Olympiad problems

1965 Miklós Schweitzer, 2

Tags: vector , algebra , polynomial , induction , ring theory , superior algebra

Let $ R$ be a finite commutative ring. Prove that $ R$ has a multiplicative identity element $ (1)$ if and only if the annihilator of $ R$ is $ 0$ (that is, $ aR\equal{}0, \;a\in R $ imply $ a\equal{}0$).

1974 Miklós Schweitzer, 4

Tags: superior algebra , ring theory , function , induction , number theory , prime number

Let $ R$ be an infinite ring such that every subring of $ R$ different from $ \{0 \}$ has a finite index in $ R$. (By the index of a subring, we mean the index of its additive group in the additive group of $ R$.) Prove that the additive group of $ R$ is cyclic. [i]L. Lovasz, J. Pelikan[/i]

2002 All-Russian Olympiad, 1

Tags: algebra , polynomial , quadratic , ring theory , equation , functional equation

The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.

2005 Romania National Olympiad, 4

Tags: superior algebra , induction , topology , ring theory

Let $A$ be a ring with $2^n+1$ elements, where $n$ is a positive integer and let \[ M = \{ k \in\mathbb{Z} \mid k \geq 2, \ x^k =x , \ \forall \ x\in A \} . \] Prove that the following statements are equivalent: a) $A$ is a field; b) $M$ is not empty and the smallest element in $M$ is $2^n+1$. [i]Marian Andronache[/i]

2017 District Olympiad, 4

Tags: superior algebra , idempotence , ring theory

Let $ A $ be a ring that is not a division ring, and such that any non-unit of it is idempotent. Show that: [b]a)[/b] $ \left( U(A) +A\setminus\left( U(A)\cup \{ 0\} \right) \right)\cap U(A) =\emptyset . $ [b]b)[/b] Every element of $ A $ is idempotent.

2024 District Olympiad, P3

Tags: ring theory , number theory

Let $k$ be a positive integer. A ring $(A,+,\cdot)$ has property $P_k$ if for any $a,b\in A$ there exists $c\in A$ such that $a^k=b^k+c^k.$[list=a] [*]Give an example of a finite ring $(A,+,\cdot)$ which [i]does not[/i] have $P_k$ for any $k\geqslant 2.$ [*]Let $n\geqslant 3$ be an integer and $M_n=\{m\in\mathbb{N}:(\mathbb{Z}_n,+,\cdot)\text{ has }P_m\}.$ Prove that all the elements of $M_n$ are odd integers and that $(M_n,\cdot)$ is a monoid. [/list]

2006 IMC, 3

Tags: linear algebra , matrix , ring theory

Let $A$ be an $n$x$n$ matrix with integer entries and $b_{1},b_{2},...,b_{k}$ be integers satisfying $detA=b_{1}\cdot b_{2}\cdot ...\cdot b_{k}$. Prove that there exist $n$x$n$-matrices $B_{1},B_{2},...,B_{k}$ with integers entries such that $A=B_{1}\cdot B_{2}\cdot ...\cdot B_{k}$ and $detB_{i}=b_{i}$ for all $i=1,...,k$.

1993 China Team Selection Test, 3

Tags: geometry , circumcircle , incenter , trigonometry , geometric transformation , reflection , ring theory

Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.

2022 Romania National Olympiad, P2

Tags: ring theory

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