2011 JBMO Shortlist

Part of JBMO ShortLists

Subcontests

(9)

2

sums with minimums in R

x_1,x_2, ..., x_n

be real numbers satisfying

\sum_{k=1}^{n-1} min(x_k; x_{k+1}) = min(x_1; x_n)

\sum_{k=2}^{n-1} x_k \ge 0

2011 in plane, distance <1, 1005 points outside unit circles

Decide if it is possible to consider

2011

points in a plane such that the distance between every two of these points is different from

1

and each unit circle centered at one of these points leaves exactly

1005

points outside the circle.

2

decipher (LARN-ACA): (CYP +RUS) = CYP RUS,

Decipher the equality

(\overline{LARN} -\overline{ACA}) : (\overline{CYP} +\overline{RUS}) = C^{Y^P} \cdot R^{U^S}

where different symbols correspond to different digits and equal symbols correspond to equal digits. It is also supposed that all these digits are different from

0

reflection of vertices wrt diagonal not outside polygon

Determine the polygons with

n

(n \ge 4)

, not necessarily convex, which satisfy the property that the reflection of every vertex of polygon with respect to every diagonal of the polygon does not fall outside the polygon. Note: Each segment joining two non-neighboring vertices of the polygon is a diagonal. The reflection is considered with respect to the support line of the diagonal.

3

&Sigma;(x + 2y)(z + 2x + 3y) <= 3/2

x, y, z

be positive real numbers. Prove that:

\frac{x + 2y}{z + 2x + 3y}+\frac{y + 2z}{x + 2y + 3z}+\frac{z + 2x}{y + 2z + 3x} \le \frac{3}{2}

2011 JBMO Shortlist G2

AD,BF

{CE}

be the altitudes of

\vartriangle ABC

. A line passing through

{D}

and parallel to

{AB}

intersects the line

{EF}

{G}

{H}

is the orthocenter of

\vartriangle ABC

, find the angle

{\angle{CGH}}

splitting an equilateral into 2011 triangles with 122 lines

Can we divide an equilateral triangle

\vartriangle ABC

2011

small triangles using

122

straight lines? (there should be

2011

triangles that are not themselves divided into smaller parts and there should be no polygons which are not triangles)

3

circles inside a square

Inside of a square whose side length is

1

there are a few circles such that the sum of their circumferences is equal to

10

. Show that there exists a line that meets at least four of these circles.

2011 JBMO Shortlist G1

ABC

be an isosceles triangle with

AB=AC

. On the extension of the side

{CA}

we consider the point

{D}

{AD<AC}

. The perpendicular bisector of the segment

{BD}

meets the internal and the external bisectors of the angle

\angle BAC

{E}

{Z}

, respectively. Prove that the points

{A, E, D, Z}

1005^x + 2011^y = 1006^z

Solve in positive integers the equation

1005^x + 2011^y = 1006^z

1

Simple inequality

\displaystyle {x_i> 1, \forall i \in \left \{1, 2, 3, \ldots, 2011 \right \}}

\displaystyle{\frac{x^2_1}{x_2-1}+\frac{x^2_2}{x_3-1}+\frac{x^2_3}{x_4-1}+\ldots+\frac{x^2_{2010}}{x_{2011}-1}+\frac{x^2_{2011}}{x_1-1}\geq 8044}

When the equality holds?

4

4

4

2

Inequality with abc=1

\boxed{\text{A7}}

a,b,c

be positive reals such that

abc=1

.Prove the inequality

\sum\frac{2a^2+\frac{1}{a}}{b+\frac{1}{a}+1}\geq 3

dimensions of max of squares filling a rectangle by half

Consider a rectangle whose lengths of sides are natural numbers. If someone places as many squares as possible, each with area

3

, inside of the given rectangle, such that the sides of the squares are parallel to the rectangle sides, then the maximal number of these squares fill exactly half of the area of the rectangle. Determine the dimensions of all rectangles with this property.