MR 6 2013 Problems (Final)

5/24/2018 MR 6 2013 Problems (Final)

1/4

Junior problems

J283. Let a, b, cbe positive real numbers. Prove that

2a + 1

b + c +

2b + 1

c + a +

2c + 1

a + b 3 + 9

2(a + b + c) .

Proposed by Zarif Ibragimov , SamSU, Samarkand, Uzbekistan

J284. Find the greatest integer that cannot be written as a sum of distinct prime numbers.

Proposed by Ivan Borsenco, Massachusetts Institute of Technology, USA

J285. Let a, b, cbe the sidelengths of a triangle. Prove that

8< (a + b + c)(2ab + 2bc + 2ca a2 b2 c2)

abc 9.

Proposed by Adithya Ganesh, Plano, USA

J286. Let ABCDbe a square inscribed in a circle. IfPis a point on the arc AB, find the minimumof the expression

PC PDPA PB .

Proposed by Panagiote Ligouras, Noci, Italy

J287. Let n be a positive integer and let a1, a2 . . . , an be real numbers in the interval

0, 1

n. Prove

that log1a1(1 na2) + log1a2(1 na3) + + log1an(1 na1) n2.

Proposed by Titu Andreescu, University of Texas at Dallas, USA

J288. Four points are given in the plane such that no three of them are collinear. These four pointsform six segments. Prove that if five of their midpoints lie on a single circle, then the sixthmidpoint lies on this circle too.

Proposed by Michal Rolinek and Josef Tkadlec, Czech Republic

Mathematical Reflections 6 (2013)


2/4

Senior problems

S283. Let a, b, cbe positive real numbers greater than or equal to 1, such that

5(a2

4a + 5)(b2

4b + 5)(c2

4c + 5)

a + b + c

1.

Prove that(a2 + 1)(b2 + 1)(c2 + 1) (a + b + c 1)3.


S284. Let Ibe the incenter of triangle ABCand let D,E,F be the feet of the angle bisectors. Theperpendicular dropped fromD ontoBCintersects the semicircle with diameter BC inA, whereAand A lie on the same side of line BC. Then angle bisector ofBACintersects BCat D .Denote by V =BE FD and W =CF ED . Prove that V W is parallel to BC.

Proposed by Ercole Suppa, Teramo, Italy

S285. Let be a, b, cbe positive real numbers such that ab + bc + ca= 1. Prove that

a

b2 + c2 + 2+

b

c2 + a2 + 2+

c

a2 + b2 + 2 1

8

1

a+

1

b+

1

c

.

Proposed by Mircea Lascu and Marius Stanean, Zalau, Romania

S286. Let ABC be a triangle and let P be a point in its interior. Lines AP,BP,CP intersectBC,CA,AB at A1, B1, C1, respectively. Prove that the circumcenters of triangles APB1,APC1, BPC1, BPA1 are concyclic if and only if lines A1B1 and AB are parallel.

Proposed by Mher Mnatsakanyan, Armenia

S287. Let mand n be positive integers such that gcd(m,n) = 1. Prove that

mk=1

nk

m

mk

=

nk=1

mk

n

nk

,

where is the Euler totient function.

Proposed by Marius Cavachi, Romania

S288. Consider triangle ABCwith vertices A, B, Cin the counterclockwise order. On the sides ABandACconstruct outwards similar rectanglesABXY andCAZTin the clockwise order. LetDbe the second point of intersection of the circumcircle of triangle AY Zand , the circumcircleof triangleABC. Denote by U=DY andV =DZ and let Wbe the midpoint ofU V.Prove that AW is perpendicular to Y Z.

Proposed by Marius Stanean, Zalau, Romania



3/4

Undergraduate problems

U283. Let a,b,c,d be nonnegative real numbers such that a+ b + c + d = 2. Prove that for anypositive integer n,

an + bn + cn + dn + 23n

3

2 a

3

n

+

2 b

3

n

+

2 c

3

n

+

2 d

3

n

.


U284. Let an =

n2 + 1

be the sequence of real numbers, where{x} denotes the fractional partofx. Find limn nan.


U285. Let (an)n>0 be the sequence defined by an= e

1

n+1

n + 1+ e

1

n+2

n + 2+ +e

1

2n

2n. Prove that sequence(an)n>0 is decreasing and find its limit.

Proposed by Angel Plaza, Universidad de Las Palmas de Gran Canaria, Spain

U286. Let (xn)n1 be a sequence of positive real numbers such that

(xn+1 xn)(xn+1xn 1) 0

for all nand limnxn+1

xn= 1. Prove that (xn)n1 is convergent.

Proposed by Mihai Piticari, Campulung Moldovenesc, Romania

U287. Let u, v : [0 + ) (0,+) be differentiable functions such that u =vv and v =uu. Provethat limx(u(x) v(x)) = 0.

Proposed by Aaron Doman, University of California, Berkeley , USA

U288. Let f : [a, b] R be an antiderivative and integrable function such that for each interval(m,n)[a, b] there is an interval (m, n) contained in (m,n) such that f(x)0 for all x in(m, n). Prove that f(x) 0 for all x in [a, b].

Proposed by Mihai Piticari and Sorin Radulescu, Romania



4/4

Olympiad problems

O283. Prove that for all positive real numbersx1, x2, . . . , xn the following inequality holds:

ni=1

x3i

x21+ + x2i1+ x2i+1+ + x2n

x1+

+ xnn 1 .

Proposed by Mircea Becheanu, Bucharest, Romania

O284. Consider a convex hexagon ABCDEF with AB||DE, BC||EF, CD||FA. The distance be-tween the lines AB and DE is equal to the distance between the lines BC and EF and isequal to the distance between the lines CD and FA. Prove thatAD+ BE+ CF does notexceed the perimeter of the hexagon ABCDEF.

Proposed by Nairi Sedrakyan, Yerevan, Armenia

O285. Let an be the sequence of integers defined as a1= 1 and an+1= 2n(2an 1) for n 1. Provethat n! divides an.


O286. LetABCbe a triangle with orthocenter H. LetHMbe the median andHSbe the symmedianin triangle BHC. Denote by P the orthogonal projection ofA onto HS. Prove that thecircumcircle of triangle MPSis tangent to the circumcircle of triangle ABC.

Proposed by Marius Stanean, Zalau, Romania

O287. We are given a 6 6 table with 36 unit cells. A 2 2 square with 4 unit cells is called a block.A set of blocks covers the table if each cell of the table is covered by at least one cell of oneblock in the set. Blocks can overlap with each other. Find the largest integer n such thatthere is a cover of the table with n blocks, but if we remove any block from the covering, theremaining set of blocks will no longer cover the table.

Proposed by Nairi Sedrakyan, Yerevan, Armenia

O288. Let ABCD be a square situated in the planeP. Find the minimum and the maximum of thefunction f : P Rdefined by

f(P) = PA + PB

PC+ PD

.

Proposed by Dorin Andrica, Babes-Bolyai University, Romania


MR 6 2013 Problems (Final)

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