MathDB

line tangent and parallel at the same time.

Source: Iranian RMM TST 2021 Day1 P1

4/16/2021

Suppose that two circles

\alpha, \beta

with centers

P,Q

, respectively , intersect orthogonally at

A

,

B

. Let

CD

be a diameter of

\beta

that is exterior to

\alpha

. Let

E,F

be points on

\alpha

such that

CE,DF

are tangent to

\alpha

, with

C,E

on one side of

PQ

and

D,F

on the other side of

PQ

. Let

S

be the intersection of

CF,AQ

and

T

be the intersection of

DE,QB

. Prove that

ST

is parallel to

CD

and is tangent to

\alpha

geometrycirclesorthocenter

an interesting $polyomina$

Source: Iranian RMM TST 2021 Day2 P1

4/16/2021

A polyomino is region with connected interior that is a union of a finite number of squares from a grid of unit squares. Do there exist a positive integer

n>4

and a polyomino

P

contained entirely within and

n

-by-

n

grid such that

P

contains exactly

3

unit squares in every row and every column of the grid?
Proposed by Nikolai Beluhov

combinatoricssquare grid

Creating polynomial with property

Source: Iranian RMM TST 2021 Day3 P1

4/16/2021

Let

P(x)=x^{2016}+2x^{2015}+...+2017,Q(x)=1399x^{1398}+...+2x+1

. Prove that there are strictly increasing sequances

a_i,b_i, i=1,...

of positive integers such that

gcd(a_i,a_{i+1})=1

for each

i

. Moreover, for each even

i

,

P(b_i) \nmid a_i, Q(b_i) | a_i

and for each odd

i

,

P(b_i)|a_i,Q(b_i) \nmid a_i

Proposed by Shayan Talaei

Integer Polynomialdividibilitynumber theory

1

Problems(3)