MathDB

2

a_n = A_n + B, M^2 smallest square, M < A +\sqrt{B}

A

and

B

be positive integers. Define the arithmetic sequence

a_0, a_1, a_2, ...

by

a_n = A_n + B

. Suppose that there exists an

n\ge 0

such that

a_n

is a square. Let

M

be a positive integer such that

M^2

is the smallest square in the sequence. Prove that

M < A +\sqrt{B}

.

functional with max, f : R \to R , f(x) = max{y\in R} (2xy - f(y)),

Find all functions

f : R \to R

which satisfy

f(x) = max_{y\in R} (2xy - f(y))

for all

x \in R

.

5

2

3(x^2 + y^2 + z^2) = 1 , x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3, system 3x3

Find all triples

(x,y, z)

of real (but not necessarily positive) numbers satisfying

3(x^2 + y^2 + z^2) = 1

,

x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3

.

for each prime p there is k : A(k),A(k + 1) are divisible by p, trinomial

The polynomial

A(x) = x^2 + ax + b

with integer coefficients has the following property: for each prime

p

there is an integer

k

such that

A(k)

and

A(k + 1)

are both divisible by

p

. Proof that there is an integer

m

such that

A(m) = A(m + 1) = 0

.

3

2

n|(p-1) and p | (n^3-1) => (4p-3) square

Let n\ge 2 be a positive integer and

p

a prime such that

n|p-1

and

p | n^3-1

. Show

4p-3

is a square.

M(a, b) = a -1/b +b(b + 3/a) is integer => square also

(a) Let

a

and

b

be positive integers such that

M(a, b) = a - \frac1b +b(b + \frac3a)

is an integer. Prove that

M(a,b)

is a square. (b) Find nonzero integers

a

and

b

such that

M(a,b)

is a positive integer, but not a square.

1

2

min a_{2010}, a_n < a_{n+1}, a_i + a_l > a_j + a_k

Consider sequences

a_1, a_2, a_3,...

of positive integers. Determine the smallest possible value of

a_{2010}

if (i)

a_n < a_{n+1}

for all

n\ge 1

, (ii)

a_i + a_l > a_j + a_k

for all quadruples

(i, j, k, l)

which satisfy

1 \le i < j \le k < l

.

<A=45^o, CD \perp AB, P\perp BC iff |AP| = |BC|.

Let

ABC

be an acute triangle such that

\angle BAC = 45^o

. Let

D

a point on

AB

such that

CD \perp AB

. Let

P

be an internal point of the segment

CD

. Prove that

AP\perp BC

if and only if

|AP| = |BC|

.

4

2

cyclic ABCD with <ABD <DBC given, cyclic from 2 midpoints wanted

Let

ABCD

be a cyclic quadrilateral satisfying

\angle ABD = \angle DBC

. Let

E

be the intersection of the diagonals

AC

and

BD

. Let

M

be the midpoint of

AE

, and

N

be the midpoint of

DC

. Show that

MBCN

is a cyclic quadrilateral.

square inscribed in circle, another tangent circle, equal segments wanted

Let

ABCD

be a square with circumcircle

\Gamma_1

. Let

P

be a point on the arc

AC

that also contains

B

. A circle

\Gamma_2

touches

\Gamma_1

in

P

and also touches the diagonal

AC

in

Q

. Let

R

be a point on

\Gamma_2

such that the line

DR

touches

\Gamma_2

. Proof that

|DR| = |DA|

.

2010 Dutch IMO TST

Subcontests

a_n = A_n + B, M^2 smallest square, M < A +\sqrt{B}

functional with max, f : R \to R , f(x) = max{y\in R} (2xy - f(y)),

3(x^2 + y^2 + z^2) = 1 , x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3, system 3x3

for each prime p there is k : A(k),A(k + 1) are divisible by p, trinomial

n|(p-1) and p | (n^3-1) => (4p-3) square

M(a, b) = a -1/b +b(b + 3/a) is integer => square also

min a_{2010}, a_n < a_{n+1}, a_i + a_l > a_j + a_k

<A=45^o, CD \perp AB, P\perp BC iff |AP| = |BC|.

cyclic ABCD with <ABD <DBC given, cyclic from 2 midpoints wanted

square inscribed in circle, another tangent circle, equal segments wanted

2010 Dutch IMO TST

Subcontests

a_n = A_n + B, M^2 smallest square, M &lt; A +\sqrt{B}

functional with max, f : R \to R , f(x) = max{y\in R} (2xy - f(y)),

3(x^2 + y^2 + z^2) = 1 , x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3, system 3x3

for each prime p there is k : A(k),A(k + 1) are divisible by p, trinomial

n|(p-1) and p | (n^3-1) =&gt; (4p-3) square

M(a, b) = a -1/b +b(b + 3/a) is integer =&gt; square also

min a_{2010}, a_n &lt; a_{n+1}, a_i + a_l &gt; a_j + a_k

&lt;A=45^o, CD \perp AB, P\perp BC iff |AP| = |BC|.

cyclic ABCD with &lt;ABD &lt;DBC given, cyclic from 2 midpoints wanted

square inscribed in circle, another tangent circle, equal segments wanted

a_n = A_n + B, M^2 smallest square, M < A +\sqrt{B}

n|(p-1) and p | (n^3-1) => (4p-3) square

M(a, b) = a -1/b +b(b + 3/a) is integer => square also

min a_{2010}, a_n < a_{n+1}, a_i + a_l > a_j + a_k

<A=45^o, CD \perp AB, P\perp BC iff |AP| = |BC|.

cyclic ABCD with <ABD <DBC given, cyclic from 2 midpoints wanted