2009 JBMO Shortlist

Part of JBMO ShortLists

Subcontests

(5)

2

tiling a table mxn with corner figures

Determine all pairs of

(m, n)

such that is possible to tile the table

m \times n

with figure ”corner” as in figure with condition that in that tilling does not exist rectangle (except

m \times n

) regularly covered with figures.

p_1^2+ p_2^2+ ... + p_{12}^2 = p_{13}^2

Determine all prime numbers

p_1, p_2,..., p_{12}, p_{13}, p_1 \le p_2 \le ... \le p_{12} \le p_{13}

p_1^2+ p_2^2+ ... + p_{12}^2 = p_{13}^2

and one of them is equal to

2p_1 + p_9

4

2

a + b + c = 15 , (a - 3)^3 + (b - 5)^3 + (c -7)^3 = 540

Determine all integers

a, b, c

satisfying identities:

a + b + c = 15

(a - 3)^3 + (b - 5)^3 + (c -7)^3 = 540

2009 JBMO Shortlist G1

{ABCD}

{AC>BD}

{O}

intersection point of

{AC}

{BD}

. Circle with center at

{O}

{OA}

intersects extensions of

{AD}

{AB}

{G}

{L}

, respectively. Let

{Z}

be intersection point of lines

{BD}

{GL}

\angle ZCA={{90}^{{}^\circ }}

3

Just Hölder

\boxed{\text{A5}}

x,y,z

be positive reals. Prove that

(x^2+y+1)(x^2+z+1)(y^2+x+1)(y^2+z+1)(z^2+x+1)(z^2+y+1)\geq (x+y+z)^6

2009 JBMO Shortlist G5

{A, B, C}

{O}

be four points in plane, such that

\angle ABC>{{90}^{{}^\circ }}

{OA=OB=OC}

.Define the point

{D\in AB}

{l}

{D\in l, AC\perp DC}

{l\perp AO}

{l}

{AC}

{E}

and circumcircle of

{ABC}

{F}

. Prove that the circumcircles of triangles

{BEF}

{CFD}

{F}

(x^2 - c)(y^2 -c) = z^2 -c , (x^2 + c)(y^2 - c) = z^2 - c

Show that there are infinitely many positive integers

c

, such that the following equations both have solutions in positive integers:

(x^2 - c)(y^2 -c) = z^2 -c

(x^2 + c)(y^2 - c) = z^2 - c

4