MathDB

Problem 8 of First round

Source: I International Festival of Young Mathematicians Sozopol 2010, Theme for 10-12 grade

12/13/2019

Find all polynomials

f(x)

with integer coefficients and leading coefficient equal to 1, for which

f(0)=2010

and for each irrational

x

,

f(x)

is also irrational.

algebrapolynomial

equal angles inside trapezoid lead to equal segments (HOMC 2016 Junior Q12)

Source:

7/20/2019

In the trapezoid

ABCD, AB // CD

and the diagonals intersect at

O

. The points

P, Q

are on

AD, BC

respectively such that

\angle AP B = \angle CP D

and

\angle AQB = \angle CQD

. Show that

OP = OQ

.

geometrytrapezoidequal segmentsequal angles

Problem 8 of Third round

Source: I International Festival of Young Mathematicians Sozopol 2010, Theme for 10-12 grade

12/14/2019

Let

m, n,

and

k

be natural numbers, where

n

is odd. Prove that

\frac{1}{m}+\frac{1}{m+n}+...+\frac{1}{m+kn}

is not a natural number.

number theorymodulo

solve equation in R #3

Source:

9/29/2010

Solve this equation with

x \in R

:

x^3-3x=\sqrt{x+2}

algebra unsolvedalgebra

Problem 8 of Finals

Source: I International Festival of Young Mathematicians Sozopol 2010, Theme for 10-12 grade

12/16/2019

Let

k

be a circle and

l

–line that is tangent to

k

in point

P

. On

l

from the two sides of

P

are chosen arbitrary points

A

and

B

. The tangents through

A

and

B

to

k

, different than

l

, intersect in point

C

. Find the geometric place of points

C

, when

A

and

B

change in such way so that

AP.BP

is a constant.

geometry

8

Problems(5)