Olympiad problems

2020 Stars of Mathematics, 4

Let $a_0 = 1, \ a_1 = 2,$ and $a_2 = 10,$ and define $a_{k+2} = a_{k+1}^3+a_k^2+a_{k-1}$ for all positive integers $k.$ Is it possible for some $a_x$ to be divisible by $2021^{2021}?$ [i]Flavian Georgescu[/i]

2021 Winter Stars of Mathematics, 3

Tags: number theory , quadratic residue , romania

Determine the primes $p$ for which the numbers $2\lfloor p/k\rfloor - 1, \ k = 1,2,\ldots, p,$ are all quadratic residues modulo $p.$ [i]Vlad Matei[/i]

2022 District Olympiad, P4

Tags: Integral , function , romania , college contests

Let $I\subseteq \mathbb{R}$ be an open interval and $f:I\to\mathbb{R}$ a strictly monotonous function. Prove that for all $c\in I$ there exist $a,b\in I$ such that $c\in (a,b)$ and \[\int_a^bf(x) \ dx=f(c)\cdot (b-a).\]

2016 Danube Mathematical Olympiad, 1

Tags: romania , algebra

Let $S=x_1x_2+x_3x_4+\cdots+x_{2015}x_{2016},$ where $x_1,x_2,\ldots,x_{2016}\in\{\sqrt{3}-\sqrt{2},\sqrt{3}+\sqrt{2}\}.$ Can $S$ be equal to $2016?$ [i]Cristian Lazăr[/i]

2021 Romania EGMO TST, P4

Tags: number theory , romania

Determine all non-negative integers $n$ for which there exist two relatively prime non-negative integers $x$ and $y$ and a positive integer $k\geqslant 2$ such that $3^n=x^k+y^k$.

2021 Stars of Mathematics, 2

Tags: combinatorics , Array , romania

Fix integers $m \geq 3$ and $n \geq 3$. Each cell of an array with $m$ rows and $n$ columns is coloured one of two colours such that: [b](1)[/b] Both colours occur on every column; and [b](2)[/b] On every two rows the cells on the same column share colour on exactly $k$ columns. Show that, if $m$ is odd, then \[\frac{n(m-1)}{2m}\leq k\leq \frac{n(m-2)}{m}\] [i]The Problem Selection Committee[/i]

2019 Romania National Olympiad, 1

Tags: inequalities , romania

If $a,b,c>0$ then $$\frac{1}{abc}+1\ge3\left(\frac{1}{a^2+b^2+c^2}+\frac{1}{a+b+c}\right)$$

2000 District Olympiad (Hunedoara), 3

Tags: limits , Sequences , calculus , real analysis , contests , romania

Let be two distinct natural numbers $ k_1 $ and $ k_2 $ and a sequence $ \left( x_n \right)_{n\ge 0} $ which satisfies $$ x_nx_m +k_1k_2\le k_1x_n +k_2x_m,\quad\forall m,n\in\{ 0\}\cup\mathbb{N}. $$ Calculate $ \lim_{n\to\infty}\frac{n!\cdot (-1)^{1+n}\cdot x_n^2}{n^n} . $

2021 Stars of Mathematics, 1

Tags: number theory , prime numbers , romania

For every integer $n\geq 3$, let $s_n$ be the sum of all primes (strictly) less than $n$. Are there infinitely many integers $n\geq 3$ such that $s_n$ is coprime to $n$? [i]Russian Competition[/i]

2020 Romania EGMO TST, P1

Tags: number theory , Sequence , romania

Let $a$ be a positive integer and $(a_n)_{n\geqslant 1}$ be a sequence of positive integers satisfying $a_n<a_{n+1}\leqslant a_n+a$ for all $n\geqslant 1$. Prove that there are infinitely many primes which divide at least one term of the sequence. [i]Moldavia Olympiad, 1994[/i]

2021 Junior Balkan Team Selection Tests - Romania, P2

Tags: number theory , Subsets , romania , Romanian TST

For any non-empty subset $X$ of $M=\{1,2,3,...,2021\}$, let $a_X$ be the sum of the greatest and smallest elements of $X$. Determine the arithmetic mean of all the values of $a_X$, as $X$ covers all the non-empty subsets of $M$.

2020 Romania EGMO TST, P4

Tags: romania , combinatorics , board

Determine the greatest positive integer $A{}$ with the following property: however we place the numbers $1,2,\ldots, 100$ on a $10\times 10$ board, each number appearing exactly once, we can find two numbers on the same row or column which differ by at least $A{}$.

2010 Romania Team Selection Test, 2

Tags: geometry , circumcircle , incenter , romania , TST

Let $ABC$ be a scalene triangle, let $I$ be its incentre, and let $A_1$, $B_1$ and $C_1$ be the points of contact of the excircles with the sides $BC$, $CA$ and $AB$, respectively. Prove that the circumcircles of the triangles $AIA_1$, $BIB_1$ and $CIC_1$ have a common point different from $I$. [i]Cezar Lupu & Vlad Matei[/i]

2021 Winter Stars of Mathematics, 1

Tags: geometry , romania

Let $ABC$ be a triangle, and let $D, E$ and $F$ be the feet of the altitudes from $A, B$ and $C,$ respectively. A circle $\omega_A$ through $B$ and $C$ crosses the line $EF$ at $X$ and $X'$. Similarly, a circle $\omega_B$ through $C$ and $A$ crosses the line $FD$ at $Y$ and $Y',$ and a circle $\omega_C$ through $A$ and $B$ crosses the line $DE$ at $Z$ and $Z'$. Prove that $X, Y$ and $Z$ are collinear if and only if $X', Y'$ and $Z'$ are collinear. [i]Vlad Robu[/i]

2022 Junior Balkan Team Selection Tests - Romania, P3

Tags: number theory , romania , Romanian TST

Determine all pairs of positive integers $(a,b)$ such that the following fraction is an integer: \[\frac{(a+b)^2}{4+4a(a-b)^2}.\]

2018 District Olympiad, 4

Tags: function , romania

Let $f:\mathbb{R} \to\mathbb{R}$ be a function. For every $a\in\mathbb{Z}$ consider the function $f_a : \mathbb{R} \to\mathbb{R}$, $f_a(x) = (x - a)f(x)$. Prove that if there exist infinitely many values $a\in\mathbb{Z}$ for which the functions $f_a$ are increasing, then the function $f$ is monotonic.

2022 Romania National Olympiad, P4

Tags: romania , algebra , inequalities

Let $a<b<c<d$ be positive integers which satisfy $ad=bc.$ Prove that $2a+\sqrt{a}+\sqrt{d}<b+c+1.$ [i]Marius Mînea[/i]

2015 District Olympiad, 2

Tags: discrete maths , real life problem , romania

At a math contest there were $ 50 $ participants, where they were given $ 3 $ problems each to solve. The results have shown that every candidate has solved correctly at least one problem, and that a total of $ 100 $ problems have been evaluated by the jury as correct. Show that there were, at most, $ 25 $ winners who got the maximum score.

2022 District Olympiad, P1

Tags: darboux s property , function , college contests , romania

Let $f,g:\mathbb{R}\to\mathbb{R}$ be functions which satisfy \[\inf_{x>a}f(x)=g(a)\text{ and }\sup_{x<a}g(x)=f(a),\]for all $a\in\mathbb{R}.$ Given that $f$ has Darboux's Property (intermediate value property), show that functions $f$ and $g$ are continuous and equal to each other. [i]Mathematical Gazette [/i]

2021 Romania EGMO TST, P2

Tags: geometry , romania

Through the midpoint $M$ of the side $BC$ of the triangle $ABC$ passes a line which intersects the rays $AB$ and $AC$ at $D$ and $E$, respectively, such that $AD=AE$. Let $F$ be the foot of the perpendicular from $A$ onto $BC$ and $P{}$ the circumcenter of triangle $ADE$. Prove that $PF=PM$.

2022 District Olympiad, P1

Tags: romania , logarithms , algebra

Determine all $x\in(0,3/4)$ which satisfy \[\log_x(1-x)+\log_2\frac{1-x}{x}=\frac{1}{(\log_2x)^2}.\]

2021 Junior Balkan Team Selection Tests - Romania, P1

Tags: algebra , inequalities , romania , Romanian TST

Let $a,b,c>0$ be real numbers with the property that $a+b+c=1$. Prove that \[\frac{1}{a+bc}+\frac{1}{b+ca}+\frac{1}{c+ab}\geq\frac{7}{1+abc}.\]

2016 Danube Mathematical Olympiad, 3

Tags: geometry , romania

Let $ABC$ be a triangle with $AB < AC,$ $I$ its incenter, and $M$ the midpoint of the side $BC$. If $IA=IM,$ determine the smallest possible value of the angle $AIM$.

2000 District Olympiad (Hunedoara), 4

Tags: functions , limits , real analysis , contests , romania , calculus , integration

Let $ f:[0,1]\longrightarrow\mathbb{R}_+^* $ be a Riemann-integrable function. Calculate $ \lim_{n\to\infty}\left(-n+\sum_{i=1}^ne^{\frac{1}{n}\cdot f\left(\frac{i}{n}\right)}\right) . $

2022 Romania Team Selection Test, 1

Tags: combinatorics , romania , Romanian TST

A finite set $\mathcal{L}$ of coplanar lines, no three of which are concurrent, is called [i]odd[/i] if, for every line $\ell$ in $\mathcal{L}$ the total number of lines in $\mathcal{L}$ crossed by $\ell$ is odd. [list=a] [*]Prove that every finite set of coplanar lines, no three of which are concurrent, extends to an odd set of coplanar lines. [*]Given a positive integer $n$ determine the smallest nonnegative integer $k$ satisfying the following condition: Every set of $n$ coplanar lines, no three of which are concurrent, extends to an odd set of $n+k$ coplanar lines. [/list]