MathDB

2016 JBMO Shortlist G2

Source: 2016 JBMO Shortlist G2

10/8/2017

Let

{ABC}

be a triangle with

\angle BAC={{60}^{{}^\circ }}

. Let

D

and

E

be the feet of the perpendiculars from

{A}

to the external angle bisectors of

\angle ABC

and

\angle ACB

, respectively. Let

{O}

be the circumcenter of the triangle

{ABC}

. Prove that the circumcircles of the triangles

{ADE}

and

{BOC}

are tangent to each other.

geometryJBMO

least cardinality from 1-50 so they have non prime sum

Source: JBMO 2016 Shortlist C2

10/14/2017

The natural numbers from

1

to

50

are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?

JBMOcombinatoricsSumprime

max number so that 11 divides sums of products of naturals

Source: JBMO 2016 Shortlist N2

10/14/2017

Find the maximum number of natural numbers

x_1,x_2, ... , x_m

satisfying the conditions: a) No

x_i - x_j , 1 \le i < j \le m

is divisible by

11

, and b) The sum

x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}

is divisible by

11

.

JBMOnumber theorydivides

2

Problems(3)