MathDB

Σ(ab + 4)/(a + 2) >=6 if abc = 8, a,b,c>0

Source: JBMO 2016 Shortlist A1

10/14/2017

Let

a, b, c

be positive real numbers such that

abc = 8

. Prove that

\frac{ab + 4}{a + 2}+\frac{bc + 4}{b + 2}+\frac{ca + 4}{c + 2}\ge 6

.

JBMOalgebrainequalities

least positive integer k so that k! S_{2016} \in Z

Source: JBMO Shortlist 2016 C1

10/14/2017

Let

S_n

be the sum of reciprocal values of non-zero digits of all positive integers up to (and including)

n

. For instance,

S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \frac{1}{1}+ \frac{1}{3}

. Find the least positive integer

k

making the number

k!\cdot S_{2016}

an integer.

Sumcombinatoricspositive integers

2016 JBMO Shortlist G1

Source: 2016 JBMO Shortlist G1

10/8/2017

Let

{ABC}

be an acute angled triangle, let

{O}

be its circumcentre, and let

{D,E,F}

be points on the sides

{BC,CA,AB}

, respectively. The circle

{(c_1)}

of radius

{FA}

, centered at

{F}

, crosses the segment

{OA}

at

{A'}

and the circumcircle

{(c)}

of the triangle

{ABC}

again at

{K}

. Similarly, the circle

{(c_2)}

of radius

DB

, centered at

D

, crosses the segment

\left( OB \right)

at

{B}'

and the circle

{(c)}

again at

{L}

. Finally, the circle

{(c_3)}

of radius

EC

, centered at

E

, crosses the segment

\left( OC \right)

at

{C}'

and the circle

{(c)}

again at

{M}

. Prove that the quadrilaterals

BKF{A}',CLD{B}'

and

AME{C}'

are all cyclic, and their circumcircles share a common point.
Evangelos Psychas (Greece)

geometryJBMO

largest n so that n | p^6 - 1 for all primes p>7

Source: JBMO 2016 Shortlist N1

10/14/2017

Determine the largest positive integer

n

that divides

p^6 - 1

for all primes

p > 7

.

JBMOnumber theoryprime

1

Problems(4)