MathDB

1

inequalities with 3-element subsets of {1,2,...,n}

n,m

be positive integers. Let

A_1,A_2,A_3, ... ,A_m

be sets such that

A_i \subseteq \{1, 2, 3, . . . , n\}

and

|A_i| = 3

for all

i

(i.e.

A_i

consists of three different positive integers each at most

n

). Suppose for all

i < j

we have

|A_i \cap A_j | \le 1

(i.e.

A_i

and

A_j

have at most one element in common). (a) Prove that

m \le \frac{n(n-1)}{ 6}

. (b) Show that for all

n \ge3

it is possible to have

m \ge \frac{(n-1)(n-2)}{ 6}

.

8

1

P(x) = ax^4 - 8ax^3 + bx^2 - 32cx + 16c has 4 positive real roots

Find all non-zero real numbers

a, b, c

such that the following polynomial has four (not necessarily distinct) positive real roots.

P(x) = ax^4 - 8ax^3 + bx^2 - 32cx + 16c

6

1

1/BD - 1/BC =1/2AB if <bAC=90^o, BC - BD = 2AB, BD bisectsd <ABC

Let triangle

ABC

be right-angled at

A

. Let

D

be the point on

AC

such that

BD

bisects angle

\angle ABC

. Prove that

BC - BD = 2AB

if and only if

\frac{1}{BD} - \frac{1}{BC} =\frac{1}{2AB}

.

5

2

a+b =n/2, and a,b both divisors of n

Find all triples

(a, b, n)

of positive integers such that

a

and

b

are both divisors of

n

, and

a+b = \frac{n}{2}

.

x + y + z is integer if x^2 = y + 2 and y^2 = z + 2 and z^2 = x + 2

Let

x, y

and

z

be real numbers such that:

x^2 = y + 2

, and

y^2 = z + 2

, and

z^2 = x + 2

. Prove that

x + y + z

is an integer.

4

2

f(x) = ax^2 + bx + c is divisible by prime p whenever x , 0 < a, b, c <= p

Let

p

be a prime and let

f(x) = ax^2 + bx + c

be a quadratic polynomial with integer coefficients such that

0 < a, b, c \le p

. Suppose

f(x)

is divisible by

p

whenever

x

is a positive integer. Find all possible values of

a + b + c

.

f(m) = f(m + 1), where f(n) is number of subsets of {1,...,n}

For any positive integer

n

, let

f(n)

be the number of subsets of

\{1, 2, . . . , n\}

whose sum is equal to

n

. Does there exist infinitely many positive integers

m

such that

f(m) = f(m + 1)

? (Note that each element in a subset must be distinct.)

3

2

9^{x+1} + 2187 = 3^{6x-x^2}

Find the sum of the smallest and largest possible values for

x

which satisfy the following equation.

9^{x+1} + 2187 = 3^{6x-x^2}.

collinearity wanted, 2 squares related

Let

ABCD

be a square (vertices labelled in clockwise order). Let

Z

be any point on diagonal

AC

between

A

and

C

such that

AZ > ZC

. Points

X

and

Y

exist such that

AXY Z

is a square (vertices labelled in clockwise order) and point

B

lies inside

AXY Z

. Let

M

be the point of intersection of lines

BX

and

DZ

(extended if necessary). Prove that

C

,

M

and

Y

are colinear

2

|DAQ|^2 = |PAQ| x |BCD| , ABCD #

Let

ABCD

be a parallelogram, and let

P

be a point on the side

AB

. Let the line through

P

parallel to

BC

intersect the diagonal

AC

at point

Q

. Prove that

|DAQ|^2 = |PAQ| \times |BCD| ,

where

|XY Z|

denotes the area of triangle

XY Z

.

at least one of a,b,c >17/10 if a,b,c>0 with a+b+c = abc

Let

a, b

and

c

be positive real numbers such that

a+b+c = abc

. Prove that at least one of

a, b

or

c

is greater than

\frac{17}{10}

.

1

2

2023 employees in the office

There are 2023 employees in the office, each of them knowing exactly

1686

of the others. For any pair of employees they either both know each other or both don’t know each other. Prove that we can find

7

employees each of them knowing all

6

others.

p! x q! x r! is perfect square if p > q > r > 1

For any positive integer

n

let

n! = 1\times 2\times 3\times ... \times n

. Do there exist infinitely many triples

(p, q, r)

, of positive integers with

p > q > r > 1

such that the product

p! \cdot q! \cdot r!

$ is a perfect square?

2023 New Zealand MO

Subcontests

inequalities with 3-element subsets of {1,2,...,n}

P(x) = ax^4 - 8ax^3 + bx^2 - 32cx + 16c has 4 positive real roots

1/BD - 1/BC =1/2AB if <bAC=90^o, BC - BD = 2AB, BD bisectsd <ABC

a+b =n/2, and a,b both divisors of n

x + y + z is integer if x^2 = y + 2 and y^2 = z + 2 and z^2 = x + 2

f(x) = ax^2 + bx + c is divisible by prime p whenever x , 0 < a, b, c <= p

f(m) = f(m + 1), where f(n) is number of subsets of {1,...,n}

9^{x+1} + 2187 = 3^{6x-x^2}

collinearity wanted, 2 squares related

|DAQ|^2 = |PAQ| x |BCD| , ABCD #

at least one of a,b,c >17/10 if a,b,c>0 with a+b+c = abc

2023 employees in the office

p! x q! x r! is perfect square if p > q > r > 1

2023 New Zealand MO

Subcontests

inequalities with 3-element subsets of {1,2,...,n}

P(x) = ax^4 - 8ax^3 + bx^2 - 32cx + 16c has 4 positive real roots

1/BD - 1/BC =1/2AB if &lt;bAC=90^o, BC - BD = 2AB, BD bisectsd &lt;ABC

a+b =n/2, and a,b both divisors of n

x + y + z is integer if x^2 = y + 2 and y^2 = z + 2 and z^2 = x + 2

f(x) = ax^2 + bx + c is divisible by prime p whenever x , 0 &lt; a, b, c &lt;= p

f(m) = f(m + 1), where f(n) is number of subsets of {1,...,n}

9^{x+1} + 2187 = 3^{6x-x^2}

collinearity wanted, 2 squares related

|DAQ|^2 = |PAQ| x |BCD| , ABCD #

at least one of a,b,c &gt;17/10 if a,b,c&gt;0 with a+b+c = abc

2023 employees in the office

p! x q! x r! is perfect square if p &gt; q &gt; r &gt; 1

1/BD - 1/BC =1/2AB if <bAC=90^o, BC - BD = 2AB, BD bisectsd <ABC

f(x) = ax^2 + bx + c is divisible by prime p whenever x , 0 < a, b, c <= p

at least one of a,b,c >17/10 if a,b,c>0 with a+b+c = abc

p! x q! x r! is perfect square if p > q > r > 1