2020 EGMO

Part of EGMO

Subcontests

(6)

1

2020 EGMO P6: m such that 3-term linear recurrence is always a square

m > 1

be an integer. A sequence

a_1, a_2, a_3, \ldots

a_1 = a_2 = 1

a_3 = 4

n \ge 4

a_n = m(a_{n - 1} + a_{n - 2}) - a_{n - 3}.

Determine all integers

m

such that every term of the sequence is a square.

1

2020 EGMO P4: n times the previous number of fresh permutations

A permutation of the integers

1, 2, \ldots, m

is called fresh if there exists no positive integer

k < m

such that the first

k

numbers in the permutation are

1, 2, \ldots, k

in some order. Let

f_m

be the number of fresh permutations of the integers

1, 2, \ldots, m

f_n \ge n \cdot f_{n - 1}

n \ge 3

.
For example, if

m = 4

, then the permutation

(3, 1, 4, 2)

is fresh, whereas the permutation

(2, 3, 1, 4)

1

2020 EGMO P3: Symmetric concurrence of angle bisectors in hexagon

ABCDEF

be a convex hexagon such that

\angle A = \angle C = \angle E

\angle B = \angle D = \angle F

and the (interior) angle bisectors of

\angle A, ~\angle C,

\angle E

are concurrent.
Prove that the (interior) angle bisectors of

\angle B, ~\angle D,

\angle F

must also be concurrent.
Note that

\angle A = \angle FAB

. The other interior angles of the hexagon are similarly described.

1

2020 EGMO P2: Sum inequality with permutations

(x_1, x_2, \ldots, x_{2020})

of non-negative real numbers such that the following three conditions are all satisfied:
[*]

x_1 \le x_2 \le \ldots \le x_{2020}

x_{2020} \le x_1 + 1

; [*] there is a permutation

(y_1, y_2, \ldots, y_{2020})

(x_1, x_2, \ldots, x_{2020})

\sum_{i = 1}^{2020} ((x_i + 1)(y_i + 1))^2 = 8 \sum_{i = 1}^{2020} x_i^3.

A permutation of a list is a list of the same length, with the same entries, but the entries are allowed to be in any order. For example,

(2, 1, 2)

is a permutation of

(1, 2, 2)

, and they are both permutations of

(2, 2, 1)

. Note that any list is a permutation of itself.

1

2020 EGMO P1: Linear recurrence is divisible by 2^2020

The positive integers

a_0, a_1, a_2, \ldots, a_{3030}

2a_{n + 2} = a_{n + 1} + 4a_n \text{ for } n = 0, 1, 2, \ldots, 3028.

Prove that at least one of the numbers

a_0, a_1, a_2, \ldots, a_{3030}

is divisible by

2^{2020}

1

2020 EGMO P5: P is the incentre of CDE

Consider the triangle

ABC

\angle BCA > 90^{\circ}

. The circumcircle

\Gamma

ABC

R

. There is a point

P

in the interior of the line segment

AB

PB = PC

and the length of

PA

R

. The perpendicular bisector of

PB

\Gamma

D

E

P

is the incentre of triangle

CDE