MathDB

1

(2n+1) times (2n+1) grid colored white and black

(2n + 1) \times (2n + 1)

grid, with

n> 0

, is colored in such a way that each of the cell is white or black. A cell is called special if there are at least

n

other cells of the same color in its row, and at least another

n

cells of the same color in its column.
(a) Prove that there are at least

2n + 1

special boxes. (b) Provide an example where there are at most

4n

special cells. (c) Determine, as a function of

n

, the minimum possible number of special cells.

5

1

Sum of 2014-th powers of 2015 integers

Prove that there exists a positive integer that can be written, in at least two ways, as a sum of

2014

-th powers of

2015

distinct positive integers

x_1 <x_2 <\cdots <x_{2015}

.

4

1

B,C,G,H are chosen on the circumference of omega

Let

\omega

be a circle with center

A

and radius

R

. On the circumference of

\omega

four distinct points

B, C, G, H

are taken in that order in such a way that

G

lies on the extended

B

-median of the triangle

ABC

, and H lies on the extension of altitude of

ABC

from

B

. Let

X

be the intersection of the straight lines

AC

and

GH

. Show that the segment

AX

has length

2R

.

3

1

gcd of {a^n+(a+1)^n+(a+2)^n | a in N}

For any positive integer

n

, let

D_n

denote the greatest common divisor of all numbers of the form

a^n + (a + 1)^n + (a + 2)^n

where

a

varies among all positive integers.
(a) Prove that for each

n

,

D_n

is of the form

3^k

for some integer

k \ge 0

. (b) Prove that, for all

k\ge 0

, there exists an integer

n

such that

D_n = 3^k

.

2

1

H is the foot of the altitude on AB, such that AH=3HB

Let

ABC

be a triangle. Let

H

be the foot of the altitude from

C

on

AB

. Suppose that

AH = 3HB

. Suppose in addition we are given that
(a)

M

is the midpoint of

AB

; (b)

N

is the midpoint of

AC

; (c)

P

is a point on the opposite side of

B

with respect to the line

AC

such that

NP = NC

and

PC = CB

.
Prove that

\angle APM = \angle PBA

.

1

Eliminate zeroes from n: ITAMO 2014 P1

For every

3

-digit natural number

n

(leading digit of

n

is nonzero), we consider the number

n_0

obtained from

n

eliminating all possible digits that are zero. For example, if

n = 207

, then

n_0 = 27

. Determine the number of three-digit positive integers

n

, for which

n_0

is a divisor of

n

different from

n

.

2014 ITAMO

Subcontests

(2n+1) times (2n+1) grid colored white and black

Sum of 2014-th powers of 2015 integers

B,C,G,H are chosen on the circumference of omega

gcd of {a^n+(a+1)^n+(a+2)^n | a in N}

H is the foot of the altitude on AB, such that AH=3HB

Eliminate zeroes from n: ITAMO 2014 P1