MathDB

[url=https://artofproblemsolving.com/community/c675693]Macedonia JBMO TST 2017
[url=http://artofproblemsolving.com/community/c6h1663908p10569198]Problem 1. Let

p

be a prime number such that

3p+10

is a sum of squares of six consecutive positive integers. Prove that

p-7

is divisible by

36

.
[url=http://artofproblemsolving.com/community/c6h1663916p10569261]Problem 2. In the triangle

ABC

, the medians

AA_1

BB_1

, and

CC_1

are concurrent at a point

T

such that

BA_1=TA_1

. The points

C_2

and

B_2

are chosen on the extensions of

CC_1

and

BB_2

, respectively, such that C_1C_2 = \frac{CC_1}{3} \text{and} B_1B_2 = \frac{BB_1}{3}. Show that

TB_2AC_2

is a rectangle.
[url=http://artofproblemsolving.com/community/c6h1663918p10569305]Problem 3. Let

x,y,z

be positive reals such that

xyz=1

. Show that

\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.

When does equality happen?
[url=http://artofproblemsolving.com/community/c6h1663920p10569326]Problem 4. In triangle

ABC

, the points

X

and

Y

are chosen on the arc

BC

of the circumscribed circle of

ABC

that doesn't contain

A

so that

\measuredangle BAX = \measuredangle CAY

. Let

M

be the midpoint of the segment

AX

. Show that

BM + CM > AY.

[url=http://artofproblemsolving.com/community/c6h1663922p10569370]Problem 5. Find all the positive integers

n

so that

n

has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of

n

Source

Problems(1)