MathDB

p1. For any number of positive reals

a, b, c

with

a + b + c = 1

, determine the value of

ab\left(\frac{a^2+b^2}{a^3+b^3}\right)+bc\left(\frac{b^2+c^2}{b^3+c^3}\right)+ca\left(\frac{c^2+a^2}{c^3+a^3}\right)+\frac{a^4+b^4}{a^3+b^3}+\frac{b^4+c^4}{b^3+c^3}+\frac{c^4+a^4}{c^3+a^3}

[url=https://artofproblemsolving.com/community/c6h2371565p19388497]p2. Given an acute triangle

ABC

with

AB <AC

. The ex-circles of triangle

ABC

opposite

B

and

C

are centered on

B_1

and

C_1

, respectively. Let

D

be the midpoint of

B_1C_1

. Suppose that

E

is the point of intersection of

AB

and

CD

, and

F

is the point of intersection of

AC

and

BD

. If

EF

intersects

BC

at point

G

, prove that

AG

is the bisector of

\angle BAC

.
p3. It is known that

X

is a set with

102

elements. Suppose

A_1, A_2, ..., A_{101}

is a set of subset of

X

such that the sum of every

50

of them has more than

100

elements. Prove that there are

1 \le i <j <k \le 101

such that

A_i \cap A_j

A_i \cap A_k

and

A_k \cap A_j

are not empty.
[url=https://artofproblemsolving.com/community/c6h2371573p19388630]p4. Let

\Gamma

be the circumcircle of triangle

ABC

. One circle

\omega

is tangent to

\Gamma

A

and tangent to

BC

N

. Suppose that the extension of

AN

crosses

\Gamma

again at

E

. Let

AD

and

AF

be respectively the line of altitude

ABC

and diameter of

\Gamma

, show that

AN \times AE = AD \times AF = AB \times AC

p5. Suppose

\{a_n\}

is a sequence of integers that satisfies

a_1 = 2

a_2 = 8

and

a_{n+2} = 3a_{n+1}-a_n + 5(-1)^n.

a) Is

a_{2014}

prime? b) Prove that for every odd number

m

, the number

\frac{a_m + a_{4m}}{a_{2m} + a_{3m}}

is an integer.

Problem Statement