MR6
MR6
MR6
(x 2y)(x 4z) = 6
(y 2z)(y 4x) = 10
(z 2x)(z 4y) = 16.
J213. For any positive integer n, let S(n) denote the sum of digits in its decimal representation.
Prove that the set of all positive integers n such that n is not divisible by 10 and S(n) >
S(n2 + 2012) is infinite.
ab + bc + cd + de + ea a2 + b2 + c2 + d2 e2
Proposed by Ion Dobrota, Romania and Adrian Zahariuc, Harvard University, USA
p6 7
J215. Prove that for any prime p > 3, 3 + 2p2 can be written as sum of two perfect cubes.
J216. Let be a circle and let M be a point outside it. Draw the lines l1 , l2 and l3 intersecting
and consider the intersections l1 = {A1 , A2 }, l2 = {B1 , B2 } and l3 = {C1 , C2 }.
Denote P = A1 B2 A2 B1 , Q = B1 C2 B2 C1 and R is one of the points of intersection
between P Q and . Prove that M R is tangent at .
S211. Let (a, b, c, d, e, f ) be a 6-tuple of positive real numbers satisfying simultaneously the
equations:
S212. Consider a circle (I, r) and let be a point inside it such that I = `. Using only the
straightedge and the compass, construct a triangle such that is its incircle and its
Gergonne point.
//Note. The Gergonne point of a triangle is the intersection point of the lines determined
by the vertices with the corresponding tangency points of the incircle with the opposite
sides of the triangle.
S214. Let x > y be positive rational numbers and R0 a rectangle of dimensions x y. By a cut
of R0 we understand a dissection of the rectangle in two pieces: a square of dimensions
y y and a rectangle R1 of dimensions (x y) y. Similarly, R2 is obtained from a
cut of R1 , and so on. Prove that after finitely many cuts the sequence of rectangles
R1 , R2 , . . . , Rk ends into the square Rk . Find k in terms of x, y and find the dimensions
of Rk .
S215. Let ABC be a given triangle and let A , B , C be the lines through the vertices A, B, C
and parallel to the Euler line OH, where O and H are the circumcenter and orthocenter
of ABC. Let X be the intersection of A with the sideline BC. The points Y , Z are
defined analogously. If Ia , Ib , Ic are the corresponding excenters of triangle ABC, then
the lines XIa , Y Ib , ZIc are concurrent on the circumcircle of triangle Ia Ib Ic .
Proposed by Cezar Lupu, University of Pittsburgh, USA, and and Tudorel Lupu,
Decebal High School Constanta, Romania
U213. Let x0 (0, ) fixed. For n N we set xn = sin xn1 . Show that
31/2 33/2 ln n
xn = + O(n3/2 ).
n1/2 10n3/2
Proposed by Anastasios Kotronis, Athens, Greece
U214. Prove that
n
cosh k 2 + k + 12 + i sinh k + 12
Y e2 1 + 2ie
lim =
cosh k 2 + k + 12 i sinh k + 12 e2 + 1
n+
k=1
Proposed by Cezar Lupu, University of Pittsburgh, USA, and and Tudorel Lupu,
Decebal High School Constanta, Romania
U216. Let N be a positive integer and let fN be the discriminant of the polynomial fN (x) =
xN x 1. Prove that for any prime p dividing fN , the reduction modulo p of fN (x)
has one double root and N 2 simple roots.
Note. The discriminant f of a polynomial f of degree n is defined as
Y
Df = an2n2 (ri rj )2 ,
i<j
where an is the leading coefficient and r1 , . . . , rn are the roots (counting multiplicity) of
the polynomial in some splitting field.
O211. Prove that for each positive integer n the number 4n + 8n + 16n + 2(6n + 9n + 12n ) has
at least three prime divisors.
O215. Prove that there are no positive integers a, b, c, d that are consecutive terms of an
arithmetic progression and also satisfy the condition that ab + 1, ac + 1, ad + 1, bc + 1,
bd + 1, cd + 1 are all perfect squares.
O216. Let f Z[X] be a monic polynomial of degree greater than 1. Suppose that f (X n ) is a
reducible polynomial in Z[X] for all n 2. Does it follow that f is reducible in Z[X]?