MathDB

a_{n+1} - a_n = n(a_n - 1)

Source: 2021 Dutch IMO TST 1.1

12/28/2021

The sequence of positive integers

a_0, a_1, a_2, . . .

is defined by

a_0 = 3

and

a_{n+1} - a_n = n(a_n - 1)

for all

n \ge 0

. Determine all integers

m \ge 2

for which

gcd (m, a_n) = 1

for all

n \ge 0

.

number theorySequencerecurrence relation

covering a m x n board with dominos

Source: 2021 Dutch IMO TST 3.1

12/28/2021

Let

m

and

n

be natural numbers with

mn

even. Jetze is going to cover an

m \times n

board (consisting of

m

rows and

n

columns) with dominoes, so that every domino covers exactly two squares, dominos do not protrude or overlap, and all squares are covered by a domino. Merlin then moves all the dominoe color red or blue on the board. Find the smallest non-negative integer

V

(in terms of

m

and

n

) so that Merlin can always ensure that in each row the number squares covered by a red domino and the number of squares covered by a blue one dominoes are not more than

V

, no matter how Jetze covers the board.

combinatoricsgamegame strategywinning strategydominos

EX=EY wanted, 3 circles related

Source: 2021 Dutch IMO TST 2.1

12/29/2021

Let

\Gamma

be the circumscribed circle of a triangle

ABC

and let

D

be a point at line segment

BC

. The circle passing through

B

and

D

tangent to

\Gamma

and the circle passing through

C

and

D

tangent to

\Gamma

intersect at a point

E \ne D

. The line

DE

intersects

\Gamma

at two points

X

and

Y

. Prove that

|EX| = |EY|

.

equal segmentsgeometry

1

Problems(3)