MathDB

1

intersection of circles is intersection of perpendicular bisectors

ABC

be a triangle with

AB>AC

. Let

D

be a point on side

AB

such that

BD=AC

. Consider the circle

\gamma

passing through point

D

and tangent to side

AC

at point

A

. Consider the circumscribed circle

\omega

of the triangle

ABC

that interesects the circle

\gamma

at points

A

and

E

. Prove that point

E

is the intersection point of the perpendicular bisectors of line segments

BC

and

AD

.

2

1

sum (a+b)/(a^2+ab+b^2) <=2 if 1/a+1/b+1/c =3 for a,b,c>0

Let

a,b,c

be positive real numbers such that

\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}=3

. Prove that

\frac{a+b}{a^2+ab+b^2}+ \frac{b+c}{b^2+bc+c^2}+ \frac{c+a}{c^2+ca+a^2}\le 2

When is the equality valid?

3

1

3a^2b+16ab^2 is perfect square for primes a,b >0

Find all pairs

(a,b)

of prime positive integers

a,b

such that number

A=3a^2b+16ab^2

equals to a square of an integer.

4

1

min sum of products of elements of 2 dijoint sets whose union is {1,...,8}

Let

A

and

B

be two non-empty subsets of

X = \{1, 2, . . . , 8 \}

with

A \cup B = X

and

A \cap B = \emptyset

. Let

P_A

be the product of all elements of

A

and let

P_B

be the product of all elements of

B

. Find the minimum possible value of sum

P_A +P_B

.
PS. It is a variation of [url=https://artofproblemsolving.com/community/c6h2267998p17621980]JBMO Shortlist 2019 A3

2020 Greece JBMO TST

Subcontests

intersection of circles is intersection of perpendicular bisectors

sum (a+b)/(a^2+ab+b^2) <=2 if 1/a+1/b+1/c =3 for a,b,c>0

3a^2b+16ab^2 is perfect square for primes a,b >0

min sum of products of elements of 2 dijoint sets whose union is {1,...,8}

2020 Greece JBMO TST

Subcontests

intersection of circles is intersection of perpendicular bisectors

sum (a+b)/(a^2+ab+b^2) &lt;=2 if 1/a+1/b+1/c =3 for a,b,c&gt;0

3a^2b+16ab^2 is perfect square for primes a,b &gt;0

min sum of products of elements of 2 dijoint sets whose union is {1,...,8}

sum (a+b)/(a^2+ab+b^2) <=2 if 1/a+1/b+1/c =3 for a,b,c>0

3a^2b+16ab^2 is perfect square for primes a,b >0