RMC2006

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March 11

th

, 2006

7thGRADE

Problem 1. Letn > 1 be an integer. Prove that the number√11. . .144. . .4

(digit “1” occursn times and digit “4” occurs 2n times) is an irrational number.

Cecilia Deaconescu, Pites¸ti

Problem 2. In triangleABC, ∠ABC = 2· ∠ACB. Prove that:

a)AC2_{= AB}2_{+ AB}

· BC;

b)AB + BC < 2· AC.

Gh. Bumb˘acea, Bus¸teni

Problem 3. A setM containing 4 positive integers is called connected, if for

everyx in M at least one of the numbers x− 1, x + 1 belongs to M. Let Un be

the number of connected subsets of the set_{{1, 2, . . . , n}.} a) EvaluateU7.

b) Determine the leastn for which Un>2006.

Lucian Dragomir, Ot¸elul Ros¸u

Problem 4. LetABC be an isosceles triangle, with AB = AC. Let D be the

midpoint of the sideBC, M the midpoint of the line segment AD and let N be

the projection ofD on BM. Prove that ∠AN C = 90◦_.

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8thGRADE

Problem 1. LetABC be a right triangle (with A = 90◦_{). Two perpendiculars}

on the triangle’s plane are erected at pointsA and B, and the points M and N

are considered on these perpendiculars, on the same side of the plane, such that

BN < AM . It is known that AC = 2a, AB = a√3, AM = a and that the angle

between the planesM N C and ABC equals 30◦_{. Find:}

a) the area of triangleM N C;

b) the distance from the pointB to the plane M N C.

Gianina Busuioc, Niculai Solomon

Problem 2. For each positive integern, denote by u(n) the largest prime

num-ber less than or equal ton and by v(n) the smallest prime number greater than n.

Prove that 1 u(2)v(2) + 1 u(3)v(3) + 1 u(4)v(4) +· · · + 1 u(2010)v(2010) = 1 2− 1 2011.

Nicolae St˘aniloiu, Bocs¸a

Problem 3. Prove that there exist infinitely many irrational numbersx and y

such thatx + y = xy∈ N.

Claudiu S¸ tefan Popa, Ias¸i

Problem 4. a) Prove that one can assign to each of the vertices of a cube one of

the numbers 1 or_{−1 such that the product of the numbers assigned to the vertices} of each face equals_−1.

b) Prove that such an assignment is impossible in the case of a regular hexag-onal prism.

9thGRADE

Problem 1. Let x, y, z be positive real numbers. Prove that the following

inequality holds: 1 x2_{+ yz} + 1 y2_{+ zx}+ 1 z2_{+ xy} 6 1 2 1 xy+ 1 yz+ 1 zx . Traian T˘amˆıian

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Problem 2. The entries of a9×9 array are all the numbers from 1 to 81. Prove

that there existsk _{∈ {1, 2, 3, . . . , 9} such that the product of the numbers in the}

linek differs from the product of the numbers in the column k.

Marius Ghergu, Slatina

Problem 3. LetABCD be a convex quadrilateral. Let M and N be the

mid-points of the line segmentsAB and BC, respectively. The line segments AN and BD intersect at E and the line segments DM and AC intersect at F . Prove that

ifBE =1₃BD and AF = 1₃AC, then ABCD is a parallelogram.

Gh. Iurea, Ias¸i

Problem 4. For each positive integern, denote by p(n) the largest prime

num-ber less than or equal ton and by q(n) the smallest prime number greater than n.

Prove that n X k=2 1 p(k)q(k) < 1 2.

Nicolae St˘aniloiu, Bocs¸a

10thGRADE

Problem 1. Consider the real numbersa, b, c ∈ (0, 1) and x, y, z ∈ (0, ∞),

such that ax= bc, by= ca, cz= ab. Prove that 1 2 + x + 1 2 + y + 1 2 + z 6 3 4.

Cezar Lupu, Bucures¸ti

Problem 2. LetABC be a triangle and consider the points M ∈ (BC), N ∈ (CA), P _{∈ (AB) such that} AP

P B = BM M C =

CN

N A. Prove that ifM N P is an

equilateral triangle, thenABC is an equilateral triangle as well.

I.V. Maftei, A. Schier, Bucures¸ti

Problem 3. A prism is called binary if one can assign to each of its vertices

a number from the set_{{−1, +1}, in such a way that the product of the numbers} assigned to the vertices of every face equals_−1.

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b) Prove that a prism with2000 vertices is binary.

Problem 4. a) Find two setsX, Y such that X_{∩ Y = ∅, X ∪ Y = Q}∗ +and

Y =_{{a · b | a, b ∈ X}.}

b) Find two setsU, V such that U_{∩ V = ∅, U ∪ V = R and V = {x + y |} x, y_{∈ U}.}

Marius Cavachi, Constant¸a

11thGRADE

Problem 1. Letx > 0 be a real number and let A be a 2_{× 2 matrix with real}

entries, such that

det(A2_{+ xI} 2) = 0.

Prove that det(A2_{+ A + xI} 2) = x.

Vasile Pop, Cluj

Problem 2. Letn, p > 2 be integer numbers and let A be a n× n real matrix

such thatAp+1_{= A.}

a) Prove that rank(A) + rank(In− Ap) =n.

b) Prove that ifp is a prime number, then

rank(In− A) = rank(In− A2) =· · · = rank(In− Ap−1).

Marius Ghergu, Slatina

Problem 3. The sequence of real numbers(xn)n>0satisfies

(xn+1− xn)(xn+1+ xn+ 1) 6 0, n > 0.

a) Prove that the sequence is bounded. b) Can such a sequence be divergent?

Mihai B˘alun˘a, Bucures¸ti

Problem 4. We say that a functionf : R_{→ R has the property (P) if for every}

realx,

sup

t6x

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a) Give an example of a function having property (P) which is discontinuous at every real point.

b) Prove that if f is continuous and has property (P) then f is the identical

function.

Mihai Piticari, Cˆampulung

12thGRADE

Problem 1. Letf1, f2, . . . , fn : [0, 1]→ (0, ∞) be continuous functions and

letσ be a permutation of the set_{{1, 2, . . . , n}. Prove that}

n Y i=1 Z 1 0 f2 i(x) fσ(i)(x) dx > n Y i=1 Z 1 0 fi(x) dx.

Cezar Lupu, Mihai Piticari

Problem 2. Let G = {A ∈ M2(R)| det(A) = ±1} and H = {A ∈

M2(C)| det A = 1}. Prove that, under matrix multiplication, G and H are

non-isomorphic groups.

Marius Cavachi, Constant¸a

Problem 3. LetA be a finite commutative ring with at least two elements.

Prove that for any positive integern > 2, there exists a polynomial f ∈ A[X] of

degreen, with no roots in A.

Marian Andronache, Bucures¸ti

Problem 4. Let_{F = {f : [0, 1] → [0, ∞) | f continuous} and let n > 2 be a}

positive integer. Determine the least real constantc, such that Z 1 0 f (√n_{x)dx 6 c} Z 1 0 f (x)dx for allf ∈ F. Gh. Iurea, Ias¸i

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FINAL ROUND

April 15

th

, 2006

7thGRADE

Problem 1. Consider the triangleABC and points M , N belonging to the

sidesAB, BC respectively, such that 2·CN_BC = AM_AB. Let P be a point on AC.

Prove that the linesM N and N P are perpendicular if and only if P N bisects the

angle ∠M P C.

Marcel Teleuc˘a

Problem 2. A square of siden is divided into n2_{unit squares each colored}

red, yellow or green. Find the minimum value ofn such that for any such coloring

we can find a row or a column containing at least three squares of the same color.

Mircea Fianu

Problem 3. In the acute triangleABC angle C equals 45◦_{. Points A}

1andB1

are the foots of the perpendiculars fromA and B respectively. Denote by H the

ortocenter ofABC. Points D and E are situated on the segments AA1andBC,

respectively, such thatA1D = A1E = A1B1. Prove that:

a)A1B1=

q

A1B2+A1C2

2 ;

b)CH = DE.

Claudiu-S¸ tefan Popa

Problem 4. Let A be a set of nonnegative integers containing at least two

elements and such that for anya, b_{∈ A, a > b, we have} [a,b]_a−b _{∈ A. Prove that the}

setA contains exactly two elements.

([a, b] denotes the least common multiple of a and b).

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8thGRADE

Problem 1. Consider a convex poliedra with 6 faces each of them being a

circumscribed quadrilaterals. Prove that all faces are circumscribed quadrilaterals.

G. Rene

Problem 2. Given a positive integern, prove that there exists an integer k, k > 2 and numbers a1, a2, . . . , ak∈ {−1, 1} such that

n = X

16i<j6k

aiaj.

Gheorghe Iurea

Problem 3. LetABCDA1B1C1D1be a cube and letP be a variable point

on the side[AB]. The plane through P , perpendicular to AB meets AC1atQ. Let

M and N be the midpoints of the segments A1P and BQ, respectively.

a) Prove that the linesM N and BC1are perpendicular if and only ifP is the

midpoint ofAB.

b) Find the minimal value of the angle between the linesM N and BC1.

Petre Simion

Problem 4. Consider real numbers a, b, c contained in the interval [1 2, 1]. Prove that 2 6 a + b 1 + c + b + c 1 + a+ c + a 1 + b 63. Mircea Lascu 9thGRADE

Problem 1. Find the maximum value of (x3+ 1)(y3+ 1),

forx, y_{∈ R such that x + y = 1.}

Dan Schwarz

Problem 2. Consider quadrilateralsABCD inscribed in a circle of radius r,

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a) Prove that there is a configuration of pointsA, B, C, D, P for which the

above configuration is possible.

b) Prove that for any such configuration we also haveP D = DA = r.

Virgil Nicula

Problem 3. Consider the trianglesABC and DBC such that AB = BC, DB = DC and ∠ABD = 90◦_{. Let}_{M be the midpoint of BC. Points E, F, P are}

such thatE_{∈ (AB), P ∈ (MC), C ∈ (AF ) and ∠BDE = ∠ADP = ∠CDF .}

Prove thatP is the midpoint of EF and DP _{⊥ EF .}

Manuela Prajea

Problem 4. A table tennis competition takes place during 4 days, the number

of participants being2n, n > 5. Every participant plays exactly one game daily

(it is possible that a pair of participants meet more times). Prove that such a com-petition can end with exactly one winner and exactly three players on the second place and such that there is no player losing all four matches. How many partic-ipants have won a single match and how many exactly two, in the given above conditions?

Radu Gologan

10thGRADE

Problem 1. Consider a setM with n elements and letP(M) denote all subsets

ofM . Find all functions f : _{P(M) → {0, 1, 2, . . . , n}, satisfying the following}

two conditions:

a)f (A)_{6= 0, for any A 6= ∅, and}

b) f (A∪ B) = f (A ∩ B) + f (A∆B) , for any A, B ∈ P(M), where A∆B = (A_{∪ B) (A ∩ B) .}

Vasile Pop

Problem 2. Prove that fora, b∈ 0,π 4 we have sinna + sinnb (sin a + sin b)n > sinn2a + sinn2b (sin 2a + sin 2b)n. Iurie Boreico

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Problem 3. Prove that the sequence given byan=n√2+n√3, n∈ N,

contains infinitely many odd numbers and infinitely many even numbers.

Marius Cavachi

Problem 4. Givenn_{∈ N, n > 2, find n disjoint sets A}i, 1 6 i 6 n, in the

plane, such that:

a) for any disk_{C and any i ∈ {1, 2, . . . .n} , we have A}i∩ Int (C) 6= ∅, and

b) for any lined and for any i_{∈ {1, 2, . . . .n} , the projection of A}iond is not

all ofd.

Severius Moldoveanu, Costel Chites¸

11th

GRADE

Problem 1. A is a two by two matrix with complex entries. Denote by A∗_its

adjoint (the matrix formed by the cofactors of the transpose). Prove that if there is an integerm > 1 such that (A∗₎m_{= 0}

n, then(A∗)2= 0n.

Marian Ionescu

Problem 2. A matrixB_{∈ M}n(C) will be called a pseudo-inverse of a matrix

A_{∈ M}n(C) if A = ABA and B = BAB.

a) Prove that any square matrix has at least one pseudo-inverse. b) Characterize the class of matrices with a unique pseudo-inverse.

Marius Cavachi

Problem 3. Consider two systems of points in the plane:A1, A2, . . . , Anand

B1, B2, . . . , Bn having different centroids. Prove that there is a pointP in the

plane such that

P A1+ P A2+· · · + P An= P B1+ P B2+· · · + P Bn.

Marius Cavachi

Problem 4. Consider a functionf : [0,∞) → R, with the property that for

anyx > 0, the sequence (f (nx))n>0is increasing.

a) Iff is also continuous on [0, 1], does it follow that it is increasing?

b) What iff is continuous on Q+?

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12thGRADE

Problem 1. LetK be a finite field. Prove that the following statements are

equivalent: a)1 + 1 = 0;

b) for anyf _{∈ K[X] with deg f > 1 the polynomial f(X}2_{) is reducible.}

Marian Andronache

Problem 2. Prove that lim n→∞n π 4 − n Z 1 0 xn 1 + x2ndx = Z 1 0 f (x)dx,

wheref (x) = arctg x_x , forx_{∈ (0, 1] and f(0) = 1.}

Dorin Andrica, Mihai Piticari

Problem 3. LetG be a group with n elements (n > 2) and let p be the smallest

prime factor ofn. Suppose G has a unique subgroup H with p elements. Prove

thatH is contained in the center of G. (The center of G is the set Z(G) = _{{a ∈} G_{| ax = xa, ∀x ∈ G}.)}

Ion Savu

Problem 4. Letf : [0, 1]_{→ R be a continuous function such that} Z 1

0

f (x)dx = 0.

Prove that there isc∈ (0, 1) such that Z c

0

xf (x)dx = 0.

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THE BMO AND IMO ROMANIAN TEAMS

FIRST SELECTION TEST

Problem 1. LetABC and AM N be two similar triangles with the same

ori-entation, such thatAB = AC, AM = AN , and having disjoint interiors. Let O

be the circumcenter of the triangleM AB. Prove that the points O, C, N , A are

concyclic if and only if the triangleABC is equilateral.

Valentin Vornicu

Problem 2. Letp > 5 be a prime number. Find the number of irreducible

polynomials inZ[X], of the form

xp+ pxk+ pxl+ 1, k > l, k, l_{∈ {1, 2, . . . , p − 1} .}

The Editors

Problem 3. Leta, b be positive integers such that for any positive integer n

we havean_{+ n}

| bn_{+ n. Prove that a = b.}

IMO Shortlist 2005

Problem 4. Let a1, a2, . . . , an be real numbers such that|ai| 6 1 for all

i = 1, 2, . . . , n, and a1+ a2+· · · + an= 0.

(a) Prove that there existsk_{∈ {1, 2, . . . , n} such that} |a1+ 2a2+· · · + kak| 6

2k + 1 4 .

(b) Prove that forn > 2 the bound above is the best possible.

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SECOND SELECTION TEST

Problem 5. Let_{an}n>1be a sequence given bya1= 1, a2 = 4, and for all

integersn > 1

an=

p

a_n−1an+1+ 1.

(a) Prove that all the terms of the sequence are positive integers.

(b) Prove that the number2anan+1 + 1 is a perfect square for all integers

n > 1.

Valentin Vornicu

Problem 6. LetABC be a triangle with ∠ABC = 30◦. Consider the closed discs of radiusAC/3 centered at A, B and C. Does there exist an equilateral

triangle whose three vertices lie one each in each of the three discs?

Radu Gologan, Dan Schwarz

Problem 7. Determine the pairs of positive integers(m, n) for which there

exists a setA such that for x, y positive integers, if_{|x − y| = m, then at least one}

of the numbersx, y belongs to the set A, while if_{|x − y| = n, then at least one of}

the numbersx, y does not belong to the set.

Adapted by the Editors from AMM

Problem 8. Letxi,1 6 i 6 n be real numbers. Prove that

X 16i<j6n |xi+ xj| > n_{− 2} 2 n X i=1 |xi|.

Adapted by the Editors from Putnam

THIRD SELECTION TEST

Problem 9. The circle of center I is inscribed in the convex quadrilateral ABCD. Let M and N be points on the segments AI and CI respectively, such

that ∠M BN = 1₂∠_{ABC. Prove that ∠M DN =}1

2∠ADC.

Problem 10. LetA be a point exterior to a circle_{C. Two lines through A meet}

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E). The parallel through D to BC meets the second time the circleC at F . The

lineAF meets _{C again at G, and the lines BC and EG meet at M. Prove that} 1 AM = 1 AB + 1 AC. Bogdan Enescu

Problem 11. Letγ be the incircle of the triangle A0A1A2. In what follows,

indices are reduced modulo3. For each i_{∈ {0, 1, 2}, let γ}ibe the circle through

Ai+1 andAi+2, and tangent toγ ; let Tibe the tangency point ofγi andγ ; and

finally, letPibe the point where the common tangent atTitoγiandγ meets the

lineAi+1Ai+2. Prove that

(a) the pointsP0,P1andP2are collinear;

(b) the linesA0T0,A1T1andA2T2are concurrent.

AMM

Problem 12. Leta, b, c be positive real numbers such that a + b + c = 3.

Prove that 1 a2 + 1 b2 + 1 c2 >a 2_{+ b}2_{+ c}2_. Vasile Cˆartoaje

FOURTH SELECTION TEST

Problem 13. Givenr, s_{∈ Q, determine all functions f : Q → Q such that} f (x + f (y)) = f (x + r) + y + s

for allx, y_{∈ Q.} Vasile Pop, Dan Schwarz

Problem 14. Find all positive integersm, n, p, q such that pm_qn_{= (p+q)}2_+1.

Adrian Stoica

Problem 15. Letn > 1 be an integer. A set S⊂ {0, 1, . . . , 4n − 1} is called

sparse if for anyk_{∈ {0, 1, . . . , n − 1} the following two conditions are satisfied:}

(1) the setS∩ {4k − 2, 4k − 1, 4k, 4k + 1, 4k + 2} has at most two elements;

(2) the setS_{∩ {4k + 1, 4k + 2, 4k + 3} has at most one element.}

Prove that the set_{{0, 1, . . . , 4n − 1} has exactly 8 · 7}n−1sparse subsets.

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Problem 16. Letp, q be two integers, q > p > 0. Let n > 2 be an integer and a0= 0, a1>0, a2, . . . , an−1,an= 1 be real numbers such that

ak 6 a_k−1+ ak+1 2 , k = 1, 2, . . . , n− 1. Prove that (p + 1) n−1 X k=1 ap_k>(q + 1) n−1_X k=1 aq_k. C˘alin Popescu

FIFTH SELECTION TEST

Problem 17. Letk > 1 be an integer and n = 4k + 1. Let A =_{a2_{+ nb}2

| a, b_{∈ Z}. Prove that there exist integers x, y such that x}n_+yn

∈ A and x+y /∈ A.

AMM

Problem 18. Let m and n be positive integers and let S be a subset with (2m

− 1)n + 1 elements of the set {1, 2, . . . , 2m_n

}. Prove that S contains m + 1

distinct numbersa0, a1, . . . , amsuch thatak−1| akfor allk = 1, 2, . . . , m.

AMM

Problem 19. Letx1= 1, x2,x3,. . . be a sequence of real numbers such that

for alln > 1 we have

xn+1 = xn+ 1 2xn. Prove that b25x625c = 625. The Editors

Problem 20. LetABC be an acute triangle with AB _{6= AC. Let D be the}

foot of the altitude fromA to BC and let ω be the circumcircle of the triangle ABC. Let ω1be the circle that is tangent toAD, BD and ω. Let ω2be the circle

that is tangent toAD, CD and ω. Finally, let ` be the common internal tangent to ω1andω2that is notAD.

Prove that ` passes through the midpoint of BC if and only if 2BC = AB + AC.

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BALKAN MATHEMATICAL OLYMPIAD

FIRST SELECTION TEST

Problem 1. LetABC be a rightangle triangle at C and consider points D, E

on the sidesBC, CA, respectively, such that BD_AC = _CDAE = k. Lines BE and AD

intersect at pointO. Show that ∠BOD = 60◦_{if and only if}_{k =}√_3.

Marcel Chirit¸˘a

Problem 2. Consider five points in the plane such that each triangle with

ver-tices at three of those points has area at most 1. Prove that the five points can be covered by a trapezoid of area at most 3.

Marcel Chirit¸˘a

Problem 3. For any positive integer n let s(n) be the sum of its digits in

decimal representation. Find all numbersn for which s(n) is the largest proper

divisor ofn.

Laurent¸iu Panaitopol

SECOND SELECTION TEST

Problem 4. Prove that a_bc3+b3

ca+ c3

ba>a + b + c, for all positive real numbers

a, b, and c.

Problem 5. Consider a circle C of center O and let A, B be points on the

circle with ∠AOB = 90◦_{. Circles C}

1(O1) and C2(O2) are internally tangent to

C at points A, B, respectively, and – moreover – are tangent to themselves.

Cir-cleC3(O3), located inside the angle ∠AOB, is externally tangent to C1, C2and

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Problem 6. A7× 7 array is divided into 49 unit squares. Find all integers n_{∈ N}∗_{for which}_{n checkers can be placed on the unit squares so that each row}

and each line contain an even number of checkers.

(0 is an even number, so empty rows or columns are not excluded. At most one checker is allowed inside a unit square.)

Dinu S¸ erb˘anescu

THIRD SELECTION TEST

Problem 7. SupposeABCD is a cyclic quadrilateral of area 8. Prove that if

there exists a pointO in the plane of the quadrilateral such that OA + OB + OC + OD = 8, then ABCD is an isosceles trapezoid (or a square).

Flavian Georgescu

Problem 8. Prove that a b + b c + c a 2 > 3 2· a + b c + b + c a + c + a b ,

for all positive real numbersa, b, and c.

Cezar Lupu

Problem 9. Find all real numbersa and b satisfying 2(a2+ 1)(b2+ 1) = (a + 1)(b + 1)(ab + 1).

Valentin Vornicu

Problem 10. Show that the set of real numbers can be partitioned into subsets

having two elements.

Dan Schwarz

FOURTH SELECTION TEST

Problem 11. LetA ={1, 2, . . . , 2006}. Find the maximal number of subsets

ofA that can be chosen such that the intersection of any two such distinct subsets

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Problem 12. LetABC be a triangle and let A1, B1, C1be the midpoints of

the sidesBC, CA, AB, respectively. Show that if M is a point in the plane of the

triangle such that

M A M A1 = M B M B1 = M C M C1 = 2,

thenM is the centroid of the triangle.

Dinu S¸ erb˘anescu

Problem 13. Suppose a, b, c are positive real numbers which sum up to 1.

Prove that a2 b + b2 c + c2 a >3(a 2_{+ b}2_{+ c}2_). Mircea Lascu

Problem 14. The set of positive integers is partitioned into subsets with

in-finitely many elements each. The following question arises: does there exist a subset in the partition such that any positive integer has a multiple in that subset?

a) Prove that if the number of subsets in the partition is finite, then the answer is “yes”.

b) Prove that if the number of subsets in the partition is infinite, then the answer can be “no” (for some partition).

FIFTH SELECTION TEST

Problem 15. LetABC be a triangle and D a point inside the triangle, located

on the median fromA. Show that if ∠BDC = 180◦_{− ∠BAC, then AB · CD =}

AC· BD.

Eduard B˘az˘avan

Problem 16. Consider the integersa1, a2, a3, a4, b1, b2, b3, b4withak 6= bk

for allk = 1, 2, 3, 4. If

{a1, b1} + {a2, b2} = {a3, b3} + {a4, b4},

show that the number_|(a1− b1)(a2− b2)(a3− b3)(a4− b4)| is a square.

Note. For any setsA and B, we denote A + B ={x + y | x ∈ A, y ∈ B}.

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Problem 17. Letx, y, z be positive real numbers such that 1 1 + x+ 1 1 + y + 1 1 + z = 2.

Prove that8xyz 6 1.

Mircea Lascu

Problem 18. For a positive integern denote by r(n) the number having the

digits ofn in reverse order; for example, r(2006) = 6002. Prove that for any

positive integersa and b the numbers 4a2_{+ r(b) and 4b}2_{+ r(a) cannot be}

simul-taneously perfect squares.

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7thGRADE

Problem 1. The bisectors of the angles of the triangleABC meet the sides BC, CA, AB in D, E, F respectively. Prove that

1 AB_{· CE} + 1 BC_{· AF} + 1 CA_{· BD} = 1 r_{· R}

Problem 2. In a triangleABC, m(∠BAC) = 110◦_{, m(∠ABC) = 50}◦_{. Let}

D be an internal point such that m(∠DBC) = 20◦_and_{m(∠DCB) = 10}◦_{. Find}

m(∠ADC).

Problem 3. The pointsM and N are taken on the sides AC, respectively AB

of triangleABC such that M A = m· MC and NA = n · NB, where m, n are

positive reals andm + n = 2. The straight lines BM and CN meet at P . Prove

that area(AM P N ) > mn₃ area(ABC).

Problem 4. LetABCDEF be a convex hexagon. Triangles ∆1s¸i∆2will be

called opposite if they are determined by consecutive vertices of the hexagon and have no common points. Prove that the straight lines joining the centroids of the three pairs of opposite triangles are concurrent.

Problem 5. LetABCDE be a convex pentagon. A straight line will be called

central if it joins the centroid of the triangle determined by three consecutive

ver-tices of the pentagon and the midpoint of the “opposite” side. Prove that the five central lines are concurrent.

Problem 6. Leta, b, c, d be four distinct positive integers whose product is a

perfect square. Prove that the numbera4_{+ b}4_{+ c}4_{+ d}4_{is the sum of five non-zero}

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Problem 7. Find three distinct positive integers with integral arithmetic,

geo-metric and harmonic means. Same problem forn > 4 distinct positive integers. Problem 8. Prove that three positive real numbersx, y, z satisfy the equality

x(y− z) y + z + y(z− x) z + x + z(x− y) x + y = 0,

if and only if at least two of these numbers are equal.

Problem 9. A tennis competition lasted three days and had 20 participants.

Every participant played a match each day (it is possible that the same pair of players met more than once). In the end there was only one winner and everybody had at least a victory. How many participants won exactly one match?

8thGRADE

Problem 10. Letm, n be integers such that m > n > 3. Prove that the roots x1, x2of the equationx2− mx + n = 0 are integers if and only if the number

bmx1c + bmx2c is a perfect square.

Problem 11. Prove that ifa, b, c are three positive real numbers then X cyc b + c a >3 + (a2_{+ b}2_{+ c}2_{)(ab + bc + ca)} abc(a + b + c) .

Problem 12. Leta, b be positive integers such that a < b and a is not a divisor

ofb. Solve the equation abxc − b{x} = 0. Problem 13. Consider the sets

A = (r 1 a+ 1 b| a, b ∈ N ∗_{, a}_{6= b} ) and B =r 1 x + 1 y + 1 z| x, y, z ∈ N ∗_{, x > y > z} .

Prove thatA_{∩ B contains infinitely many rational and infinitely many irrational}

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Problem 14. Prove that ifa, b, c are positive real numbers then X cyc a2 3a2_{+ b}2_{+ 2ac}6 1 2.

Problem 15. Find all positive integersn and x1, x2, . . . , xnsuch that

x1+ x2+· · · + xn= 3n and 1 x1 + 1 x2 +_{· · · +} 1 xn = 1 + 1 4n. Problem 16. Letp, q be integers. Prove that if a set A has p2

− q elements

thenA cannot have exactly q2

− p subsets.

Problem 17. Find all integersx, y, z, t such that x + y + z = t2_and_x2_{+ y}2₊

z2_{= t}3_.

Problem 18. The bases of a right prismABCDEF A1B1C1D1E1F1are

reg-ular hexagons. Prove that:

a)AE1⊥ B1E if and only if AA1= AB√3;

b) ifAE1 ⊥ B1E then the distance between the straight lines AE1andB1E

is√₁₄42AB.

Problem 19. Consider a tetrahedronABCD of volume 2 and points M , N , P , Q, R, S on the edges AB, BC, CD, DA, AC and BD, respectively, such that

the segmentsM P, N Q and RS be concurrent. Prove that the volume of the

poly-hedronM N P QRS is at most 1.

Problem 20. The cubeABCDA0_B0_C0_D0 _{has edges of length 2. The two}

triangles having as vertices the midpoints of the edges starting fromB and C have

centroidsE and F respectively. Let P = A0_E_{∩ D}0_{F . Compute the cosine of the}

angle ∠A0_{P D}0_{and the distances from}_A0_{to the planes of the two triangles.}

9thGRADE

Problem 21. Consider an integern > 2 and positive real numbers a1, a2, . . . ,

a2n, with sums. Prove that

a1 s + an+1− a1 +· · ·+_{s + a}an 2n− an + an+1 s + a1− an+1 +· · ·+_{s + a}a2n n− a2n >1

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Problem 22. Leta, b, c be positive real numbers such that a2_{+ b}2_{+ c}2 _{= 3.}

Prove that, for every positive numbersx, y, z, x a + y b + z c > √xy + √_{yz +}√ zx.

Problem 23. Leta, b, c be positive numbers such that abc = 1. Prove that X cyc 1 a2_{+ 2b}2_{+ 3} 6 1 2.

Problem 24. With each functionf :_{{1, 2, . . . , n} → {1, 2, . . . , n} associate}

a functionf :_{{1, 2, . . . , n} → {0, 1, . . . , n − 1}, by letting}

f (k) = f (1) +· · · + f(k) − n f(1) + · · · + f(k)_n

,

for eachk = 1, 2, . . . , n. Prove that a necessary and sufficient condition for a pair (f, f) of one-to-one associated functions to exist is that n be even.

Problem 25. A functionf : [0,∞) → [0, ∞) will be said to have property P if

f (xf (y2)) = f (y)f (f (x2))

for allx, y∈ [0, ∞).

a) Show that there exist infinitely many functions which have property_P. b) Prove that there exists an unique function with property_{P, whose range} contains an open interval centered at1.

Problem 26. Find all integersx, y such that x = q

y2₋p_y2_{+ x.}

Problem 27. Leta, b, c be three positive real numbers such that a + b + c >

1 a + 1 b + 1 c. Prove that a + b + c > 3 a + b + c + 2 abc.

Problem 28. Leta, b, c be three positive real numbers such that P1 a 6 3.

Prove that

X a2_{+ 1}

√

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Problem 29. Given that the real numbersa, b, c satisfy|ax2_{+ bx + c}_{| 6 1 for}

allx_{∈ [−1, 1] and all α ∈ [0, 1], prove that}

α(1 + α)_{|b| + (1 − α}2)_{|c| 6 1 + α}2.

Find the cases of equality.

Problem 30. An acute-angled triangleABC has orthocenter H and altitudes (AM ), (BN ), (CP ). Let Q and R be the midpoints of the segments (BH) and (CH), respectively, and let U = M Q∩ AB, V = MR ∩ AC, T = AH ∩ P N.

Prove that: a)M H_{M A} = T H_{T A};

b)T is the orthocenter of triangle U AV .

Problem 31. Consider a triangleABC, the point M on the side (BC) such

thatM B_{M C} = c(b+c)_b2 and the pointN on (AM ) such that ∠BN M = ∠BAC. Prove that2∠CN M = ∠BAC.

Problem 32. LetI be the incenter of triangle ABC and A1, B1, C1the

incen-ters of trianglesIBC, ICA, IAB, respectively. Prove that AA1, BB1, CC1 are

concurrent.

Problem 33. In a competition there were 18 teams. Each pair of teams met at

most once, and within each group of 12 teams there were at least 6 matches. Find the minimum number of matches that have been played.

10thGRADE

Problem 34. Ifa1, a2, . . . , an∈ {−1, 1} and a1+ a2+· · · + an= 0, prove

that there existsk_{∈ {1, 2, . . . , n} such that}

|a1+ 2a2+· · · + kak| 6 k

2

.

Problem 35. Prove that(22n_{+ 2}n+m_{+ 2}2m_{)! is divisible by (2}n_!)2n₊₂m−1

· (2m_!)2m₊₂n−1

for everyn, m∈ N∗_.

Problem 36. Leta_{∈ N, a > 2. Define the sequence (x}n)n>0by

x0= a2 4, x1= a 4(2a 3 − 4a2− a + 4), xn+1− (4a2− 2)xn+ xn−1= 0

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forn > 1. Prove that 2xn−a

2

−2

2 is a perfect square for everyn∈ N. Problem 37. Solve the equation

log3

2log3(2x+1)_{+ 1}

= log2(3x− 1) . Problem 38. Considerp, n _{∈ N}∗ _{and nonnegative integers}_x

1, x2, . . . , xn. Prove that 2p−1 n X i=1 xi !p 6p 1 n X i=1 x2p−1_i +p 3 n X i=1 x2p−3_i +_{· · · +} p 2m + 1 n X i=1 x2p−2m−1_i , wherem = p − 1 2 .

Problem 39. Find all positive integersp, q such that p is prime, p > q > q2

and p2 q −q_p = 1.

Problem 40. A quadrilateralA1A2A3A4has an incircle of radiusr.

a) Prove that there exist circles_Ci = (Ai, ri), centered at Ai and radiiri,

i = 1, 2, 3, 4, such that_Ciis tangent toCi+1(whereC5=C1).

b) If, in addition 4 X i=1 1 ri = 4 r,

prove that the quadrilateral is a square.

Problem 41. Consider a straight line d in space. For every n points A1,

A2, . . . , An not outsided, the union of the halfplanesSk = (dAk, 1 6 k 6 n

will be called an-fan if, when expressed in degrees, the measure of the dihedral

angle between any two halfplanes_SiandSj,1 6 i < j 6 n, is a positive integer.

a) Prove that every91-fan has two perpendicular or two mutually estending

halfplanes.

b) For each1 6 n 6 360, determine the number of n-fans containing two

perpendicular or two prolongating halfplanes (two n-fans are considered to be

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Problem 42. Show that ifa, b, c are the lengths of the sides of a triangle, R is

its circumradius andS is its area, then a2_{+ b}2_{+ c}2_{= 4}

6R2+ S2>3.

Problem 43. Given a pointP inside triangle ABC, let r1,r2,r3, respectively,

denote the inradii of trianglesP BC, P CA and P AB. Prove that a r1+ b r2 + c r3 >₆₍₂₋√_3).

Problem 44. Determine all complex numbersa, b, c such that a3_{+ b}3_{+ c}3₌

24, (a + b)(b + c)(c + a) = 64, and|a + b| = |b + c| = |c + a|.

Problem 45. With reference to the standard notations in a triangle, prove that

s 6 r

3(ma+ mb+ mc)

2R .

11thGRADE

Problem 46. a) Prove that if a matrix A ∈ M2(R) has the property that

rang (A + XY ) = rang (A + Y X) for every invertible matrices X, Y _{∈ M}2(R),

then there existsa∈ R such that A = aI2.

b) LetA_{∈ M}n(R) (n > 2) be a matrix which is not of the form aIn, a∈ R.

Prove that there existX, Y ∈ Mn(R), with X invertible and rang (A + XY ) <

rang (A + Y X).

Problem 47. Determine the largest integern > 2 with the following property:

ifA _{∈ M}n(C), A 6= λIn, for anyλ∈ C, then B ∈ Mn(C) and AB = BA

implies the existence ofa0, a1, . . . , an−1∈ C such that B = a0In+ a1A +· · · +

a_n−1An−1_.

Problem 48. Let n > 2 be an integer. Find the largest integer k > 1

with the following property: for anyk matrices A1, A2, . . . , Ak ∈ Mn(C), if

In − A1A2· · · Ak is invertible, then so is In − Aτ (1)Aτ (2)· · · Aτ (k) for every

permutationτ _{∈ S}k.

Problem 49. Consider a matrixA _{∈ M}3(R) such that det(A2+ I3) = 0.

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a)det(A + I3)− det(A − I3) = 4;

b) tr(A3_{) = tr}3_(A).

Problem 50. LetA, B, C be matrices with real entries and let

X = AB + BC + CA, Y = BA + CB + AC, Z = A2+ B2+ C2.

Prove that

det(2Z− X − Y ) > 3 det(X − Y ).

Problem 51. A functionf : R _{→ R}+ is continuous and has an irrational

period. LetM = max f . Evaluate lim

n→∞

f (1)f (2)_{· · · f(n)}

Mn .

Problem 52. A polynomialp_{∈ R[X] has the following properties: p(Q) ⊂ Q}

andp(R_{\ Q) ⊂ R \ Q. Prove that deg p = 1.}

Problem 53. Find all continuous functionsf : [0, 1]_{→ R with the following}

property: for every integern > 3 and every arithmetic sequence a1, a2, . . . , an,

the sequencef (a1), f (a2), . . . , f (an) is a geometric sequence.

Problem 54. Letf, g : R _{→ R be such that f has the intermediate value}

property and, for everyx_{∈ R, the limit} lim

h→0

f (x + h)− g(x)

h ,

exists and is finite. Prove thatf = g.

Problem 55. The functionf : (0,_{∞) → R has the property that for every} a, b∈ R, a < b, there exists c ∈ (a, b) such that f is continuous at c. Given that f (nx) < f ((n + 1)x) for every x _{∈ (0, ∞) and every n ∈ N}∗_{, prove that}_{f is}

strictly increasing.

Problem 56. Letf : (0,_{∞) → (0, ∞) be a twice differentiable function such}

that

f00(x) + f0(x) > f2(x),

for allx > 0. Prove that the limit lim

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Problem 57. A functionf : R→ R satisfies the following condition: at every x0∈ R: sup x<x0 f (x)_{− f(x}0) x− x0 = inf x>x0 f (x)_{− f(x}0) x− x0 .

Prove thatf is convex and differentiable.

Problem 58. The sequences(an)n,(bn)nand(xn)nof positive numbers

sat-isfy the conditions:

lim

n→∞a1a2· · · an= 0; n→∞lim

bn

1− an

= 0; and xn+16anxn+ bn

for everyn > 1. Prove that lim

n→∞xn = 0.

Problem 59. Consider an ellipse that is tangent to the sides of a rhombus ABCD at their midpoints. Let A0_{, B}0_{, C}0_{, D}0 _{respectively, denote the}

orthogo-nal projections ofA, B, C, D onto a variable tangent to the ellipse. Prove that AA0_{· CC}0_{= BB}0_{· DD}0_.

Problem 60. For each pointL inside a given triangle ABC, consider the

in-tersectionsE and F of the pairs of straight lines (AC, BL) and (AB, CL). Find

the locus ofL for which the quadrilateral AELF has an inscribed circle.

12thGRADE

Problem 61. SupposeA is a ring such that 1 + 1 + 1 + 1 + 1 = 0 and x4_y3_{= y}3_x4_{for all}_{x, y}

∈ A. Prove that A is commutative.

Problem 62. LetZ[α] ={a + αb | a, b ∈ Z}, where α ∈ C \ Q and |α| = 1.

Prove that exactly two of the setsZ[α] are rings under the usual operations with

complex numbers.

Problem 63. Define a sequence(an)nby

an=

Z 1 0

x2n

1 + xdx, n > 1.

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Problem 64. Letf : [0, 1] → R+ be a continuous function withf (1) = 1 and let an= Z 1 0 f (x) 1 + xn dx, n > 1. Prove that lim n→∞ 1 n Z 1 0 f (x)dx− an = ln 2. Problem 65. Find all natural numbersn such that the integral

Z n 0

x[x]{x}dx

is an integer.

Problem 66. Letf : [0,_{∞) → R be a continuous bounded function, with} f (0) = 0.

a) Prove that lim

n→∞ Z 1 0 f (nxn) dx = 0. b) Evaluate lim n→∞ Z 1 0 p 1 + n2_x2n_dx.

Problem 67. Prove that for any continuous functionf : [0, 1]_{→ R,} Z 1 0 f (x) dx_· Z 1 0 x4f (x) dx 6 4 15 Z 1 0 f2(x)dx.

Also, find the cases of equality.

Problem 68. Find all integrable functionsf : R_{→ R such that} Z x+1/n 0 f (t) dt = Z x 0 f (t) dt + 1 nf (x),

for allx_{∈ R and all n ∈ N}∗_.

Problem 69. Letf : [0,∞) → R be a function such that |f(x) − f(y)| 6 |x − y| for all x, y > 0. Prove that

Z b a f (x) dx 6 bf (b) +b 2 2 − af(a) − a2 2,

for all a, b _{∈ [0, ∞), a < b. For f differentiable, also consider the cases of}

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Authors of the problems: S¸tefan Alexe (36), Cristian Alexandrescu (16), Cristina and Claudiu Andone (13), Dumitru Barac (29, 38), Vasile Berinde (26, 58), Petru Braica (18), Dumitru Bus¸neag (48), Narcisa Bˆand˘a (68), Constantin Bus¸e (50), Vasile Cˆırtoaje, Costel Chites¸ (25, 39), Ioan Ucu Cris¸an (12), Marius Damian (22), Lucian Dragomir (15, 17), Farkas Csaba (53), Dragos¸ Fr˘at¸il˘a (68), Romant¸a and Ioan Ghit¸˘a (20), Flavian Georgescu (19), Marius Ghergu (46), Radu Gologan (9, 34, 43), Dana Heuberger (35), Marin Ionescu (52), Mircea Lascu (14, 23), Cezar Lupu (11, 27, 36, 49), I.V. Maftei (28, 61), Dorin M˘arghidan (65), Cristinel Mortici (41), Nicolae Mus¸uroia (44, 63), Dan Nedeianu (6), Vir-gil Nicula (3, 8, 31, 59, 60), Mihai Piticari (54), Dorian Popa (68), Manuela Prajea (30), Ioan Ras¸a (68), Vicent¸iu R˘adulescu (56), M. R˘adulescu (61), Alexandru Ros¸oiu (68, 69), Adrian Stoica (25, 39), Nicolae St˘aniloiu (45), Traian T˘amˆıian (1, 21), Marcel (¸27) Tena (62), Emil Vasile (24), Valentin Vornicu (27, 33)

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PART TWO

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PROBLEMS AND SOLUTIONS

DISTRICT ROUND

7thGRADE

Problem 1. Letn > 1 be an integer. Prove that the number√11. . .144. . .4

(digit “1” occursn times and digit “4” occurs 2n times) is an irrational number. Solution. We have to prove that the number11 . . . 144 . . . 4 is not a square.

Leta be the n-digit number 11 . . . 1. We have 11 . . . 144 . . . 4 = a_{· 10}2n_{+ 4a}

· 10n_{+ 4a = a(10}n_{+ 2)}2_.

Since the remainder ofa = 11 . . . 1 when divided by 4 equals 3, a is not a

square, therefore neither is11 . . . 144 . . . 4.

Problem 2. In triangleABC, we have ∠ABC = 2· ∠ACB. Prove that:

a)AC2_{= AB}2_{+ AB}

· BC;

b)AB + BC < 2_{· AC.}

Solution. a) LetBM be the angle bisector of ∠ABC.

The angle bisector theorem gives AB_BC = AM M C, hence

AM AC =

AB

AB+BC, which

impliesAM = _AB+BCAB·AC .

Since ∠ABM = ∠ACB, it follows that ∆ABM _{∼ ∆ACB, therefore}AB_AC =

AM

AB, that is,AM = AB2

AC.

It follows that _AB+BCAB·AC = AB_AC2 hence the conclusion.

b) Suppose that the parallel throughA to BM intersects BC at P . We have ∠AP M = ∠M BC = ∠ABM = ∠P AB = ∠C, hence AB = BP and AP = AC. It follows that AB + BC = P B + BC = P C < AP + AC = 2· AC, as

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Problem 3. A setM containing 4 positive integers is called connected, if for

everyx in M at least one of the numbers x_{− 1, x + 1 belongs to M. Let U}n be

the number of connected subsets of the set_{{1, 2, . . . , n}.} a) EvaluateU7.

b) Determine the leastn for which Un>2006.

Solution. Leta < b < c < d be the elements of a connected set M . Since a_{− 1 does not belong to the set, it follows that a + 1 ∈ M, hence b = a + 1.}

Similarly, sinced+1 /∈ M we deduce that d−1 ∈ M, hence c = d−1. Therefore,

a connected set has the form_{{a, a + 1, d − 1, d}, with d − a > 2.} a) There are 10 connected subsets of the set_{{1, 2, 3, 4, 5, 6, 7}:}

{1, 2, 3, 4};{1, 2, 4, 5}; {1, 2, 5, 6}; {1, 2, 6, 7}; {2, 3, 4, 5}; {2, 3, 5, 6}; {2, 3, 6, 7};

{3, 4, 5, 6}; {3, 4, 6, 7} and {4, 5, 6, 7}.

b) CallD = d_{− a + 1 the diameter of the set {a, b = a + 1, c = d − 1, d}.}

Clearly,D > 3 and D 6 n− 1 + 1 = n. For D = 4 there are n − 3 connected

sets, forD = 5 there are n_{− 4 connected sets, etc. Finally, for D = n there is one}

connected set.

Adding up yieldsUn= 1 + 2 + 3 +· · · + (n − 3) = (n−3)(n−2)₂ .

Consequently, we have to find the leastn such that (n_{− 3)(n − 2) > 4012. By}

inspection, we obtainn = 66.

Problem 4. LetABC be an isosceles triangle, with AB = AC. Let D be the

midpoint of the sideBC, M the midpoint of the line segment AD and let N be

the projection ofD on BM. Prove that ∠AN C = 90◦_.

Solution. Consider the pointS such that ABDS is a parallelogram. Clearly, ADCS is a rectangle and let R be the point of intersection of its diagonals. In the

right triangleDN S the line segment N R is the median from the right angle and

thereforeN R = 1

2· SD = 1 2· AC.

SinceR is the midpoint of AC and N R = 1

2· AC, it follows that the triangle

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36 THE57 ROMANIANMATHEMATICALOLYMPIAD

8thGRADE

Problem 1. LetABC be a right triangle (with A = 90◦). Two perpendiculars on the triangle’s plane are erected at pointsA and B, and the points M and N

are considered on these perpendiculars, on the same side of the plane, such that

BN < AM . It is known that AC = 2a, AB = a√3, AM = a and that the angle

between the planesM N C and ABC equals 30◦_{. Find:}

a) the area of triangleM N C;

b) the distance from the pointB to the plane M N C. Solution. a) The area of triangleABC equals a2

·√3. On the other hand, we

havearea[ABC] = area[M N C]_{· cos α, where α = 30}◦_{is the angle between the}

planesM N C and ABC. It follows that area[M N C] = 2· a2_.

b) Suppose that the linesM N and AB intersect at P . Let T be the projection

of the pointA on P C. Using the theorem of the three perpendiculars, we obtain

thatM T _{⊥ P C, hence ∠MT A = α = 30}◦_.

SinceAB = a, in triangle M AT we find AT = a√3, so ∠ACT = 60◦_,

henceAP = 2a√3. It follows that B is the midpoint of the line segment AP .

ProjectB on P C in Q. Using again the theorem of the three perpendiculars,

we obtainN Q⊥ P C. Then BN = a 2,BQ =

a√3

2 ,N Q = a and the altitude BS

of the right triangleBN Q equals BN ·BQ_{N Q} =a√₄3. This is the requested distance. Problem 2. For each positive integern, denote by u(n) the largest prime

num-ber less than or equal ton and by v(n) the smallest prime number greater than n.

Prove that 1 u(2)v(2) + 1 u(3)v(3) + 1 u(4)v(4) +· · · + 1 u(2010)v(2010) = 1 2− 1 2011. Solution. Letp and q be consecutive prime numbers. Then there are q− p

numbersn such that p 6 n < q and, for each such number, we have u (n) = p

andv (n) = q. It follows that the term 1

pqappears in the sum exactlyq− p times.

Since2003 and 2011 are consecutive primes, the sum becomes 3_{− 2} 2· 3 + 5_{− 3} 3· 5 +· · · + 2011_{− 2003} 2003· 2011 = =1 2− 1 3+ 1 3− 1 5+· · · + 1 2003− 1 2011 = 1 2− 1 2011.

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Problem 3. Prove that there exist infinitely many irrational numbersx and y

such thatx + y = xy_{∈ N.}

Solution. Letn = x + y = xy. Then y = n− x, so n = x (n − x) . We obtain x = n±

√ n2_{− 4n}

2 .

Since forn > 5 we have

(n− 3)2< n2− 4n < (n − 2)2,

it follows that the number√n2_{− 4n is irrational. Consequently, we can choose}

x = n+√n2 −4n 2 andy = n− √ n2 −4n 2 .

Problem 4. a) Prove that one can assign to each of the vertices of a cube one of

the numbers 1 or_{−1 such that the product of the numbers assigned to the vertices} of each face equals_−1.

b) Prove that such an assignment is impossible in the case of a regular hexag-onal prism.

Solution. a) Let ABCDA0_B0_C0_D0 _{be the cube. A possible labeling is the}

following: assign+1 to the vertices A, B, D, A0_and_{−1 to the other vertices.}

b) A contradiction is obtained by considering on one hand the product of the numbers assigned to all lateral faces and, on the other hand, the product of the numbers assigned to every second lateral face.

9thGRADE

Problem 1. Let x, y, z be positive real numbers. Prove that the following

inequality holds: 1 x2_{+ yz}+ 1 y2_{+ zx}+ 1 z2_{+ xy} 6 1 2 1 xy + 1 yz + 1 zx .

Solution. Using the AM-GM inequality we obtainx2_{+ yz > 2}p_x2_{yz, and}

therefore 1 x2_{+ yz} 6 1 2x√yz = √_yz 2xyz.

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It follows thatP_x2_+yz1 6

1 2xyz

P√

yz butP√yz 6Py+z₂ =P x

(AM-GM again). The equality holds whenx = y = z.

Problem 2. The entries of a9_{×9 array are all the numbers from 1 to 81. Prove}

that there existsk _{∈ {1, 2, 3, . . . , 9} such that the product of the numbers in the}

linek differs from the product of the numbers in the column k.

Solution. Suppose, by way of contradiction, that for eachi ∈ {1, 2, . . . , 9},

the product of the elements in linei equals the product of the elements in column i.

Between 40 and 81 there are exactly 10 prime numbers, namely 41, 43, 47, 53, 59, 61, 67, 71, 73, and 79. We prove that these numbers belong to the main diagonal of the table. Indeed, if40 < p < 81 is a prime number, then it is the only multiple

ofp in the table. If p lies on line i, by the assumption it follows that it lies on

columni, as well, that is, it lies on the main diagonal.

Therefore, on the main diagonal are all the 10 prime numbers, a contradiction.

Problem 3. LetABCD be a convex quadrilateral. Let M and N be the

mid-points of the line segmentsAB and BC, respectively. The line segments AN and BD intersect at E and the line segments DM and AC intersect at F . Prove that

ifBE = 1₃BD and AF =1₃AC, then ABCD is a parallelogram.

Solution. Denote−−→AB = −→u ,−−→BC =−→v ,−−→CD = a−→u + b−→v with a, b_{∈ R. It}

follows that−−→AD = (a + 1)−→u + (b + 1)−→v ,−−→AN =−→u +1 2

− →

v .

Since−−→BD = 3−−→BE, we obtain−→AE = 2−AB+−→ −−→AD

3 = (a+3) 3 − → u +b+1 3 − → v .

Because−→AE s¸i−−→AN are collinear vectors, we deduce that a_{− 2b + 1 = 0.}

Similarly, we obtain2a + b + 2 = 0 hence a =_{−1, b = 0, that is,}−−→CD =₋−−→AB = −−→

BA, whence ABCD is a parallelogram.

Problem 4. For each positive integern, denote by p(n) the largest prime

num-ber less than or equal ton and by q(n) the smallest prime number greater than n.

Prove that n X k=2 1 p(k)q(k) < 1 2.

Solution. Denote by2 = p1 < p2 < · · · < pm < · · · the sequence of the

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Suppose thatpm= q(n). Then n X k=2 1 p(k)q(k) 6 pm−1 X k=2 1 p(k)q(k) = m−1_X i=1 pi+1−1 X k=pi 1 p(k)q(k) m−1_X i=1 pi+1− pi pipi+1 = m−1_X i=1 1 pi − 1 pi+1 = 1 2− 1 pm <1 2. 10thGRADE

Problem 1. Consider the real numbersa, b, c ∈ (0, 1) and x, y, z ∈ (0, ∞),

such that ax= bc, by= ca, cz= ab. Prove that 1 2 + x + 1 2 + y + 1 2 + z 6 3 4. Solution. DenoteA = log1

2 a, B = log 1 2 b, C = log 1 2c. Then x = B+C A , y =C+A_B ,z =A+B_C . The inequality becomes

X 1 2 + B+C_A 6 3 2, or, denotingS = A + B + C, X A S + A 6 3 4.

The latter is equivalent to

−X_{S + A}A >₋3 4 or X 1−_{S + A}A > 9 4, or 4SX 1 S + A >9,

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Problem 2. LetABC be a triangle and consider the points M ∈ (BC), N ∈ (CA), P _{∈ (AB) such that} AP_{P B} = BM_{M C} = CN_{N A}. Prove that if M N P is an

equilateral triangle, thenABC is an equilateral triangle as well. Solution. Letλ =AP_AB =BM_BC = CN_CA.

We use complex numbers and we choose the pointM as origin. Furthermore,

we can assume that the complex numbers corresponding to the pointsN and P are

1 andε = cosπ₃ + i sinπ₃, respectively.

Suppose that the complex numbers corresponding to the pointsA, B, C are a, b, c, respectively. We have then

ε = (1_{− λ)a + λb, 0 = (1 − λ)b + λc, and 1 = (1 − λ)c + λa.}

It follows thatc−a

b−a = ε. Therefore, AC = AB and A = π 3.

Problem 3. A prism is called binary if one can assign to each of its vertices

a number from the set_{{−1, +1}, in such a way that the product of the numbers} assigned to the vertices of every face equals_−1.

a) Prove that the number of vertices of every binary prism is divisible by 8. b) Prove that there are binary prisms with 2000 entries.

Solution. a) Suppose the base of the prism is a polygon withn vertices. Then

the product of the numbers assigned to the vertices of the lateral faces equals

(₋₁₎n_{, but in the same time it must be equal to 1, since every vertex is counted}

twice. It follows thatn is an even number.

Now, ifn = 4k + 2, for some k, then we consider the product of the numbers

assigned to the vertices of every second lateral face. We obtain(₋₁₎2k+1₌

−1.

This equals the product of all numbers, that is 1, which is a contradiction. This proves the result.

b) Label the verticesA1, A3, A5, . . . , A997with−1 and label the rest of the

base vertices with 1. For the upper base, label all with 1, exceptA999, labelled−1. Problem 4. a) Find two setsX, Y such that X_{∩ Y = ∅, X ∪ Y = Q}∗

+and

Y =_{{a · b | a, b ∈ X}.}

b) Find two setsU, V such that U ∩ V = ∅, U ∪ V = R and V = {x + y | x, y∈ U}.

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Solution. a) As an example, we can chooseX as the set of all products of

the typepα1

1 pα22· · · pαkk, wherep1, p2, . . . , pk are distinct prime numbers,αi are

integers and

n

P

i=1

αiis odd. Finally, we setY = Q∗+\ X.

b) Choose

U = [

k∈Z

[3k + 1, 3k + 2) and V = R\ U.

It is not difficult to check that these sets satisfy the requested conditions.

11th

GRADE

Problem 1. Letx > 0 be a real number and let A be a 2_{× 2 matrix with real}

entries, such that

det(A2+ xI2) = 0.

Prove that

det(A2_{+ A + xI} 2) = x.

Solution. We have det(A + i√xI2)· det(A − i√xI2) = 0; therefore, denoting

byd the determinant of A and by t its trace, it results d = x and t = 0, hence A2+ xI2= 02. It follows that det(A2+ A + xI2) = det(A) = x.

Problem 2. Letn, p > 2 be integer numbers and let A be a n_{× n real matrix}

such thatAp+1 = A.

a) Prove that rank(A) + rank(In− Ap) =n.

b) Prove that ifp is a prime number, then

rank(In− A) = rank(In− A2) =· · · = rank(In− Ap−1).

Solution. a) The Sylvester inequality yields rank(A) + rank(In − Ap) 6

rank(A(In− Ap)) + n = n.

On the other hand, rank(A) + rank(In− Ap) > rank(Ap) + rank(In− Ap) >

rank(Ap_{+ (I}

n− Ap)) = n.

b) Observe that ifk, m_{∈ N}∗_and_k_{|m then rank (I}

n− Ak) > rank(In− Am).

Indeed,In− Amcan be written as a product of two matrices, one of them

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Letk ∈ N, 1 6 k 6 p − 1. We have Akp+1 _{= A for all k} _{∈ N. Since p is}

a prime number, the remainders of the numbersp + 1, 2p + 1, . . . , kp + 1 when

divided byk are pairwise distinct. Therefore, one of these numbers, say t = qp+1,

is divisible byk. Thus, rank(In− A) > rank(In − Ak) > rank(In− At) =

rank(In− Apq+1) = rank(In− A).

Problem 3. The sequence of real numbers(xn)n>0satisfies

(xn+1− xn)(xn+1+ xn+ 1) 6 0, n > 0.

a) Prove that the sequence is bounded. b) Can such a sequence be divergent?

Solution. a) The hypothesis impliesx2

n+1+ xn+1 6 x2n+ xn, whence the

sequenceyn = x2n+ xnis decreasing.

Since(yn) is clearly bounded from below, it is a convergent sequence.

There-fore,(xn) is bounded.

b) The answer is “yes”; an example is the sequencexn = −1+(−1)

n

2 .

Problem 4. We say that a functionf : R_{→ R has the property (P) if for every}

realx,

sup

t6x

f (t) = x.

a) Give an example of a function having property (P) which is discontinuous at every real point.

b) Prove that iff is continuous and has property (P) then f is the identical

function. Solution. a) An example is f (x) = ( x ifx_{∈ Q;} x_{− 1 if x ∈ R \ Q.}

b) Observe thatsup

t6xf (t) = supy6t6xf (t), for all y 6 x.

Sincef is continuous, for each n_{∈ N}∗_{, there exists}_x

n < x, such that

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for allt∈ [xn, x]. Consequently, sup xn6t6x f (t)− f(x) 6 1 n,

that is,_{|x − f(x)| 6} _n1, for alln_{∈ N}∗_{. It follows that}_{f (x) = x.}

12thGRADE

Problem 1. Letf1, f2, . . . , fn : [0, 1]→ (0, ∞) be continuous functions and

letσ be a permutation of the set{1, 2, . . . , n}. Prove that

n Y i=1 Z 1 0 f2 i(x) fσ(i)(x) dx > n Y i=1 Z 1 0 fi(x) dx.

Solution. Sincefi(x) > 0 for x∈ [0, 1], i = 1, 2, . . . , n, we can use

Cauchy-Schwartz inequality: Z 1 0 f2 i(x) fσ(i)(x) dx Z 1 0 fσ(i)(x) dx > Z 1 0 fi(x) dx 2 ,

for eachi = 1, 2, . . . , n. Taking the product of these inequalities yields the result.

Problem 2. Let G = {A ∈ M2(R)| det(A) = ±1} and H = {A ∈

M2(C)| det A = 1}. Prove that, under matrix multiplication, G and H are

non-isomorphic groups.

Solution. It is not difficult to show thatG and H are groups. If they were

isomorphic, then the equationX2_{= I}

2should have the same number of solutions

in both groups. Cayley theorem implies that this equation has exactly two solutions inH, namely_±I2.

Since the equation has other solutions inG_{\ H, e.g.,}

X = 0 a

1/a 0 !

, a_{∈ C}∗,

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Problem 3. LetA be a finite commutative ring having at least two elements.

Prove that for every positive integern > 2, there exists a polynomial f _{∈ A[X],}

withdeg f = n, having no roots in A.

Solution. Observe that the functionϕ : A → A, ϕ(x) = xn _{− x, is not}

one-to-one, sinceϕ(0) = 0 = ϕ(1).

BecauseA is a finite set, it follows that ϕ is not onto either.

Therefore, one can finda∈ A \ Im ϕ. But then, the polynomial f = Xn₋

X− a has no roots in A.

Problem 4. Let_{F = {f : [0, 1] → [0, ∞) | f continuous} and let n > 2 be a}

positive integer. Determine the least real constantc, such that Z 1 0 f (√n_{x)dx 6 c} Z 1 0 f (x)dx for everyf _{∈ F.}

Solution. Substitute √n_{x = t to obtain}

Z 1 0 f (√n_{x)dx = n} Z 1 0 tn−1f (t)dt 6 Z 1 0 f (t)dt, hencec 6 n.

Forp > 0, the function fp: [0, 1]→ [0, 1], fp(x) = xp, belongs toF.

R1 0 x p ndx 6 cR1 0 x p_{dx implies} n n+p 6 c p+1, thereforec > pn+n p+n. Finally,c > lim p→∞ pn+n

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FINAL ROUND

7th

GRADE

Problem 1. Consider the triangleABC and points M , N belonging to the

sidesAB, BC respectively, such that 2·CN_BC = AM_AB. Let P be a point on AC.

Prove that the linesM N and N P are perpendicular if and only if P N bisects the

angle ∠M P C.

Solution. LetT be the intersection point of the parallel to AC which contains N with line AB. FromCN

BC = AT

AB we getAM = 2· AT , thus T is the midpoint

ofAM . A M T B N Q P

Denote by Q the intersection point of M N and AC. In triangle P M Q the

pointN is the midpoint of M Q.

In the triangleP M Q, P N is a median, thus P N is perpendicular to M N if

and only ifP N bisects the angle ∠M P C.

Problem 2. A square of siden is divided into n2 _{unit squares each colored}

red, yellow or green. Find the minimum value ofn such that for any such coloring

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Solution. The number is 7. Forn = 7, at least 17 squares have the same color

by the PGH principle (49 = 3_{· 16 + 1).}

As17 = 7_{· 2 + 3, we get, by the same principle, that among the 7 rows there}

is one containing three squares of the same color. The same argument works for columns.

The fact that forn = 6 the result is no more valid is given by the following

example. r g a r g a g a r q a r a r g a r g r g a r g a g a r g a r a r g a r g

The same table can be used to find counterexamples for anyn 6 6. Problem 3. In the acute triangleABC angle C equals 45◦_{. Points A}

1andB1

are the foots of the perpendiculars fromA and B respectively. Denote by H the

ortocenter ofABC. Points D and E are situated on the segments AA1andBC,

respectively, such thatA1D = A1E = A1B1. Prove that:

a)A1B1=

q

A1B2+A1C2

2 ;

b)CH = DE.

Solution. a) As the triangleABC is acute, we have ∠ABC > 45◦_{, so the}

midpointM of BC is situated on the segment A1C. We get B1M = BC₂ = A1B+A1C 2 andA1M = M B− A1B = BC 2 − A1B = A1C−A1B 2 .

In the right triangleM A1B1we also have

A1B21= A1M2+B1M2= A1B + A1C 2 2 + A1C− A1B 2 2 =A1B 2_{+ A} 1C2 2 , thus A1B1= r A1B2+ A1C2 2 .

b) As the right triangleDA1E is isosceles we have succesively

DE = A1E· √ 2 = A1B1· √ 2 =pA1B2+ A1C2= p A1B2+ A1A2= AB.

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The equality of trianglesAA1B and CA1H implies AB = CH, and, as a

consequenceCH = DE.

Problem 4. Let A be a set of nonnegative integers containing at least two

elements and such that for anya, b_{∈ A, a > b, we have} [a,b]_a−b _{∈ A. Prove that the}

setA contains exactly two elements.

([a, b] denotes the least common multiple of a and b).

Solution. We begin by proving thatA is finite. For, if b = min A and a _∈ A\ {b}, then from (a − b)|[a, b] we get (a − b)|ab. As (a − b)|(a − b) we get (a_{− b)|ab − b(a − b), thus a − b|b}2_{, and, in turn, a 6 b + b}2_{. But a}

∈ A was

arbitrarily chosen, soA is finite.

Puta = max A and b = min A. If d = (a, b), then b = dx, a = dy, with x, y_{∈ N}∗_and_{(x, y) = 1. Then}[a,b]

a−b = xy

y−x∈ N∗. As x, y and x− y are mutually

coprime, we deducey− x = 1 or y = x + 1, implying a = d(x + 1) and b = dx.

Then[a,b]

a−b = x(x + 1)∈ A, from which b 6 x(x + 1) 6 a or d ∈ {x, x + 1}.

First case. d = x.

We havea = x(x + 1) and b = x2_{. We show that A has no other elements.}

By contradiction ifc = min A\ {b}, we get, as before,d0, z∈ N∗_{such that}

a = d0_{(z + 1) and c = d}0_{z. Then}[a,c]

a−c = z(z + 1)∈ A. As z(z + 1) 6= x2= b, or

c 6 z(z + 1) 6 a we obtain d0 _{∈ {z, z + 1}.}

Ifd0 _{= z, then a = z(z + 1) and, as a = x(x + 1), we would have x = z, a}

contradiction (this would leads tob = c).

Ifd0 _{= z + 1, then a = (z + 1)}2_and_{a = x(x + 1), a contradiction. Thus, in}

this caseA has exactly two elements.

Second case.d = x + 1.

We havea = (x + 1)2_and_{b = x(x + 1). As in the previous case it is easy to}

show thatA has no other elements.

8th

GRADE

Problem 1. Consider a convex poliedra with 6 faces each of them being a

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Solution. It is known that a quadrilateral is circumscriptible if the sums of

opposite sides are the same. If 5 of the faces of the convex body are circumscribed quadrilaterals, we can suppose that the body isABCDA0_B0_C0_D0 _with

quadri-lateralsABCD and A0_B0_C0_D0 _{opposite. Denote by}_{x, y, z, t and x}0_{, y}0_{, z}0_{, t}0_and

a, b, c, d the sides AB, BC, CD, DA and A0_B0_{, B}0_C0_{, C}0_D0_{, D}0_A0 _and

AA0_{, BB}0_,_CC0_{, DD}0 _{respectively. Suppose all faces except}_A0_B0_C0_D0 _are

cir-cumscribed quadrilaterals. Then

x0+ y0= (x_{− a − b) + (z − c − d) = (x + z) − a − b − c − d} = (t + y)− a − b − c − d = t0_{+ y}0_,

thusA0_B0_C0_D0_{is also circumscribed.}

Problem 2. Given a positive integern, prove that there exists an integer k, k > 2 and numbers a1, a2, . . . , ak ∈ {−1, 1} such that

n = X

16i<j6k

aiaj. Solution. Consider the identity

(a1+ a2+· · · + ak)2= a21+ a22+· · · + a2k+ 2

X

16i<j6k

aiaj.

Thus, the problem amounts to finding an integerk > 2, and a1, a2, . . . , ak ∈

{−1, 1} such that 2n = (a1+ a2+· · · + ak)2− (a12+ a22+· · · + a2k) = (a1+

a2+· · · + ak)2− k.

Letm be the number of ones in the sequence a1, a2, . . . , akandp = k− m the

number of minus ones. We have2n = (m_{− p)}2

− k, or, denoting by l = m − p, 2n = l2+ l− 2m. We have to find l, m ∈ N.

Takel_{∈ N, l > 2 with l}2_{+ l > 2n and m =} l2

+l−2n 2 ∈ N.

Thenk =_{−l + 2m = l}2

− 2n satisfies the given condition with a1 = a2 =

· · · = am= 1 and the remaining ones equal to−1.

Problem 3. LetABCDA1B1C1D1be a cube and letP be a variable point

on side[AB]. The plane through P , perpendicular to AB meets AC at Q. Let M

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a) Prove that the linesM N and BC1are perpendicular if and only ifP is the

midpoint ofAB.

b) Find the minimal value of the angle between the linesM N and BC1. Solution. a) Denote by O the center of the square BCC1B1. If P is the

midpoint ofAB, then Q is the midpoint of AC1, thusP BOQ is a parallelogram.

This means that the pointsP, N and O are collinear and M N is parallel to A1O.

As the triangleA1BC1is equilateral, we getA1O⊥ BC1, thusM N ⊥ BC1.

For the converse,M N is perpendicular to BC1, and, asBC1is also

perpen-dicular toA1O we have that A1O k MN, or BC1 ⊥ (A1OP ). But as BC1 is

not perpendicular toOP , we must have A1O k MN. This means that N is the

midpoint ofOP.

It follows thatP BOQ is a parallelogram and as a consequence of the fact that Q is the midpoint of AC1, we get thatP is also the midpoint of AB.

b) LetU be the point where the parallel through Q to AB meets the line BC1.

AsQP BU is a parallelogram we get P N = N U , thus M N bisects the sides A1P

andA1U of the triangle. As a consequence, the angle between M N and BC1

equals the angle betweenA1U and BC1. The triangleA1BC1is equilateral. This

implies that the angle between the linesA1U and BC1is at least60◦. Equality

occurs forP = A or P = B.

Problem 4. Consider real numbers a, b, c contained in the interval [1 2, 1]. Prove that 2 6 a + b 1 + c + b + c 1 + a+ c + a 1 + b 63.

Solution. We begin by proving the lefthand-side inequality. Sincea, b > 1₂,

we havea + b > 1, thus

a + b 1 + c >

a + b a + b + c

and the like.

Summing up the three we obtain

2 = (a + b) + (b + c) + (c + a) a + b + c 6 a + b 1 + c + b + c 1 + a+ c + a 1 + b.

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For the second inequality, observe that the considered expression can be writ-ten X a 1 + c+ c 1 + a .

Asa, c 6 1, we have_1+ca 6 _a+ca and_1+ac 6 _c+ac , so a 1 + c+ c 1 + a 6 a a + c+ c c + a= 1

and the like. Summing up the three we get the desired result.

9thGRADE

Problem 1. Find the maximum value of (x3_{+ 1)(y}3_{+ 1),}

forx, y∈ R such that x + y = 1.

Solution. Putxy = t; as x + y = 1 we get (x3_{+ 1)(y}3_{+ 1) = t}3

− 3t + 2.

Fromx + y = 1 we obtain t = xy 6 x+y₂ 2 = 1₄. It is easy to prove that

t3_{− 3t + 2 6 4 for t 6} 1

4, with equality if and only ift =−1.

We infer that(x3_{+ 1)(y}3 _{+ 1) 6 4 for x, y}

∈ R with x + y = 1 and (φ3_{+ 1)(}

−1/φ3_{+ 1) = 4, where φ is one of the roots of z}2

− z − 1 = 0. Remark. In fact, forx, y∈ R, we have

[x3+ (x + y)3][y3+ (x + y)3] 6 4(x + y)6,

with equality if and only ifx2_{+ 3xy + y}2_{= 0.}

Problem 2. Consider the trianglesABC and DBC such that AB = BC, DB = DC and ∠ABD = 90◦_{. Let}_{M be the midpoint of BC. Points E, F, P are}

such thatE _{∈ (AB), P ∈ (MC), C ∈ (AF ) and ∠BDE = ∠ADP = ∠CDF .}

Prove thatP is the midpoint of EF and DP ⊥ EF .

Solution. Putu = ∠BDE = ∠M DP = ∠CDF . In the right triangles DBE, DM P, DCP we have cos ∠BDE = BD DE, cos ∠M DP = DM DP , cos ∠CDF = DC DF.