MathDB

1

sums a_i + a_j form consecutive terms of an arithmetic progression

n

be a positive integer greater than or equal to

6

, and suppose that

a_1, a_2, ...,a_n

are real numbers such that the sums

a_i + a_j

for

1 \le i<j\le n

, taken in some order, form consecutive terms of an arithmetic progression

A

,

A + d

,

...

,

A + (k-1)d

, where

k = n(n-1)/2

. What are the possible values of

d

?

6

1

every positive integer <= \sqrt{a} is a near factor of a

In many computer languages, the division operation ignores remainders. Let’s denote this operation by

//

, so for instance

13//3 = 4

. If, for some

b

,

a//b = c

, then we say that

c

is a near factor of

a

. Thus, the near factors of

13

are

1

,

2

,

3

,

4

, and

6

. Let

a

be a positive integer. Prove that every positive integer less than or equal to

\sqrt{a}

is a near factor of

a

.

9

1

a salesperson to Elbonia who needs to visit every town in to Elbonia

Consolidated Megacorp is planning to send a salesperson to Elbonia who needs to visit every town there. It is possible to travel between any two towns of Elbonia directly either by barge or by mule cart (the same type of travel is available in either direction, and these are the only types of travel available). Show that it is possible to choose a starting town so that the salesperson can complete a round trip visiting each town exactly once and returning to her starting point, while changing the type of transportation used at most one time (this is desirable, since it’s hard to arrange for the merchandise to be transferred from barge to cart or vice versa).

8

1

n has a divisor d with the property that dn + 1 is a divisor of d^2 + n^2

Determine all positive integers

n

which have a divisor

d

with the property that

dn + 1

is a divisor of

d^2 + n^2

.

4

1

diophantine (ab)^{2015} = (a^2 + b^2)^m

For which positive integers

m

does the equation:

(ab)^{2015} = (a^2 + b^2)^m

have positive integer solutions?

2

1

a mathematics competition had 9 easy and 6 difficult problems

A mathematics competition had

9

easy and

6

difficult problems. Each of the participants in the competition solved

14

of the

15

problems. For each pair, consisting of an easy and a difficult problem, the number of participants who solved both those problems was recorded. The sum of these recorded numbers was

459

. How many participants were there?

1

is it possible to construct one that ends with the number 9,876,543,210 ?

Starting from the number

1

we write down a sequence of numbers where the next number in the sequence is obtained from the previous one either by doubling it, or by rearranging its digits (not allowing the first digit of the rearranged number to be

0

). For instance we might begin:

1, 2, 4, 8, 16, 61, 122, 212, 424,...

Is it possible to construct such a sequence that ends with the number

1,000,000,000

? Is it possible to construct one that ends with the number

9,876,543,210

?

7

1

circumcenter of APD is foot of perpendicular from A on BC, NP = NQ, cyclic BPQC

Let

ABC

be an acute-angled scalene triangle. Let

P

be a point on the extension of

AB

past

B

, and

Q

a point on the extension of

AC

past

C

such that

BPQC

is a cyclic quadrilateral. Let

N

be the foot of the perpendicular from

A

to

BC

. If

NP = NQ

then prove that

N

is also the centre of the circumcircle of

APQ

.

3

1

concurrency of reflected arcs of circumcircle

Let

ABC

be an acute angled triangle. The arc between

A

and

B

of the circumcircle of

ABC

is reflected through the line

AB

, and the arc between

A

and

C

of the circumcircle of

ABC

is reflected over the line

AC

. Obviously these two reflected arcs intersect at the point

A

. Prove that they also intersect at another point inside the triangle

ABC

.

2015 NZMOC Camp Selection Problems

Subcontests

sums a_i + a_j form consecutive terms of an arithmetic progression

every positive integer <= \sqrt{a} is a near factor of a

a salesperson to Elbonia who needs to visit every town in to Elbonia

n has a divisor d with the property that dn + 1 is a divisor of d^2 + n^2

diophantine (ab)^{2015} = (a^2 + b^2)^m

a mathematics competition had 9 easy and 6 difficult problems

is it possible to construct one that ends with the number 9,876,543,210 ?

circumcenter of APD is foot of perpendicular from A on BC, NP = NQ, cyclic BPQC

concurrency of reflected arcs of circumcircle

2015 NZMOC Camp Selection Problems

Subcontests

sums a_i + a_j form consecutive terms of an arithmetic progression

every positive integer &lt;= \sqrt{a} is a near factor of a

a salesperson to Elbonia who needs to visit every town in to Elbonia

n has a divisor d with the property that dn + 1 is a divisor of d^2 + n^2

diophantine (ab)^{2015} = (a^2 + b^2)^m

a mathematics competition had 9 easy and 6 difficult problems

is it possible to construct one that ends with the number 9,876,543,210 ?

circumcenter of APD is foot of perpendicular from A on BC, NP = NQ, cyclic BPQC

concurrency of reflected arcs of circumcircle

every positive integer <= \sqrt{a} is a near factor of a