

a * c = b *c

, then

a = b

, (3) There exists an element

e

of

A

such that

a * e = a

for all

a

in

A

, and (4) If a and b are distinct elements of

A-\{e\}

, then

a^3 * b = b^3 * a^2

, where

x^k = x * x^{k-1}

for all integers

k \ge 2

and all

x

in

A

. Determine the largest cardinality

A

may have.
proposed by Bojan Basic, Serbia

N2

1

3-digit palindrome and binary expansion \overline {xyx}

Let

x, y

and

k

be three positive integers. Prove that there exist a positive integer

N

and a set of

k + 1

positive integers

\{b_0,b_1, b_2, ... ,b_k\}

, such that, for every

i = 0, 1, ... , k

, the

b_i

-ary expansion of

N

is a

3

-digit palindrome, and the

b_0

-ary expansion is exactly \overline{\mbox{xyx}}.
proposed by Bojan Basic, Serbia

C2

1

count partitions of sieves, rmm 2017 p5 the shortlist version

Fix an integer

n \ge 2

and let

A

be an

n\times n

array with

n

cells cut out so that exactly one cell is removed out of every row and every column. A stick is a

1\times k

or

k\times 1

subarray of

A

, where

k

is a suitable positive integer. (a) Determine the minimal number of sticks

A

can be dissected into. (b) Show that the number of ways to dissect

A

into a minimal number of sticks does not exceed

100^n

.
proposed by Palmer Mebane and Nikolai Beluhov
[hide=a few comments]a variation of part a, was [url=https://artofproblemsolving.com/community/c6h1389637p7743073]problem 5 a variation of part b, was posted [url=https://artofproblemsolving.com/community/c6h1389663p7743264]here this post was made in order to complete the post collection of RMM Shortlist 2017

G2

1

incentre of ABC lies on segment PQ iff AB + AC = 3BC, mixtilinear related

Let

ABC

be a triangle. Consider the circle

\omega_B

internally tangent to the sides

BC

and

BA

, and to the circumcircle of the triangle

ABC

, let

P

be the point of contact of the two circles. Similarly, consider the circle

\omega_C

internally tangent to the sides

CB

and

CA

, and to the circumcircle of the triangle

ABC

, let

Q

be the point of contact of the two circles. Show that the incentre of the triangle

ABC

lies on the segment

PQ

if and only if

AB + AC = 3BC

.
proposed by Luis Eduardo Garcia Hernandez, Mexico

N1

1

Sum of Digits

For each positive integer

k

, let

S(k)

the sum of digits of

k

in decimal system. Show that there is an integer

k

, with no

9

in it's decimal representation, such that:

S(2^{24^{2017}}k)=S(k)

G3

1

Hard geometry problem

Let

ABCD

be a convex quadrilateral and let

P

and

Q

be variable points inside this quadrilateral so that

\angle APB=\angle CPD=\angle AQB=\angle CQD

. Prove that the lines

PQ

obtained in this way all pass through a fixed point , or they are all parallel.

C1

1

Country with cities and flights

A planar country has an odd number of cities separated by pairwise distinct distances. Some of these cities are connected by direct two-way flights. Each city is directly connected to exactly two ther cities, and the latter are located farthest from it. Prove that, using these flights, one may go from any city to any other city

G1

1

Concurrency problem

Let

ABCD

be a trapezium,

AD\parallel BC

, and let

E,F

be points on the sides

AB

and

CD

, respectively. The circumcircle of

AEF

meets

AD

again at

A_1

, and the circumcircle of

CEF

meets

BC

again at

C_1

. Prove that

A_1C_1,BD,EF

are concurrent.

2017 Romanian Master of Mathematics Shortlist

Subcontests

largest cardinality of a set with a given binary operation

3-digit palindrome and binary expansion \overline {xyx}

count partitions of sieves, rmm 2017 p5 the shortlist version

incentre of ABC lies on segment PQ iff AB + AC = 3BC, mixtilinear related

Sum of Digits

Hard geometry problem

Country with cities and flights

Concurrency problem