2015 JBMO Shortlist

Part of JBMO ShortLists

Subcontests

(16)

1

integers 0-n, into unit squares such that sum of integers 2x2 differ

n \times n

n \ge 3

) is divided into

n^2

unit squares. Integers from

O

n

included, are written down: one integer in each unit square, in such a way that the sums of integers in each

2\times 2

square of the board are different. Find all

n

for which such boards exist.

1

find position of 2015 in a table

Positive integers are put into the following table.
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|} \hline 1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & & \\ \hline 2 & 5 & 9 & 14 & 20 & 27 & 35 & 44 & & \\ \hline 4 & 8 & 13 & 19 & 26 & 34 & 43 & 53 & & \\ \hline 7 & 12 & 18 & 25 & 33 & 42 & & & & \\ \hline 11 & 17 & 24 & 32 & 41 & & & & & \\ \hline 16 & 23 & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline \end{tabular}
Find the number of the line and column where the number

2015

1

number of parallelograms inside a square grid under conditions

n\ge 1

be a positive integer. A square of side length

n

is divided by lines parallel to each side into

n^2

squares of side length

1

. Find the number of parallelograms which have vertices among the vertices of the

n^2

squares of side length

1

, with both sides smaller or equal to

2

, and which have tha area equal to

2

1

504 points on a unit disc, from any 5 points we can choose 2 with distance <=1

2015

points are given in a plane such that from any five points we can choose two points with distance less than

1

unit. Prove that

504

of the given points lie on a unit disc.

1

exists k such 41 / repunit with n digits iff k / n (also 37 / n repunit iff 3/n)

A positive integer is called a repunit, if it is written only by ones. The repunit with

n

digits will be denoted as

\underbrace{{11\cdots1}}_{n}

. Prove that: α) the repunit

\underbrace{{11\cdots1}}_{n}

is divisible by

37

n

is divisible by

3

b) there exists a positive integer

k

such that the repunit

\underbrace{{11\cdots1}}_{n}

is divisible by

41

n

is divisible by

k

1

34560 / product of all differences of possible couples of 6 given pos. integers

a) Show that the product of all differences of possible couples of six given positive integers is divisible by

960

b) Show that the product of all differences of possible couples of six given positive integers is divisible by

34560

PS. a) original from Albania b) modified by problem selecting committee

1

max no of 2015 consecutive int. , no sum of any 2 is divided by their difference

What is the greatest number of integers that can be selected from a set of

2015

consecutive numbers so that no sum of any two selected numbers is divisible by their difference?

1

given x^3-3\sqrt3 x^2 +9x - 3\sqrt3 -64=0

x^3-3\sqrt3 x^2 +9x - 3\sqrt3 -64=0

find the value of

x^6-8x^5+13x^4-5x^3+49x^2-137x+2015

1

2015 JBMO Shortlist G5

ABC

be an acute triangle with

{AB\neq AC}

{\omega}

of the triangle touches the sides

{BC, CA}

{AB}

{D, E}

{F}

, respectively. The perpendicular line erected at

{C}

{BC}

{EF}

{M}

, and similarly the perpendicular line erected at

{B}

{BC}

{EF}

{N}

{DM}

{\omega}

{P}

{DN}

{\omega}

{Q}

{DP=DQ}

.
Ruben Dario & Leo Giugiuc (Romania)

1

2015 JBMO Shortlist G3

{c\equiv c\left(O, R\right)}

be a circle with center

{O}

{R}

{A, B}

be two points on it, not belonging to the same diameter. The bisector of angle

{\angle{ABO}}

intersects the circle

{c}

{C}

, the circumcircle of the triangle

AOB

{c_1}

{K}

and the circumcircle of the triangle

AOC

{{c}_{2}}

{L}

. Prove that point

{K}

is the circumcircle of the triangle

AOC

{L}

is the incenter of the triangle

AOB

.
Evangelos Psychas (Greece)

1

2015 JBMO Shortlist G2

{P}

is outside the circle

{\Omega}

. Two tangent lines, passing from the point

{P}

touch the circle

{\Omega}

{A}

{B}

{AM \left(M\in BP\right)}

intersects the circle

{\Omega}

{C}

{PC}

intersects again the circle

{\Omega}

{D}

. Prove that the lines

{AD}

{BP}

are parallel.
(Moldova)

1

2015 JBMO Shortlist G1

Around the triangle

ABC

the circle is circumscribed, and at the vertex

{C}

{t}

to this circle is drawn. The line

{p}

, which is parallel to this tangent intersects the lines

{BC}

{AC}

{D}

{E}

, respectively. Prove that the points

A,B,D,E

belong to the same circle.
(Montenegro)

1

Inequality very interesting.

a,b,c

are positive real numbers prove that:

\frac{a}{b}+\sqrt{\frac{b}{c}}+\sqrt[3]{\frac{c}{a}}>2.

1

Number theory easy

Check if there exists positive integers

a, b

and prime number

p

a^3-b^3=4p^2

1

Simple inequality

The positive real

x, y, z

x^2+y^2+z^2 = 3

\frac{x^2+yz}{x^2+yz +1}+\frac{y^2+zx}{y^2+zx+1}+\frac{z^2+xy}{z^2+xy+1}\leq 2

1

inequality

Let x; y; z be real numbers, satisfying the relations

x \ge 20

y \ge 40 z \ge 1675 x + y + z = 2015 Find the greatest value of the product P = xy z