MathDB

3

numbers 1, 2,...,2012 in a circle

Determine whether it is possible to place the integers

1, 2,...,2012

in a circle in such a way that the

2012

products of adjacent pairs of numbers leave pairwise distinct remainders when divided by

2013

.

each pos. integer exactly once in |a_1- a_2|,|a_2 - a_3|, ...,|a_k- a_{k+1}|,...

Determine if there exists an infinite sequence of positive integers

a_1,a_2, a_3, ...

such that (i) each positive integer occurs exactly once in the sequence, and (ii) each positive integer occurs exactly once in the sequence

|a_1 - a_2|, |a_2 - a_3|, ..., |a+k - a_{k+1}|, ...

2^n - 1 is divisible by p(n)

Find all polynomials

p(x)

with integer coefficients such that for each positive integer

n

, the number

2^n - 1

is divisible by

p(n)

.

3

a Saudi company has 2 offices, no of correspondents from each office

A Saudi company has two offices. One office is located in Riyadh and the other in Jeddah. To insure the connection between the two offices, the company has designated from each office a number of correspondents so that : (a) each pair of correspondents from the same office share exactly one common correspondent from the other office. (b) there are at least

10

correspondents from Riyadh. (c) Zayd, one of the correspondents from Jeddah, is in contact with exactly

8

correspondents from Riyadh. What is the minimum number of correspondents from Jeddah who are in contact with the correspondent Amr from Riyadh?

max of p% of all divisors of n with their unit digit equal to 3

For a positive integer

n

, we consider all its divisors (including

1

and itself). Suppose that

p\%

of these divisors have their unit digit equal to

3

. (For example

n = 117

, has six divisors, namely

1,3,9,13,39,117

. Two of these divisors namely

3

and

13

, have unit digits equal to

3

. Hence for

n = 117

,

p =33.33...

). Find, when

n

is any positive integer, the maximum possible value of

p

.

2 circumcircles and a line concurrent

Let

ABC

be an acute triangle,

M

be the midpoint of

BC

and

P

be a point on line segment

AM

. Lines

BP

and

CP

meet the circumcircle of

ABC

again at

X

and

Y

, respectively, and sides

AC

at

D

and

AB

at

E

, respectively. Prove that the circumcircles of

AXD

and

AYE

have a common point

T \ne A

on line

AM

.

2

3

n + f(f(n)) <= 2f(n) , strictly increasing

Let

S = f\{0.1. 2.3,...\}

be the set of the non-negative integers. Find all strictly increasing functions

f : S \to S

such that

n + f(f(n)) \le 2f(n)

for every

n

in

S

concyclic wanted, altitudes related

Let

ABC

be an acute triangle, and let

AA_1, BB_1

, and

CC_1

be its altitudes. Segments

AA_1

and

B_1C_1

meet at point

K

. The perpendicular bisector of segment

A_1K

intersects sides

AB

and

AC

at

L

and

M

, respectively. Prove that points

A,A_1, L

, and

M

lie on a circle.

a_1 + a_2 + .. + a_n >= n^2 and a_1^2 + a_2^2 + ... + a_n^2 <= n^3 + 1

Given an integer

n \ge 2

, determine the number of ordered

n

-tuples of integers

(a_1, a_2,...,a_n)

such that (a)

a_1 + a_2 + .. + a_n \ge n^2

and (b)

a_1^2 + a_2^2 + ... + a_n^2 \le n^3 + 1

1

3

Concurrent Lines

Triangle

ABC

is inscribed in circle

\omega

. Point

P

lies inside triangle

ABC

.Lines

AP,BP

and

CP

intersect

\omega

again at points

A_1

,

B_1

and

C_1

(other than

A, B, C

), respectively. The tangent lines to

\omega

at

A_1

and

B_1

intersect at

C_2

.The tangent lines to

\omega

at

B_1

and

C_1

intersect at

A_2

. The tangent lines to

\omega

at

C_1

and

A_1

intersect at

B_2

. Prove that the lines

AA_2,BB_2

and

CC_2

are concurrent.

min, max of (1-x_1)(1-y_1)+(1-x_2)(1-y_2) when x_1^2+x_2^2=y_1^2+y_2^2=2013

Find the maximum and the minimum values of

S = (1 - x_1)(1 -y_1) + (1 - x_2)(1 - y_2)

for real numbers

x_1, x_2, y_1,y_2

with

x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013

.

colsed path along lattice points on a m x n grid of points

Adel draws an

m \times n

grid of dots on the coordinate plane, at the points of integer coordinates

(a,b)

where

1 \le a \le m

and

1 \le b \le n

. He proceeds to draw a closed path along

k

of these dots,

(a_1, b_1)

,

(a_2,b_2)

,...,

(a_k,b_k)

, such that

(a_i,b_i)

and

(a_{i+1}, b_{i+1})

(where

(a_{k+1}, b_{k+1}) = (a_1, b_1)

) are

1

unit apart for each

1 \le i \le k

. Adel makes sure his path does not cross itself, that is, the

k

dots are distinct. Find, with proof, the maximum possible value of

k

in terms of

m

and

n

.

2013 Saudi Arabia IMO TST

Subcontests

numbers 1, 2,...,2012 in a circle

each pos. integer exactly once in |a_1- a_2|,|a_2 - a_3|, ...,|a_k- a_{k+1}|,...

2^n - 1 is divisible by p(n)

a Saudi company has 2 offices, no of correspondents from each office

max of p% of all divisors of n with their unit digit equal to 3

2 circumcircles and a line concurrent

n + f(f(n)) <= 2f(n) , strictly increasing

concyclic wanted, altitudes related

a_1 + a_2 + .. + a_n >= n^2 and a_1^2 + a_2^2 + ... + a_n^2 <= n^3 + 1

Concurrent Lines

min, max of (1-x_1)(1-y_1)+(1-x_2)(1-y_2) when x_1^2+x_2^2=y_1^2+y_2^2=2013

colsed path along lattice points on a m x n grid of points

2013 Saudi Arabia IMO TST

Subcontests

numbers 1, 2,...,2012 in a circle

each pos. integer exactly once in |a_1- a_2|,|a_2 - a_3|, ...,|a_k- a_{k+1}|,...

2^n - 1 is divisible by p(n)

a Saudi company has 2 offices, no of correspondents from each office

max of p% of all divisors of n with their unit digit equal to 3

2 circumcircles and a line concurrent

n + f(f(n)) &lt;= 2f(n) , strictly increasing

concyclic wanted, altitudes related

a_1 + a_2 + .. + a_n &gt;= n^2 and a_1^2 + a_2^2 + ... + a_n^2 &lt;= n^3 + 1

Concurrent Lines

min, max of (1-x_1)(1-y_1)+(1-x_2)(1-y_2) when x_1^2+x_2^2=y_1^2+y_2^2=2013

colsed path along lattice points on a m x n grid of points

n + f(f(n)) <= 2f(n) , strictly increasing

a_1 + a_2 + .. + a_n >= n^2 and a_1^2 + a_2^2 + ... + a_n^2 <= n^3 + 1