RO-NMO 2021

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 $DN\cap BM=\{P\}$ . [list=a] [] 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 $(ABC)$ . [/list]

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

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

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: [list=1] [] $f$ is surjective; [] $c=0$ , $b<d$ and $0<a\le d$ . [/list]

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

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 $2^n=1(mod n)$