MathDB

3

a^2_1 + a^2_2 + ... + a^2_100 = 1

a_1, a_2, \ldots, a_{100}

be nonnegative real numbers such that a^2_1 \plus{} a^2_2 \plus{} \ldots \plus{} a^2_{100} \equal{} 1. Prove that a^2_1 \cdot a_2 \plus{} a^2_2 \cdot a_3 \plus{} \ldots \plus{} a^2_{100} \cdot a_1 < \frac {12}{25}. Author: Marcin Kuzma, Poland

Find the smallest positive real number

Determine the smallest positive real number

k

with the following property. Let

ABCD

be a convex quadrilateral, and let points

A_1

,

B_1

,

C_1

, and

D_1

lie on sides

AB

,

BC

,

CD

, and

DA

, respectively. Consider the areas of triangles

AA_1D_1

,

BB_1A_1

,

CC_1B_1

and

DD_1C_1

; let

S

be the sum of the two smallest ones, and let

S_1

be the area of quadrilateral

A_1B_1C_1D_1

. Then we always have

kS_1\ge S

.
Author: Zuming Feng and Oleg Golberg, USA

Select an easier sub-problem and you get an IMO problem

Let

k

be a positive integer. Prove that the number (4 \cdot k^2 \minus{} 1)^2 has a positive divisor of the form 8kn \minus{} 1 if and only if

k

is even. [url=http://www.mathlinks.ro/viewtopic.php?p=894656#894656]Actual IMO 2007 Problem, posed as question 5 in the contest, which was used as a lemma in the official solutions for problem N6 as shown above. Author: Kevin Buzzard and Edward Crane, United Kingdom

3

4

7

3

2p pairwise distinct subsets s.t. intersection non-empty

Let \alpha < \frac {3 \minus{} \sqrt {5}}{2} be a positive real number. Prove that there exist positive integers

n

and

p > \alpha \cdot 2^n

for which one can select

2 \cdot p

pairwise distinct subsets

S_1, \ldots, S_p, T_1, \ldots, T_p

of the set

\{1,2, \ldots, n\}

such that

S_i \cap T_j \neq \emptyset

for all

1 \leq i,j \leq p

Author: Gerhard Wöginger, Austria

circumradius, incenter, ...

Given an acute triangle

ABC

with

\angle B > \angle C

. Point

I

is the incenter, and

R

the circumradius. Point

D

is the foot of the altitude from vertex

A

. Point

K

lies on line

AD

such that AK \equal{} 2R, and

D

separates

A

and

K

. Lines

DI

and

KI

meet sides

AC

and

BC

at

E,F

respectively. Let IE \equal{} IF. Prove that

\angle B\leq 3\angle C

. Author: Davoud Vakili, Iran

Prime factorisation of n!

For a prime

p

and a given integer

n

let

\nu_p(n)

denote the exponent of

p

in the prime factorisation of

n!

. Given

d \in \mathbb{N}

and

\{p_1,p_2,\ldots,p_k\}

a set of

k

primes, show that there are infinitely many positive integers

n

such that

d\mid \nu_{p_i}(n)

for all

1 \leq i \leq k

.
Author: Tejaswi Navilarekkallu, India

4

3

2007 SL A4

Find all functions f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} } satisfying f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right) for all pairs of positive reals

x

and

y

. Here, \mathbb{R}^{ \plus{} } denotes the set of all positive reals.
Proposed by Paisan Nakmahachalasint, Thailand

Minumum difference of sums

Let A_0 \equal{} (a_1,\dots,a_n) be a finite sequence of real numbers. For each

k\geq 0

, from the sequence A_k \equal{} (x_1,\dots,x_k) we construct a new sequence A_{k \plus{} 1} in the following way. 1. We choose a partition \{1,\dots,n\} \equal{} I\cup J, where

I

and

J

are two disjoint sets, such that the expression \left|\sum_{i\in I}x_i \minus{} \sum_{j\in J}x_j\right| attains the smallest value. (We allow

I

or

J

to be empty; in this case the corresponding sum is 0.) If there are several such partitions, one is chosen arbitrarily. 2. We set A_{k \plus{} 1} \equal{} (y_1,\dots,y_n) where y_i \equal{} x_i \plus{} 1 if

i\in I

, and y_i \equal{} x_i \minus{} 1 if

i\in J

. Prove that for some

k

, the sequence

A_k

contains an element

x

such that

|x|\geq\frac n2

. Author: Omid Hatami, Iran

2^{3k} divides difference of binomials but not 2^{3k+1}

For every integer

k \geq 2,

prove that

2^{3k}

divides the number \binom{2^{k \plus{} 1}}{2^{k}} \minus{} \binom{2^{k}}{2^{k \minus{} 1}} but 2^{3k \plus{} 1} does not.
Author: Waldemar Pompe, Poland

8

2

Convex polygon and equilateral triangles

Given is a convex polygon

P

with

n

vertices. Triangle whose vertices lie on vertices of

P

is called good if all its sides are unit length. Prove that there are at most

\frac {2n}{3}

good triangles.
Author: Vyacheslav Yasinskiy, Ukraine

A nice collinearity problem

Point

P

lies on side

AB

of a convex quadrilateral

ABCD

. Let

\omega

be the incircle of triangle

CPD

, and let

I

be its incenter. Suppose that

\omega

is tangent to the incircles of triangles

APD

and

BPC

at points

K

and

L

, respectively. Let lines

AC

and

BD

meet at

E

, and let lines

AK

and

BL

meet at

F

. Prove that points

E

,

I

, and

F

are collinear. Author: Waldemar Pompe, Poland

2

3

f(m + n) >= f(m) + f(f(n)) - 1

Consider those functions

f: \mathbb{N} \mapsto \mathbb{N}

which satisfy the condition f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1 for all

m,n \in \mathbb{N}.

Find all possible values of

f(2007).

Author: Nikolai Nikolov, Bulgaria

MTB - CTM does not depend on choice of X

Denote by

M

midpoint of side

BC

in an isosceles triangle

\triangle ABC

with

AC = AB

. Take a point

X

on a smaller arc \overarc{MA} of circumcircle of triangle

\triangle ABM

. Denote by

T

point inside of angle

BMA

such that

\angle TMX = 90

and

TX = BX

.
Prove that

\angle MTB - \angle CTM

does not depend on choice of

X

.
Author: Farzan Barekat, Canada

Integer a_k such that b - a^n_k is divisible by k

Let

b,n > 1

be integers. Suppose that for each

k > 1

there exists an integer

a_k

such that

b - a^n_k

is divisible by

k

. Prove that

b = A^n

for some integer

A

.
Author: Dan Brown, Canada

1

2

Find all sequences satisfying two conditions

Let

n > 1

be an integer. Find all sequences a_1, a_2, \ldots a_{n^2 \plus{} n} satisfying the following conditions: \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n;
\text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n. Author: Dusan Dukic, Serbia

7^a - 3^b divides a^4 + b^2 (from IMO Shortlist 2007)

Find all pairs of natural numbers

(a, b)

such that 7^a \minus{} 3^b divides a^4 \plus{} b^2.
Author: Stephan Wagner, Austria

5

4

2007 IMO Shortlist

Subcontests

a^2_1 + a^2_2 + ... + a^2_100 = 1

Find the smallest positive real number

Select an easier sub-problem and you get an IMO problem

2p pairwise distinct subsets s.t. intersection non-empty

circumradius, incenter, ...

Prime factorisation of n!

2007 SL A4

Minumum difference of sums

2^{3k} divides difference of binomials but not 2^{3k+1}

Convex polygon and equilateral triangles

A nice collinearity problem

f(m + n) >= f(m) + f(f(n)) - 1

MTB - CTM does not depend on choice of X

Integer a_k such that b - a^n_k is divisible by k

Find all sequences satisfying two conditions

7^a - 3^b divides a^4 + b^2 (from IMO Shortlist 2007)

2007 IMO Shortlist

Subcontests

a^2_1 + a^2_2 + ... + a^2_100 = 1

Find the smallest positive real number

Select an easier sub-problem and you get an IMO problem

2p pairwise distinct subsets s.t. intersection non-empty

circumradius, incenter, ...

Prime factorisation of n!

2007 SL A4

Minumum difference of sums

2^{3k} divides difference of binomials but not 2^{3k+1}

Convex polygon and equilateral triangles

A nice collinearity problem

f(m + n) &gt;= f(m) + f(f(n)) - 1

MTB - CTM does not depend on choice of X

Integer a_k such that b - a^n_k is divisible by k

Find all sequences satisfying two conditions

7^a - 3^b divides a^4 + b^2 (from IMO Shortlist 2007)

f(m + n) >= f(m) + f(f(n)) - 1