Olympiad problems

Russian TST 2017, P3

Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$, there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$.

1987 Traian Lălescu, 2.1

Tags: extension rings , superrings , polynomial , ring theory , algebra , abstract algebra

Any polynom, with coefficients in a given division ring, that is irreducible over it, is also irreducible over a given extension skew ring of it that's finite. Prove that the ring and its extension coincide.

1986 Traian Lălescu, 1.1

Tags: ring theory , field theory , abstract algebra , morphism , characteristic , superior algebra

Let be two nontrivial rings linked by an application ($ K\stackrel{\vartheta }{\mapsto } L $) having the following properties: $ \text{(i)}\quad x,y\in K\implies \vartheta (x+y) = \vartheta (x) +\vartheta (y) $ $ \text{(ii)}\quad \vartheta (1)=1 $ $ \text{(iii)}\quad \vartheta \left( x^3\right) =\vartheta^3 (x) $ [b]a)[/b] Show that if $ \text{char} (L)\ge 4, $ and $ K,L $ are fields, then $ \vartheta $ is an homomorphism. [b]b)[/b] Prove that if $ K $ is a noncommutative division ring, then it’s possible that $ \vartheta $ is not an homomorphism.

2008 District Olympiad, 4

Tags: polynomial , ring theory , finite field , wedderburn , superior algebra , algebra

Let be a finite field $ K. $ Say that two polynoms $ f,g $ from $ K[X] $ are [i]neighbours,[/i] if the have the same degree and they differ by exactly one coefficient. [b]a)[/b] Show that all the neighbours of $ 1+X^2 $ from $ \mathbb{Z}_3[X] $ are reducible in $ \mathbb{Z}_3[X] . $ [b]b)[/b] If $ |K|\ge 4, $ show that any polynomial of degree $ |K|-1 $ from $ K[X] $ has a neighbour from $ K[X] $ that is reducible in $ K[X] , $ and also has a neighbour that doesn´t have any root in $ K. $

2008 Romania National Olympiad, 3

Tags: number theory , least common multiple , ring theory , search , linear algebra , group theory , matrix

Let $ A$ be a unitary finite ring with $ n$ elements, such that the equation $ x^n\equal{}1$ has a unique solution in $ A$, $ x\equal{}1$. Prove that a) $ 0$ is the only nilpotent element of $ A$; b) there exists an integer $ k\geq 2$, such that the equation $ x^k\equal{}x$ has $ n$ solutions in $ A$.

2022 Romania National Olympiad, P4

Tags: ring theory , group theory

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 \[|G|\geq\frac{|U|^2}{|Z\cap U|}.\][i]Dragoș Crișan[/i]

1951 Miklós Schweitzer, 10

Tags: ring theory , algebra , polynomial , modular arithmetic

Let $ f(x)$ be a polynomial with integer coefficients and let $ p$ be a prime. Denote by $ z_1,...,z_{p\minus{}1}$ the $ (p\minus{}1)$th complex roots of unity. Prove that $ f(z_1)\cdots f(z_{p\minus{}1})\equiv f(1)\cdots f(p\minus{}1) \pmod{p}$.

2016 District Olympiad, 1

Tags: fta , ring theory , superior algebra , group theory , abstract algebra

A ring $ A $ has property [i](P),[/i] if $ A $ is finite and there exists $ (\{ 0\}\neq R,+)\le (A,+) $ such that $ (U(A),\cdot )\cong (R,+) . $ Show that: [b]a)[/b] If a ring has property [i](P),[/i] then, the number of its elements is even. [b]b)[/b] There are infinitely many rings of distinct order that have property [i](P).[/i]

2001 Miklós Schweitzer, 3

Tags: ring theory , matrices , abstract algebra

How many minimal left ideals does the full matrix ring $M_n(K)$ of $n\times n$ matrices over a field $K$ have?

Gheorghe Țițeica 2024, P3

Tags: ring theory , abstract algebra

Determine all commutative rings $R$ with at least four elements that are not fields, such that for any pairwise distinct and nonzero elements $a,b,c\in R$, $ab+bc+ca$ is invertible. [i]Vlad Matei[/i]

2014 PUMaC Individual Finals A, 1

Tags: ring theory , barycentric , geometry , incenter , analytic geometry , circumcircle , geometric transformation

Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) for which $AB+AC=3BC$. Let the point where $AC$ is tangent to $\gamma$ be $D$. Let the incenter of $I$. Let the intersection of the circumcircle of $\triangle BCI$ with $\gamma$ that is closer to $B$ be $P$. Show that $PID$ is collinear.

2003 Romania National Olympiad, 4

Tags: limit , ring theory , group theory , abstract algebra

$ i(L) $ denotes the number of multiplicative binary operations over the set of elements of the finite additive group $ L $ such that the set of elements of $ L, $ along with these additive and multiplicative operations, form a ring. Prove that [b]a)[/b] $ i\left( \mathbb{Z}_{12} \right) =4. $ [b]b)[/b] $ i(A\times B)\ge i(A)i(B) , $ for any two finite commutative groups $ B $ and $ A. $ [b]c)[/b] there exist two sequences $ \left( G_k \right)_{k\ge 1} ,\left( H_k \right)_{k\ge 1} $ of finite commutative groups such that $$ \lim_{k\to\infty }\frac{\# G_k }{i\left( G_k \right)} =0 $$ and $$ \lim_{k\to\infty }\frac{\# H_k }{i\left( H_k \right)} =\infty. $$ [i]Barbu Berceanu[/i]

1990 Vietnam Team Selection Test, 1

Tags: least common multiple , greatest common divisor , ring theory , number theory , relatively prime

Let $ T$ be a finite set of positive integers, satisfying the following conditions: 1. For any two elements of $ T$, their greatest common divisor and their least common multiple are also elements of $ T$. 2. For any element $ x$ of $ T$, there exists an element $ x'$ of $ T$ such that $ x$ and $ x'$ are relatively prime, and their least common multiple is the largest number in $ T$. For each such set $ T$, denote by $ s(T)$ its number of elements. It is known that $ s(T) < 1990$; find the largest value $ s(T)$ may take.

2018 Ramnicean Hope, 3

Tags: automorphism , morphism , ring theory , endomorphism , polynomial ring , function

[b]a)[/b] Let $ u $ be a polynom in $ \mathbb{Q}[X] . $ Prove that the function $ E_u:\mathbb{Q}[X]\longrightarrow\mathbb{Q}[X] $ defined as $ E_u(P)=P(u) $ is an endomorphism. [b]b)[/b] Let $ E $ be an injective endomorphism of $ \mathbb{Q} [X] . $ Show that there exists a nonconstant polynom $ v $ in $ \mathbb{Q}[X] $ such that $ E(P)=P(v) , $ for any $ P $ in $ \mathbb{Q}[X] . $ [b]c)[/b] Let $ A $ be an automorphism of $ \mathbb{Q}[X] . $ Demonstrate that there is a nonzero constant polynom $ w $ in $ \mathbb{Q}[X] $ which has the property that $ A(P)=P(w) , $ for any $ P $ in $ \mathbb{Q}[X] . $ [i]Marcel Țena[/i]

1974 Miklós Schweitzer, 4

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

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]

Gheorghe Țițeica 2025, P4

Tags: abstract algebra , ring theory

Let $R$ be a ring. Let $x,y\in R$ such that $x^2=y^2=0$. Prove that if $x+y-xy$ is nilpotent, so is $xy$. [i]Janez Šter[/i]

2002 Romania National Olympiad, 1

Tags: superior algebra , ring theory

Let $A$ be a ring. $a)$ Show that the set $Z(A)=\{a\in A|ax=xa,\ \text{for all}\ x\in A\}$ is a subring of the ring $A$. $b)$ Prove that, if any commutative subring of $A$ is a field, then $A$ is a field.

2004 Alexandru Myller, 3

Tags: superior algebra , abstract algebra , ring theory , nilpotence

Prove that the number of nilpotent elements of a commutative ring with an order greater than $ 8 $ and congruent to $ 3 $ modulo $ 6 $ is at most a third of the order of the ring.

2006 Iran Team Selection Test, 1

Tags: polynomial , ring theory , combinatorial geometry , algebra , combinatorics

We have $n$ points in the plane, no three on a line. We call $k$ of them good if they form a convex polygon and there is no other point in the convex polygon. Suppose that for a fixed $k$ the number of $k$ good points is $c_k$. Show that the following sum is independent of the structure of points and only depends on $n$ : \[ \sum_{i=3}^n (-1)^i c_i \]

1985 Traian Lălescu, 1.4

Tags: abstract algebra , superior algebra , ring theory

Let $ A $ be a ring in which $ 1\neq 0. $ If $ a,b\in A, $ then the following affirmations are equivalent: $ \text{(i)}\quad aba=a\wedge ba^2b=1 $ $ \text{(ii)}\quad ab=ba=1 $ $ \text{(iii)}\quad \exists !b\in A\quad aba=a $

2024 District Olympiad, P1

Tags: ring theory , number theory

Determine the integers $n\geqslant 2$ for which the equation $x^2-\hat{3}\cdot x+\hat{5}=\hat{0}$ has a unique solution in $(\mathbb{Z}_n,+,\cdot).$

1993 All-Russian Olympiad, 4

Tags: binomial theorem , induction , ring theory , vector , function , algebra , polynomial

If $ \{a_k\}$ is a sequence of real numbers, call the sequence $ \{a'_k\}$ defined by $ a_k' \equal{} \frac {a_k \plus{} a_{k \plus{} 1}}2$ the [i]average sequence[/i] of $ \{a_k\}$. Consider the sequences $ \{a_k\}$; $ \{a_k'\}$ - [i]average sequence[/i] of $ \{a_k\}$; $ \{a_k''\}$ - average sequence of $ \{a_k'\}$ and so on. If all these sequences consist only of integers, then $ \{a_k\}$ is called [i]Good[/i]. Prove that if $ \{x_k\}$ is a [i]good[/i] sequence, then $ \{x_k^2\}$ is also [i]good[/i].

2006 Iran Team Selection Test, 1

Tags: combinatorics , combinatorial geometry , ring theory , polynomial , algebra

We have $n$ points in the plane, no three on a line. We call $k$ of them good if they form a convex polygon and there is no other point in the convex polygon. Suppose that for a fixed $k$ the number of $k$ good points is $c_k$. Show that the following sum is independent of the structure of points and only depends on $n$ : \[ \sum_{i=3}^n (-1)^i c_i \]

1973 Miklós Schweitzer, 2

Tags: superior algebra , ring theory

Let $ R$ be an Artinian ring with unity. Suppose that every idempotent element of $ R$ commutes with every element of $ R$ whose square is $ 0$. Suppose $ R$ is the sum of the ideals $ A$ and $ B$. Prove that $ AB\equal{}BA$. [i]A. Kertesz[/i]

2010 N.N. Mihăileanu Individual, 4

Tags: totient , ring theory , modular arithmetic , abstract algebra

If $ p $ is an odd prime, then the following characterization holds. $$ 2^{p-1}\equiv 1\pmod{p^2}\iff \sum_{2=q}^{(p-1)/2} q^{p-2}\equiv -1\pmod p $$ [i]Marius Cavachi[/i]