MathDB

1

prime divisors of n^{2}+n+1

Prove that there are infinitely many positive integers

n

for which all the prime divisors of n^{2}\plus{}n\plus{}1 are not more then

\sqrt{n}

. Stronger one. Prove that there are infinitely many positive integers

n

for which all the prime divisors of n^{3}\minus{}1 are not more then

\sqrt{n}

.

10

1

combinatoric geometry

Find all positive integers

n

such that acute-angled

\triangle ABC

with

\angle BAC<\frac{\pi}{4}

could be divided into

n

quadrilateral. Every quadrilateral is inscribed in circle and radiuses of circles are in geometric progression. be carefull ! :lol:

8

1

minimum and local maximum

F(x)

is polynomial with real coefficients. F(x) \equal{} x^{4}\plus{}a_{1}x^{3}\plus{}a_{2}x^{2}\plus{}a_{1}x^{1}\plus{}a_{0}.

M

is local maximum and

m

is minimum. Prove that \frac{3}{10}(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}< M\minus{}m < 3(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}

7

1

nice easy problem

There are 25 people. Every two of them are use some language to speak between. They use only one language even if they both know another one. Among every three of them there is one who speaking with two other on the same language. Prove that there exist one who speaking with 10 other on the same language.

4

1

function from Q to Q

Find all functions

f: \mathbb Q \to \mathbb Q

such that f(x^{2}\plus{}y\plus{}f(xy)) \equal{} 3\plus{}(x\plus{}f(y)\minus{}2)f(x) for all

x,y \in \mathbb Q

.

3

1

(x-1)^{k} | P(x)

It is known that

k

and

n

are positive integers and k \plus{} 1\leq\sqrt {\frac {n \plus{} 1}{\ln(n \plus{} 1)}}. Prove that there exists a polynomial

P(x)

of degree

n

with coefficients in the set \{0,1, \minus{} 1\} such that (x \minus{} 1)^{k} divides

P(x)

.

5

1

some constructions with altitudes and parallel lines

AA_{3}

and

BB_{3}

are altitudes of acute-angled

\triangle ABC

. Points

A_{1}

and

B_{1}

are second points of intersection lines

AA_{3}

and

BB_{3}

with circumcircle of

\triangle ABC

respectively.

A_{2}

and

B_{2}

are points on

BC

and

AC

respectively.

A_{1}A_{2}\parallel AC

,

B_{1}B_{2}\parallel BC

. Point

M

is midpoint of

A_{2}B_{2}

. \angle BCA \equal{} x. Find

\angle A_{3}MB_{3}

.

2

1

angle bisectors in trapezoid

ABCD

is convex

AD\parallel BC

,

AC\perp BD

.

M

is interior point of

ABCD

which is not a intersection of diagonals

AC

and

BD

such that \angle AMB \equal{}\angle CMD \equal{}\frac{\pi}{2} .

P

is intersection of angel bisectors of

\angle A

and

\angle C

.

Q

is intersection of angel bisectors of

\angle B

and

\angle D

. Prove that \angle PMB \equal{}\angle QMC.

1

Ukraine 2007

\{a,b,c\}\subset\left(\frac{1}{\sqrt6},+\infty\right)

such that

a^{2}+b^{2}+c^{2}=1.

Prove that

\frac{1+a^{2}}{\sqrt{2a^{2}+3ab-c^{2}}}+\frac{1+b^{2}}{\sqrt{2b^{2}+3bc-a^{2}}}+\frac{1+c^{2}}{\sqrt{2c^{2}+3ca-b^{2}}}\ge2(a+b+c).

6

1

Everything is sum of two cubes...

Find all primes

p

for that there is an integer

n

such that there are no integers

x,y

with x^3 \plus{} y^3 \equiv n \mod p (so not all residues are the sum of two cubes). E.g. for p \equal{} 7, one could set n \equal{} \pm 3 since

x^3,y^3 \equiv 0 , \pm 1 \mod 7

, thus x^3 \plus{} y^3 \equiv 0 , \pm 1 , \pm 2 \mod 7 only.

2007 Ukraine Team Selection Test

Subcontests

prime divisors of n^{2}+n+1

combinatoric geometry

minimum and local maximum

nice easy problem

function from Q to Q

(x-1)^{k} | P(x)

some constructions with altitudes and parallel lines

angle bisectors in trapezoid

Ukraine 2007

Everything is sum of two cubes...