Let $a,b$ be positive integers. Prove that the equation $x^2+(a+b)^2x+4ab=1$ has rational solutions if and only if $a=b$. Mihai Opincariu

Let $ABC$ be a right triangle with $\angle A=90^\circ.$ Let $A'$ be the midpoint of $BC,$ $M$ be the midpoint of the height $AD$ and $P$ be the intersection of $BM$ and $AA'.$ Prove that $\tan\angle PCB=\sin C\cdot\cos C.$ Daniel Văcărețu

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ for which there exists a function $g:\mathbb{R}\to\mathbb{R}$ such that $f(x)+f(y)=\lfloor g(x+y)\rfloor$ for all real numbers $x$ and $y$. Emil Vasile

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

Let $a\neq 1$ be a positive real number. Find all real solutions to the equation $a^x=x^x+\log_a(\log_a(x)).$ Mihai Opincariu

Let $z_1$ and $z_2$ be complex numbers. Prove that \[|z_1+z_2|+|z_1-z_2|\leqslant |z_1|+|z_2|+\max\{|z_1|,|z_2|\}.\]Vlad Cerbu and Sorin Rădulescu

Let $Z\subset \mathbb{C}$ be a set of $n$ complex numbers, $n\geqslant 2.$ Prove that for any positive integer $m$ satisfying $m\leqslant n/2$ there exists a subset $U$ of $Z$ with $m$ elements such that\[\Bigg|\sum_{z\in U}z\Bigg|\leqslant\Bigg|\sum_{z\in Z\setminus U}z\Bigg|.\]Vasile Pop

Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that $$|f(A)\cap f(B)|=|A\cap B|$$whenever $A$ and $B$ are two distinct subsets of $X$. (Sergiu Novac)

Let $f:[0,1]\to(0,1)$ be a surjective function. Prove that $f$ has at least one point of discontinuity. Given that $f$ admits a limit in any point of the interval $[0,1],$ show that is has at least two points of discontinuity. Mihai Piticari and Sorin Rădulescu

Let $\mathcal{F}$ be the set of pairs of matrices $(A,B)\in\mathcal{M}_2(\mathbb{Z})\times\mathcal{M}_2(\mathbb{Z})$ for which there exists some positive integer $k$ and matrices $C_1,C_2,\ldots, C_k\in\{A,B\}$ such that $C_1C_2\cdots C_k=O_2.$ For each $(A,B)\in\mathcal{F},$ let $k(A,B)$ denote the minimal positive integer $k$ which satisfies the latter property. Let $(A,B)\in\mathcal{F}$ with $\det(A)=0,\det(B)\neq 0$ and $k(A,B)=p+2$ for some $p\in\mathbb{N}^*.$ Show that $AB^pA=O_2.$ Prove that for any $k\geq 3$ there exists a pair $(A,B)\in\mathcal{F}$ such that $k(A,B)=k.$ Bogdan Blaga

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ which are differentiable in $0$ and satisfy the following inequality for all real numbers $x,y$ \[f(x+y)+f(xy)\geq f(x)+f(y).\]Dan Ștefan Marinescu and Mihai Piticari

Let $A,B\in\mathcal{M}_n(\mathbb{C})$ such that $A^2+B^2=2AB.$ Prove that for any complex number $x$\[\det(A-xI_n)=\det(B-xI_n).\]Mihai Opincariu and Vasile Pop

Let $\mathcal{F}$ be the set of functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(2x)=f(x)$ for all $x\in\mathbb{R}.$ Determine all functions $f\in\mathcal{F}$ which admit antiderivatives on $\mathbb{R}.$ Give an example of a non-constant function $f\in\mathcal{F}$ which is integrable on any interval $[a,b]\subset\mathbb{R}$ and satisfies \[\int_a^bf(x) \ dx=0\]for all real numbers $a$ and $b.$ Mihai Piticari and Sorin Rădulescu

Determine all rings $(A,+,\cdot)$ such that $x^3\in\{0,1\}$ for any $x\in A.$ Mihai Opincariu

Let $f,g:\mathbb{R}\to\mathbb{R}$ be two nondecreasing functions. Show that for any $a\in\mathbb{R},$ $b\in[f(a-0),f(a+0)]$ and $x\in\mathbb{R},$ the following inequality holds \[\int_a^xf(t) \ dt\geq b(x-a).\] Given that $[f(a-0),f(a+0)]\cap[g(a-0),g(a+0)]\neq\emptyset$ for any $a\in\mathbb{R},$ prove that for any real numbers $a<b$\[\int_a^b f(t) \ dt=\int_a^b g(t) \ dt.\] Note: $h(a-0)$ and $h(a+0)$ denote the limits to the left and to the right respectively of a function $h$ at point $a\in\mathbb{R}.$

Let $(R,+,\cdot)$ be a ring with center $Z=\{a\in\mathbb{R}:ar=ra,\forall r\in\mathbb{R}\}$ with the property that the group $U=U(R)$ of its invertible elements is finite. Given that $G$ is the group of automorphisms of the additive group $(R,+),$ prove that \[|G|\geq\frac{|U|^2}{|Z\cap U|}.\]Dragoș Crișan