2016 JBMO Shortlist

Part of JBMO ShortLists

Subcontests

(7)

1

2016 JBMO Shortlist G7

{AB}

be a chord of a circle

{(c)}

{O}

{K}

be a point on the segment

{AB}

{AK<BK}

. Two circles through

{K}

, internally tangent to

{(c)}

{A}

{B}

, respectively, meet again at

{L}

{P}

be one of the points of intersection of the line

{KL}

{(c)}

, and let the lines

{AB}

{LO}

{M}

. Prove that the line

{MP}

is tangent to the circle

{(c)}

.
Theoklitos Paragyiou (Cyprus)

1

2016 JBMO Shortlist G6

Given an acute triangle

{ABC}

, erect triangles

{ABD}

{ACE}

externally, so that

{\angle ADB= \angle AEC=90^o}

{\angle BAD= \angle CAE}

{{A}_{1}}\in BC,{{B}_{1}}\in AC

{{C}_{1}}\in AB

be the feet of the altitudes of the triangle

{ABC}

K

{K,L}

be the midpoints of

[ B{{C}_{1}} ]

{BC_1, CB_1}

, respectively. Prove that the circumcenters of the triangles

AKL,{{A}_{1}}{{B}_{1}}{{C}_{1}}

{DEA_1}

are collinear.
(Bulgaria)

3

2016 JBMO Shortlist G2

{ABC}

be a triangle with

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

D

E

be the feet of the perpendiculars from

{A}

to the external angle bisectors of

\angle ABC

\angle ACB

, respectively. Let

{O}

be the circumcenter of the triangle

{ABC}

. Prove that the circumcircles of the triangles

{ADE}

{BOC}

are tangent to each other.

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

The natural numbers from

1

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?

max number so that 11 divides sums of products of naturals

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

4

2

Find all the pairs of integers $ (m, n)$

Find all the pairs of integers

(m, n)

\sqrt {n +\sqrt {2016}} +\sqrt {m-\sqrt {2016}} \in \mathbb {Q}.

(2^{4n+2}+1)/65 (i) integer, (ii) prime

Find all positive integers

n

such that the number

A_n =\frac{ 2^{4n+2}+1}{65}

is a) an integer, b) a prime.

3

Inequality

If the non-negative reals

x,y,z

x^2+y^2+z^2=x+y+z

\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.

When does the equality occur?
Proposed by Dorlir Ahmeti, Albania

splitting of planar polygons, largest integer of triangles

A splitting of a planar polygon is a finite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer

n \ge 3

, determine the largest integer

m

such that no planar

n

-gon splits into less than

m

2016 JBMO Shortlist G4

{ABC}

be an acute angled triangle whose shortest side is

{BC}

. Consider a variable point

{P}

{BC}

{D}

{E}

{AB}

{AC}

, respectively, such that

{BD=BP}

{CP=CE}

. Prove that, as

{P}

{BC}

, the circumcircle of the triangle

{ADE}

passes through a fixed point.

3

Inequality From JBMO

x,y,z

be positive real numbers such that

x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.

x+y+z\geq \sqrt{\frac{xy+1}{2}}+\sqrt{\frac{yz+1}{2}}+\sqrt{\frac{zx+1}{2}} \ .

Proposed by Azerbaijan
[hide=Second Suggested Version]Let

x,y,z

be positive real numbers such that

x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.

x+y+z\geq \sqrt{\frac{x^2+1}{2}}+\sqrt{\frac{y^2+1}{2}}+\sqrt{\frac{z^2+1}{2}} \ .

2016 JBMO Shortlist G5

ABC

be an acute angled triangle with orthocenter

{H}

and circumcenter

{O}

. Assume the circumcenter

{X}

{BHC}

lies on the circumcircle of

{ABC}

O

{X}

{O'}

, and let the lines

{XH}

{O'A}

{K}

L,M

N

be the midpoints of

\left[ XB \right],\left[ XC \right]

\left[ BC \right]

, respectively. Prove that the points

K,L,M

{N}

(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) =\overline{abcd}

Determine all four-digit numbers

\overline{abcd}

(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) =\overline{abcd}