MathDB

1

a_{n+2} = xa_{n+1} - a_n , a_1 = x + 1, a_0 = 1

x

, define a sequence

a_0, a_1, a_2, . . .

according to the following rules:

a_0 = 1

,

a_1 = x + 1

and

a_{n+2} = xa_{n+1} - a_n

for all

n \ge 0

.
Prove that there exist infinitely many positive integers x such that this sequence does not contain a prime number.

7

1

2 player game on a 20x 20 chessboard

Josie and Ross are playing a game on a

20 \times 20

chessboard. Initially the chessboard is empty. The two players alternately take turns, with Josie going first. On Josie’s turn, she selects any two different empty cells, and places one white stone in each of them. On Ross’ turn, he chooses any one white stone currently on the board, and replaces it with a black stone. If at any time there are

8

consecutive cells in a line (horizontally or vertically) all of which contain a white stone, Josie wins. Is it possible that Ross can stop Josie winning - regardless of how Josie plays?

3

2

unlimited supply of square tiles and equilateral triangle tiles

You have an unlimited supply of square tiles with side length

1

and equilateral triangle tiles with side length

1

. For which n can you use these tiles to create a convex

n

-sided polygon? The tiles must fit together without gaps and may not overlap.

3 of 13 points on circle of r=1 form a triangle with area <13

There are

13

marked points on the circumference of a circle with radius

13

. Prove that we can choose three of the marked points which form a triangle with area less than

13

.

1

2

max n such that 50!/2^n is an integer

What is the maximum integer

n

such that

\frac{50!}{2^n}

is an integer?

if P(r) = 0 then Q(r^2) = 0, P(x) = x^3 - 2x + 1, Q(x) = x^3 - 4x^2 + 4x - 1

Let

P(x) = x^3 - 2x + 1

and let

Q(x) = x^3 - 4x^2 + 4x - 1

. Show that if

P(r) = 0

then

Q(r^2) = 0

.

2

midpoint of XY lies on diagonal BD of square ABCD (2020 NZMO 1.2)

Let

ABCD

be a square and let

X

be any point on side

BC

between

B

and

C

. Let

Y

be the point on line

CD

such that

BX = YD

and

D

is between

C

and

Y

. Prove that the midpoint of

XY

lies on diagonal

BD

.

min N = 5mod7, 8 mod6, 9 mod7

Find the smallest positive integer

N

satisfying the following three properties.

\bullet

N leaves a remainder of

5

when divided by

7

.

\bullet

N leaves a remainder of

6

when divided by

8

.

\bullet

N leaves a remainder of

7

when divided by

9

.

6

1

PY bisects AX, feet of altitudes, intersections of circumcircles AXC and PQC

Let

\vartriangle ABC

be an acute triangle with

AB > AC

. Let

P

be the foot of the altitude from

C

to

AB

and let

Q

be the foot of the altitude from

B

to

AC

. Let

X

be the intersection of

PQ

and

BC

. Let the intersection of the circumcircles of triangle

\vartriangle AXC

and triangle

\vartriangle PQC

be distinct points:

C

and

Y

. Prove that

PY

bisects

AX

.

4

2

p^2 - 6 and p^2 + 6 are both prime numbers

Determine all prime numbers

p

such that

p^2 - 6

and

p^2 + 6

are both prime numbers.

angle bisector wanted, internally tangent circles

Let

\Gamma_1

and

\Gamma_2

be circles internally tangent at point

A

, with

\Gamma_1

inside

\Gamma_2

. Let

BC

be a chord of

\Gamma_2

which is tangent to

\Gamma_1

at point

D

. Prove that line

AD

is the angle bisector of

\angle BAC

.

5

2

NZMO Function Equation

Find all functions

f:\mathbb R \to \mathbb R

such that for all

x,y\in \mathbb R

f(x+f(y))=2x+2f(y+1)

sum of the costs of all antipalindromic sequences of length 2020

A sequence of

A

s and

B

s is called antipalindromic if writing it backwards, then turning all the

A

s into

B

s and vice versa, produces the original sequence. For example

ABBAAB

is antipalindromic. For any sequence of

A

s and

B

s we define the cost of the sequence to be the product of the positions of the

A

s. For example, the string

ABBAAB

has cost

1\cdot 4 \cdot 5 = 20

. Find the sum of the costs of all antipalindromic sequences of length

2020

.

2020 New Zealand MO

Subcontests

a_{n+2} = xa_{n+1} - a_n , a_1 = x + 1, a_0 = 1

2 player game on a 20x 20 chessboard

unlimited supply of square tiles and equilateral triangle tiles

3 of 13 points on circle of r=1 form a triangle with area <13

max n such that 50!/2^n is an integer

if P(r) = 0 then Q(r^2) = 0, P(x) = x^3 - 2x + 1, Q(x) = x^3 - 4x^2 + 4x - 1

midpoint of XY lies on diagonal BD of square ABCD (2020 NZMO 1.2)

min N = 5mod7, 8 mod6, 9 mod7

PY bisects AX, feet of altitudes, intersections of circumcircles AXC and PQC

p^2 - 6 and p^2 + 6 are both prime numbers

angle bisector wanted, internally tangent circles

NZMO Function Equation

sum of the costs of all antipalindromic sequences of length 2020

2020 New Zealand MO

Subcontests

a_{n+2} = xa_{n+1} - a_n , a_1 = x + 1, a_0 = 1

2 player game on a 20x 20 chessboard

unlimited supply of square tiles and equilateral triangle tiles

3 of 13 points on circle of r=1 form a triangle with area &lt;13

max n such that 50!/2^n is an integer

if P(r) = 0 then Q(r^2) = 0, P(x) = x^3 - 2x + 1, Q(x) = x^3 - 4x^2 + 4x - 1

midpoint of XY lies on diagonal BD of square ABCD (2020 NZMO 1.2)

min N = 5mod7, 8 mod6, 9 mod7

PY bisects AX, feet of altitudes, intersections of circumcircles AXC and PQC

p^2 - 6 and p^2 + 6 are both prime numbers

angle bisector wanted, internally tangent circles

NZMO Function Equation

sum of the costs of all antipalindromic sequences of length 2020

3 of 13 points on circle of r=1 form a triangle with area <13