MathDB

1

max no of marked cells in 20 x 20 board

20 \times 20

board are adjacent if they have a common edge (a cell is not considered adjacent to itself). What is the maximum number of cells that can be marked in a

20 \times 20

board such that every cell is adjacent to at most one marked cell?

7

1

max ac if (a + b)/(b + c) = (c + d)/(d + a)

Let

a, b, c, d

be integers such that

a > b > c > d \ge -2021

and

\frac{a + b}{b + c}=\frac{c + d}{d + a}

(and

b + c \ne 0 \ne d + a

). What is the maximum possible value of

ac

?

6

1

positive integers in every cell of a 10 x10 array

Is it possible to place a positive integer in every cell of a

10 \times 10

array in such a way that both the following conditions are satisfied?

\bullet

Each number (not in the top row) is a proper divisor of the number immediately above.

\bullet

Each row consists of 1

0

consecutive positive integers (but not necessarily in order).

5

2

DE//AB wanted, isosceles ABC, BE=CD NZMO 2021 R1 p5

Let

ABC

be an isosceles triangle with

AB = AC

. Point

D

lies on side

AC

such that

BD

is the angle bisector of

\angle ABC

. Point

E

lies on side

BC

between

B

and

C

such that

BE = CD

. Prove that

DE

is parallel to

AB

.

y^5 + 2xy = x^2 + 2y^4

Find all pairs of integers

x, y

such that

y^5 + 2xy = x^2 + 2y^4.

.

4

2

diophantine 2x(x + 5) = p^n + 3(x - 1)

Find all triples

(x, p, n)

of non-negative integers such that

p

is prime and

2x(x + 5) = p^n + 3(x - 1)

.

3 Circles

Let

AB

be a chord of circle

\Gamma

. Let

O

be the centre of a circle which is tangent to

AB

at

C

and internally tangent to

\Gamma

at

P

. Point

C

lies between

A

and

B

. Let the circumcircle of triangle

POC

intersect

\Gamma

at distinct points

P

and

Q

. Prove that

\angle{AQP}=\angle{CQB}

.

3

2

max no of golden terms in a permutation of 1, 2, 3, . . . , 2021?

In a sequence of numbers, a term is called golden if it is divisible by the term immediately before it. What is the maximum possible number of golden terms in a permutation of

1, 2, 3, . . . , 2021

?

sum of any 3 of n is a prime, max n wanted

Let

\{x_1, x_2, x_3, ..., x_n\}

be a set of

n

distinct positive integers, such that the sum of any

3

of them is a prime number. What is the maximum value of

n

?

2

bisector of <BAC _|_AD, trapezium, AB = BE, AC = DE NZMO 2021 R1 p2

Let

ABCD

be a trapezium such that

AB\parallel CD

. Let

E

be the intersection of diagonals

AC

and

BD

. Suppose that

AB = BE

and

AC = DE

. Prove that the internal angle bisector of

\angle BAC

is perpendicular to

AD

.

x^2 +8/xy + y^2 >= 8 New Zealand MO 2021 Round 2 p2 NZMO

Prove that

x^2 +\frac{8}{xy}+ y^2 \ge 8

for all positive real numbers

x

and

y

.

1

2

school offers three subjects: Mathematics, Art and Science

A school offers three subjects: Mathematics, Art and Science. At least

80\%

of students study both Mathematics and Art. At least

80\%

of students study both Mathematics and Science. Prove that at least

80\%

of students who study both Art and Science, also study Mathematics.

BD=? AB + BC = 2021, AD = CD, <ABC = <CDA = 90^o

Let

ABCD

be a convex quadrilateral such that

AB + BC = 2021

and

AD = CD

. We are also given that

\angle ABC = \angle CDA = 90^o

. Determine the length of the diagonal

BD

.

2021 New Zealand MO

Subcontests

max no of marked cells in 20 x 20 board

max ac if (a + b)/(b + c) = (c + d)/(d + a)

positive integers in every cell of a 10 x10 array

DE//AB wanted, isosceles ABC, BE=CD NZMO 2021 R1 p5

y^5 + 2xy = x^2 + 2y^4

diophantine 2x(x + 5) = p^n + 3(x - 1)

3 Circles

max no of golden terms in a permutation of 1, 2, 3, . . . , 2021?

sum of any 3 of n is a prime, max n wanted

bisector of <BAC _|_AD, trapezium, AB = BE, AC = DE NZMO 2021 R1 p2

x^2 +8/xy + y^2 >= 8 New Zealand MO 2021 Round 2 p2 NZMO

school offers three subjects: Mathematics, Art and Science

BD=? AB + BC = 2021, AD = CD, <ABC = <CDA = 90^o

2021 New Zealand MO

Subcontests

max no of marked cells in 20 x 20 board

max ac if (a + b)/(b + c) = (c + d)/(d + a)

positive integers in every cell of a 10 x10 array

DE//AB wanted, isosceles ABC, BE=CD NZMO 2021 R1 p5

y^5 + 2xy = x^2 + 2y^4

diophantine 2x(x + 5) = p^n + 3(x - 1)

3 Circles

max no of golden terms in a permutation of 1, 2, 3, . . . , 2021?

sum of any 3 of n is a prime, max n wanted

bisector of &lt;BAC _|_AD, trapezium, AB = BE, AC = DE NZMO 2021 R1 p2

x^2 +8/xy + y^2 &gt;= 8 New Zealand MO 2021 Round 2 p2 NZMO

school offers three subjects: Mathematics, Art and Science

BD=? AB + BC = 2021, AD = CD, &lt;ABC = &lt;CDA = 90^o

bisector of <BAC _|_AD, trapezium, AB = BE, AC = DE NZMO 2021 R1 p2

x^2 +8/xy + y^2 >= 8 New Zealand MO 2021 Round 2 p2 NZMO

BD=? AB + BC = 2021, AD = CD, <ABC = <CDA = 90^o