Let $\mathcal C$ be a circle centered at $O$ and $A\ne O$ be a point in its interior. The perpendicular bisector of the segment $OA$ meets $\mathcal C$ at the points $B$ and $C$, and the lines $AB$ and $AC$ meet $\mathcal C$ again at $D$ and $E$, respectively. Show that the circles $(OBC)$ and $(ADE)$ have the same centre. Ion Pătrașcu, Ion Cotoi

Solve the system in reals: $\frac{4-a}{b}=\frac{5-b}{a}=\frac{10}{a^2+b^2}$.

Let $ABC$ be a scalene triangle with $\angle BAC>90^\circ$. Let $D$ and $E$ be two points on the side $BC$ such that $\angle BAD=\angle ACB$ and $\angle CAE=\angle ABC$. The angle-bisector of $\angle ACB$ meets $AD$ at $N$, If $MN\parallel BC$, determine $\angle (BM, CN)$. Petru Braica

Determine the smallest non-negative integer $n$ such that \[\sqrt{(6n+11)(6n+14)(20n+19)}\in\mathbb Q.\] Mihai Bunget

In the cuboid $ABCDA'B'C'D'$ with $AB=a$, $AD=b$ and $AA'=c$ such that $a>b>c>0$, the points $E$ and $F$ are the orthogonal projections of $A$ on the lines $A'D$ and $A'B$, respectively, and the points $M$ and $N$ are the orthogonal projections of $C$ on the lines $C'D$ and $C'B$, respectively. Let $DF\cap BE=\{G\}$ and $DN\cap BM=\{P\}$. Show that $(A'AG)\parallel (C'CP)$ and determine the distance between these two planes; Show that $GP\parallel (ABC)$ and determine the distance between the line $GP$ and the plane $(ABC)$. Petre Simion, Nicolae Victor Ioan

Prove that for all positive real numbers $a,b,c$ the following inequality holds: \[(a+b+c)\left(\frac1a+\frac1b+\frac1c\right)\ge\frac{2(a^2+b^2+c^2)}{ab+bc+ca}+7\]and determine all cases of equality. Lucian Petrescu

Solve the system in reals: $(x+\sqrt{x^2+1})(y+\sqrt{y^2+1})=2022$ and $x+y=\frac{2021}{\sqrt{2022}}$

Students in a class of $n$ students had to solve $2^{n-1}$ problems on an exam. It turned out that for each pair of distinct problems: • there is at least one student who has solved both • there is at least one student who has solved one of them, but not the other. Show that there is a problem solved by all the students in the class.

Let $ABC$ be an acute-angled triangle with the circumcenter $O$. Let $D$ be the foot of the altitude from $A$. If $OD\parallel AB$, show that $\sin 2B = \cot C$. Mădălin Mitrofan

Let $P_0, P_1,\ldots, P_{2021}$ points on the unit circle of centre $O$ such that for each $n\in \{1,2,\ldots, 2021\}$ the length of the arc from $P_{n-1}$ to $P_n$ (in anti-clockwise direction) is in the interval $\left[\frac{\pi}2,\pi\right]$. Determine the maximum possible length of the vector: \[\overrightarrow{OP_0}+\overrightarrow{OP_1}+\ldots+\overrightarrow{OP_{2021}}.\] Mihai Iancu

If $a,b,c>0,a+b+c=1$,then: $\frac{1}{abc}+\frac{4}{a^{2}+b^{2}+c^{2}}\geq\frac{13}{ab+bc+ca}$

Let $A$ be a finite set of non-negative integers. Determine all functions $f:\mathbb{Z}_{\ge 0} \to A$ such that \[f(|x-y|)=|f(x)-f(y)|\]for each $x,y\in\mathbb Z_{\ge 0}$. Andrei Bâra

Find the complex numbers $x,y,z$,with $\mid x\mid=\mid y\mid=\mid z\mid$,knowing that $x+y+z$ and $x^{3}+y^{3}+z^{3}$ are be real numbers.

Let $a,b,c,d\in\mathbb{Z}_{\ge 0}$, $d\ne 0$ and the function $f:\mathbb{Z}_{\ge 0}\to\mathbb Z_{\ge 0}$ defined by \[f(n)=\left\lfloor \frac{an+b}{cn+d}\right\rfloor\text{ for all } n\in\mathbb{Z}_{\ge 0}.\]Prove that the following are equivalent: $f$ is surjective; $c=0$, $b<d$ and $0<a\le d$. Tiberiu Trif

Let $n\ge 2$ be a positive integer such that the set of $n$th roots of unity has less than $2^{\lfloor\sqrt n\rfloor}-1$ subsets with the sum $0$. Show that $n$ is a prime number. Cristi Săvescu

Determine all nonzero integers $a$ for which there exists two functions $f,g:\mathbb Q\to\mathbb Q$ such that \[f(x+g(y))=g(x)+f(y)+ay\text{ for all } x,y\in\mathbb Q.\]Also, determine all pairs of functions with this property. Vasile Pop

Let $f:[a,b] \rightarrow \mathbb{R}$ a function with Intermediate Value property such that $f(a) * f(b) < 0$. Show that there exist $\alpha$, $\beta$ such that $a < \alpha < \beta < b$ and $f(\alpha) + f(\beta) = f(\alpha) * f(\beta)$.

Let $n \ge 2$ and $ a_1, a_2, \ldots , a_n $, nonzero real numbers not necessarily distinct. We define matrix $A = (a_{ij})_{1 \le i,j \le n} \in M_n( \mathbb{R} )$ , $a_{i,j} = max \{ a_i, a_j \}$, $\forall i,j \in \{ 1,2 , \ldots , n \} $. Show that $\mathbf{rank}(A) $= $\mathbf{card} $ $\{ a_k | k = 1,2, \ldots n \} $

Let $f :\mathbb R \to\mathbb R$ a function $ n \geq 2$ times differentiable so that: $ \lim_{x \to \infty} f(x) = l \in \mathbb R$ and $ \lim_{x \to \infty} f^{(n)}(x) = 0$. Prove that: $ \lim_{x \to \infty} f^{(k)}(x) = 0 $ for all $ k \in \{1, 2, \dots, n - 1\} $, where $f^{(k)}$ is the $ k $ - th derivative of $f$.

Let $n \ge 2$ and matrices $A,B \in M_n(\mathbb{R})$. There exist $x \in \mathbb{R} \backslash \{0,\frac{1}{2}, 1 \}$, such that $ xAB + (1-x)BA = I_n$. Show that $(AB-BA)^n = O_n$.

Find all continuous functions $f:\left[0,1\right]\rightarrow[0,\infty)$ such that: $\int_{0}^{1}f\left(x\right)dx\cdotp\int_{0}^{1}f^{2}\left(x\right)dx\cdotp...\cdotp\int_{0}^{1}f^{2020}\left(x\right)dx=\left(\int_{0}^{1}f^{2021}\left(x\right)dx\right)^{1010}$

Determine all non-trivial finite rings with am unit element in which the sum of all elements is invertible. Mihai Opincariu

Given is an positive integer $a>2$ a) Prove that there exists positive integer $n$ different from $1$, which is not a prime, such that $a^n=1(mod n)$ b) Prove that if $p$ is the smallest positive integer, different from $1$, such that $a^p=1(mod p)$, then $p$ is a prime. c) There does not exist positive integer $n$, different from $1$, such that $2^n=1(mod n)$

Let be $f:\left[0,1\right]\rightarrow\left[0,1\right]$ a continuous and bijective function,such that : $f\left(0\right)=0$.Then the following inequality holds: $\left(\alpha+2\right)\cdotp\int_{0}^{1}x^{\alpha}\left(f\left(x\right)+f^{-1}\left(x\right)\right)\leq2,\forall\alpha\geq0 $