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10 Years of Romanian Mathematical competitions / Radu Gologan, Dan $tefanMarinescu, Mihai Monea. - Pitegti :Paralela45,2Al8

IndexrsBN 978-973 -47 -27 20-9

I. Marinescu. Dan-$tefanlI. Monea, Mihai

51

Copyright @ Editura Paralela 45,20lB

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Radu Gologan

Dan $tefan Marinescu

Mihai Monea

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Editura Paralela 45

Contents

Preface . . ...2Members of the RMC Editorial Board .. . . .8Abbreviations and Notations . ........ g

Algebra ... 111.1. Problems ... . .. 11

1.2. Solutions.. ....18Combinatorics .......87

2.1. Problems.. .... JT

2.2. Solutions .. .... 44

Geometry.... ...693.1. Problems . . .... Og

3.2. Solutions.. ....76NumberTheory ......99

4.1. Problems .. .... gg

4.2. Solutions.. ...10bCalculus .. L2L

5.1. Problems.. ...I2I5.2. Solutions . . .. .I2g

Higher Algebra .... lb16.1. Problems .. . .. 151

6.2. Solutions . . .. .ISTAuthors Index .... LTB

-dbb

Algebra

1.1. Problems

A1. We say that the function / ' Ql ---+ Q has property 2 if

f@a):f(*)+f(a),

for anv r.u e Oi.sf

a) Prove that there are no one-to-one functions with property P;b) Does there exist onto functions with property P?

Bogdan Moldovan - ONM 2017

A2. Determineallintegers n)2 suchthat a+J2 €Qand a"+12€Q,for some real number o depending on n.

Mihai BXIund - IIMO 2017

A3. Let o € N, a 2 2. Prove that the following statements are equivalents:

a) One can find b, c € N* such that a2 : b2 + c2;

b) One can find d € N*, such that the equations 12 - ar ld : 0 and12 - ar - d:0 have integer roots.

Vasile Pop - OJM 2016

A4. a) Prove

that

forallr€iR;b) Prove that any function g : IR --+ IR such that

g (r +1) + g (, - D :'[3s 1r1,

for all ,r € lR, is periodical.

that there exist non-periodical functions ,f , R ---+ lR. such

f ("+1)+f ("-1) :JEf @),

Vasile Pop - OJM 2016

A5.

10 Years of Romanian Mathematical Competi,ti,ons

Let f : lR. --+ IR be a function with the following properties:

(P1) : f(r+s) </(r)+f(a)

- t) a) ( r/ (r) + (1 - t) f (il,

1(i<l(n

Nicolae Bourbdcu! - ONM 2016

A6. Determine all functions / : N* ---+ N*, such that f (m) 2 m andf (* +n) divides f (m) + f (n), for all m,n € N*.

Marius Cavachi - IIMO 2016

A7. Find all real numbers a and b such that the equality

lar -r ba) + lbr + aa) : @ + b)1, + a)

holds, for every r, y € IR.

Lucian Dragomir - OJM 2015

A8. A quadratic function / sends any interval 1 of length 1 to an interval/(1) of length at ieast 1. Prove that for any interval .r of length 2, thelengthof the intervai /(/) is at least 4.

Mihai Bdlund, - ONM 201b

f(tr+(1,€[0, 1] .

f@+fd such that

and

(P2):

for all r,?J e IR and

a) Prove that

foranya(b(c(b) Prove that

,(r'-)\t:r )

foralln€N,n23,

f (n-r, (I f k-))

and r1r12r...r/?? € ]R.

&npetiti,ons

8t),

Alsebra

A9. Let a be a natural odd number which is not a perfect square. IfTrt,n e N* prove that:

") {*@+ t/d} I {"@ - t/")};b) L-@+1G))ll"@-J')l

Vasile Pop - ONM 2014

A10. Consider the function y' : N* --+ N* which satisfies the properties:

a) f(1):1;u) f (p): 1 t f (p - 1), for anv prime number p;

c) f (ptpr...p) : f (pr) + f (pz) +.'.+ f (p,), for any prime numbers

Pt. pz. ..., p,. not necessary distinct.prove 1tnu12f @) l nr --3/("); for aly n € N, n ) 2.

George Stoica - ONM 2014

A11. Find all functions ./,.q ' Q --+ Q such that

f (g(*) + g(a)): f (g(r)) +a

and

for all r, g e Q.

g (f (r) + f (a)) : s (f (d) + a,

Vasile Pop - ONM 2015

AI2. Let n € N,n ) 2. Prove that there exist n*l pairwise distinctnumbers rr, fr2, ...,, fin, fin+r e Q \ Z such that

t'?) + {"8} +.. + {'i} : {r'.*.},where {r} denotes the fractional part of the real number r.

orel Mihet - IIMO 2014

A13. Given an odd prime p, determine all polynomials / and g, withintegral coefficients, satisfying the condition f (g(X)) : t;:lX*

Cezar Lupu & Vlad Matei - IIMO 2014

13

- oNM 2016

)mand

i - IIMO 2016

- oJM 20i5

! to an interval2- the length

- oNlr 2015

74 10 Years of Romani,an Mathemati,cal Competi,ti,ons

Al4. Let f : IR. ---+ lR be a function and g : lR. ---+ IR. be a quadratic functionsuch that, for any m,n € lR, the equation f (r): mr-fn has soiutions ifand only if the equation g(r):mr I n has soiutions. prove that f : g.

Vasile Pop - ONM 2013

A15. Letn € N* and at,a2,...,enelR suchthat a1 loz+...Iat lk,forany k € {I,2, . . .,n}. Prove that

a102an111r + t +... + ; S t+ r*... -r -.

Marius Cavachi - OJM 2013

A16. Let n € N, n ) 2, and an,bn, cn e Z such that

/ "- \n(72-t) :antb,W*cn(4.

Prove that cn: 1 (mod 3) if and only if n:2(mod 3).

Dorin Andrica & Mihai Piticari - IIMO 2013

AL7. Let o,,b e (0,oo) such that a I b andn € N*. Prove that a and b are both integer.

[na] divides [nbJ, for every

Marius Cavachi - IIMO 2013

A18. Let f and g be two polynomials with integral coefficients such thatd"s(/) > deg (9) and d"s (/) > 2. rf the polynomiar pf * 9 has a rationalroot, for infinitely many primes p, prove that / has a rational root.

Marius cavachi & Radu Gologan - IMo 2012 short List & tIMo 2013

A19. An irrational number r from (0, 1) is called sui,table if its first 4 de-cimals, in the decimal representation, are equals. Find the smallest rz € N*such that any t € (0,1) may be written as a sum of n distinct sui,tablenumbers.

Lucian lurea - TJBMO 2007