AoPS Community 2009 JBMO Shortlist
JBMO Shortlist 2009
www.artofproblemsolving.com/community/c584835
by Snakes, parmenides51, tenplusten, delegat
– Algebra
1 Determine all integers a, b, c satisfying identities: a+b+c = 15 (a−3)3 +(b−5)3 +(c−7)3 = 540
2 A2 Find the maximum value of z + x if x, y, z are satisfying the given conditions.x2 + y 2 = 4
z 2 + t2 = 9 xt + yz ≥ 6
3 Find all values of the real parameter a, for which the system (|x| + |y| − 2)2 = 1 y = ax + 5
has exactly three solutions
4 Let x, y, z be real numbers such that 0 < x, y, z < 1 and xyz = (1 − x)(1 − y)(1 − z). Show that
at least one of the numbers (1 − x)y, (1 − y)z, (1 − z)x is greater than or equal to 14
5 A5 Let x, y, z be positive reals. Prove that (x2 + y + 1)(x2 + z + 1)(y 2 + x + 1)(y 2 + z + 1)(z 2 +
x + 1)(z 2 + y + 1) ≥ (x + y + z)6
– Combinatorics
1 Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-
centered unit circle there are exactly two red points. Find the gratest possible number of blue
points.
2 Five players (A, B, C, D, E) take part in a bridge tournament. Every two players must play (as
partners) against every other two players. Any two given players can be partners not more
than once per a day. What is the least number of days needed for this tournament?
3 a) In how many ways can we read the word SARAJEVO from the table below, if it is allowed to
jump from cell to an adjacent cell (by vertex or a side) cell?
b) After the letter in one cell was deleted, only 525 ways to read the word SARAJEVO remained.
Find all possible positions of that cell.
4 Determine all pairs of (m, n) such that is possible to tile the table m × n with figure corner
as in figure with condition that in that tilling does not exist rectangle (except m × n) regularly
covered with figures.
– Geometry
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AoPS Community 2009 JBMO Shortlist
1 Parallelogram ABCD is given with AC > BD, and O point of intersection of AC and BD.
Circle with center at Oand radius OA intersects extensions of ADand ABat points G and L,
◦
respectively. Let Z be intersection point of lines BDand GL. Prove that ∠ZCA = 90 .
◦
2 In right trapezoid ABCD (AB k CD) the angle at vertex B measures 75 . Point His the foot of
the perpendicular from point A to the line BC. If BH = DC andAD + AH = 8, find the area of
ABCD.
3 Parallelogram ABCDwith obtuse angle ∠ABC is given. After rotation of the triangle ACD
around the vertex C, we get a triangle CD0 A0 , such that points B, C and D0 are collinear. Ex-
tensions of median of triangle CD0 A0 that passes through D0 intersects the straight line BDat
point P . Prove that P Cis the bisector of the angle ∠BP D0 .
4 Let ABCDE be a convex pentagon such that AB + CD = BC + DE and k a circle with center
on side AE that touches the sides AB, BC, CD and DE at points P , Q, R and S (different from
vertices of the pentagon) respectively. Prove that lines P S and AE are parallel.
◦
5 Let A, B, C and O be four points in plane, such that ∠ABC > 90 and OA = OB = OC.Define
the point D ∈ AB and the line l such that D ∈ l, AC ⊥ DC and l ⊥ AO. Line l cuts ACat E and
circumcircle of ABC at F . Prove that the circumcircles of triangles BEF and CF Dare tangent
at F .
– Number Theory
1 Solve in non-negative integers the equation 2a 3b + 9 = c2
2 A group of n > 1 pirates of different age owned total of 2009 coins. Initially each pirate (except
the youngest one) had one coin more than the next younger.
a) Find all possible values of n.
b) Every day a pirate was chosen. The chosen pirate gave a coin to each of the other pirates.
If n = 7, find the largest possible number of coins a pirate can have after several days.
3 Find all pairs (x, y) of integers which satisfy the equation (x + y)2 (x2 + y 2 ) = 20092
4 Determine all prime numbers p1 , p2 , ..., p12 , p13 , p1 ≤ p2 ≤ ... ≤ p12 ≤ p13 , such
that p21 + p22 + ... + p212 = p213 and one of them is equal to 2p1 + p9 .
5 Show that there are infinitely many positive integers c, such that the following equations both
have solutions in positive integers: (x2 − c)(y 2 − c) = z 2 − c and (x2 + c)(y 2 − c) = z 2 − c.
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