MathDB

1

(a + b)^3/ 4 >= a^2b + ab^2 for a,b>0

Prove that for all positive real numbers

a

and

b

:

\frac{(a + b)^3}{4} \ge a^2b + ab^2

10

1

100 letter words in 2letter languages

In the land of Microbablia the alphabet has only two letters, ‘A’ and ‘B’. Not surprisingly, the inhabitants are obsessed with the band ABBA. Words in the local dialect with a high ABBA-factor are considered particularly lucky. To compute the ABBA-factor of a word you just count the number of occurrences of ABBA within the word (not necessarily consecutively). So for instance AABA has ABBA-factor

0

, ABBA has ABBA-factor

1

, AABBBA has ABBA-factor

6

, and ABBABBA has ABBA factor

8

. What is the greatest possible ABBA-factor for a

100

letter word?

8

1

doubles tennis tournament among Michael's friends

Michael wants to arrange a doubles tennis tournament among his friends. However, he has some peculiar conditions: the total number of matches should equal the total number of players, and every pair of friends should play as either teammates or opponents in at least one match. The number of players in a single match is four. What is the largest number of people who can take part in such a tournament?

7

1

system x^3 + y^3 = 2, kx + d = y, has no real solutions

Determine all pairs of real numbers

(k, d)

such that the system of equations

\begin{cases} x^3 + y^3 = 2 \\ kx + d = y\end{cases}

has no solutions

(x, y)

with

x

and

y

real numbers.

6

1

lcm(a,b,c) =a+b+c

Determine all triples of positive integers

a

,

b

and

c

such that their least common multiple is equal to their sum.

4

1

min no of line segments, 2014 points in plane

Given

2014

points in the plane, no three of which are collinear, what is the minimum number of line segments that can be drawn connecting pairs of points in such a way that adding a single additional line segment of the same sort will always produce a triangle of three connected points?

3

1

(x + y)(x^2 + 9y) perfect cube of prime

Find all pairs

(x, y)

of positive integers such that

(x + y)(x^2 + 9y)

is the cube of a prime number.

9

1

AX x BY = AI x BI, circle with center a midpoint, triangle with that incircle

Let

AB

be a line segment with midpoint

I

. A circle, centred at

I

has diameter less than the length of the segment. A triangle

ABC

is tangent to the circle on sides

AC

and

BC

. On

AC

a point

X

is given, and on

BC

a point

Y

is given such that

XY

is also tangent to the circle (in particular

X

lies between the point of tangency of the circle with

AC

and

C

, and similarly

Y

lies between the point of tangency of the circle with

BC

and

C

. Prove that

AX \cdot BY = AI \cdot BI

.

5

1

concyclic wanted, altitudes and midpoints related

Let

ABC

be an acute angled triangle. Let the altitude from

C

to

AB

meet

AB

at

C'

and have midpoint

M

, and let the altitude from

B

to

AC

meet

AC

at

B'

and have midpoint

N

. Let

P

be the point of intersection of

AM

and

BB'

and

Q

the point of intersection of

AN

and

CC'

. Prove that the point

M, N, P

and

Q

lie on a circle.

2

1

concurrency of 2 perp. and 1 perp. bisector (2014 NZOMC Camp Sel. p2)

Let

ABC

be a triangle in which the length of side

AB

is

4

units, and that of

BC

is

2

units. Let

D

be the point on

AB

at distance

3

units from

A

. Prove that the line perpendicular to

AB

through

D

, the angle bisector of

\angle ABC

, and the perpendicular bisector of

BC

all meet at a single point.

2014 NZMOC Camp Selection Problems

Subcontests

(a + b)^3/ 4 >= a^2b + ab^2 for a,b>0

100 letter words in 2letter languages

doubles tennis tournament among Michael's friends

system x^3 + y^3 = 2, kx + d = y, has no real solutions

lcm(a,b,c) =a+b+c

min no of line segments, 2014 points in plane

(x + y)(x^2 + 9y) perfect cube of prime

AX x BY = AI x BI, circle with center a midpoint, triangle with that incircle

concyclic wanted, altitudes and midpoints related

concurrency of 2 perp. and 1 perp. bisector (2014 NZOMC Camp Sel. p2)

2014 NZMOC Camp Selection Problems

Subcontests

(a + b)^3/ 4 &gt;= a^2b + ab^2 for a,b&gt;0

100 letter words in 2letter languages

doubles tennis tournament among Michael's friends

system x^3 + y^3 = 2, kx + d = y, has no real solutions

lcm(a,b,c) =a+b+c

min no of line segments, 2014 points in plane

(x + y)(x^2 + 9y) perfect cube of prime

AX x BY = AI x BI, circle with center a midpoint, triangle with that incircle

concyclic wanted, altitudes and midpoints related

concurrency of 2 perp. and 1 perp. bisector (2014 NZOMC Camp Sel. p2)

(a + b)^3/ 4 >= a^2b + ab^2 for a,b>0