Olympiad problems

2007 Grigore Moisil Intercounty, 3

Find the natural numbers $ a $ that have the property that there exists a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ such that $ f(f(n))=a+n, $ for any natural number $ n, $ and the function $ g:\mathbb{N}\longrightarrow\mathbb{N} $ defined as $ g(n)=f(n)-n $ is injective.

2023 4th Memorial "Aleksandar Blazhevski-Cane", P2

Tags: injectivity , positive real , surjectivity , fixed point , function , algebra

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all functions $f:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that for all $x,y>0$ we have $$f(xy+f(x))=yf(x)+x.$$ [i]Proposed by Nikola Velov[/i]

2021 Alibaba Global Math Competition, 14

Tags: analysis , metric space , injectivity , geometry

Let $f$ be a smooth function on $\mathbb{R}^n$, denote by $G_f=\{(x,f(x)) \in \mathbb{R}^{n+1}: x \in \mathbb{R}^n\}$. Let $g$ be the restriction of the Euclidean metric on $G_f$. (1) Prove that $g$ is a complete metric. (2) If there exists $\Lambda>0$, such that $-\Lambda I_n \le \text{Hess}(f) \le \Lambda I_n$, where $I_n$ is the unit matrix of order $n$, and $\text{Hess}8f)$ is the Hessian matrix of $f$, then the injectivity radius of $(G_f,g)$ is at least $\frac{\pi}{2\Lambda}$.

2004 Nicolae Păun, 1

Tags: decomposition , injectivity , algebra , function

Prove that any function that maps the integers to themselves is a sum of any finite number of injective functions that map the integers to themselves. [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

2013 Bogdan Stan, 2

Tags: floor function , function , injectivity , surjectivity , bijectivity , integer part , algebra

Consider the parametric function $ f_k:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=x+k\lfloor x \rfloor . $ [b]a)[/b] For which integer values of $ k $ the above function is injective? [b]b)[/b] For which integer values of $ k $ the above function is surjective? [b]c)[/b] Given two natural numbers $ n,m, $ create two bijective functions: $$ \phi : f_m (\mathbb{R} )\cap [0,\infty )\longrightarrow f_n(\mathbb{R})\cap [0,\infty ) $$ $$ \psi : \left(\mathbb{R}\setminus f_m (\mathbb{R})\right)\cap [0,\infty )\longrightarrow\left(\mathbb{R}\setminus f_n (\mathbb{R})\right)\cap [0,\infty ) $$ [i]Cristinel Mortici[/i]

2020 Bulgaria EGMO TST, 2

Tags: algebra , functional equation , injectivity , bijection , surjectivity , function

The function $f:\mathbb{R} \to \mathbb{R}$ is such that $f(f(x+1)) = x^3+1$ for all real numbers $x$. Prove that the equation $f(x) = 0 $ has exactly one real root.

2004 Alexandru Myller, 1

Tags: function , combinatorics , injectivity

Find the number of self-maps of a set of $ 5 $ elements having the property that the preimage of any element of this set has $ 2 $ elements at most. [i]Adrian Zanoschi[/i]

2008 Alexandru Myller, 4

Tags: superior algebra , ring theory , cauchy , injectivity

In a certain ring there are as many units as there are nilpotent elements. Prove that the order of the ring is a power of $ 2. $ [i]Dinu Şerbănescu[/i]

2006 Petru Moroșan-Trident, 1

Tags: abstract algebra , group theory , endomorphism , injectivity , surjectivity

Let be a natural number $ n\ge 4, $ and a group $ G $ for which the applications $ \iota ,\eta : G\longrightarrow G $ defined by $ \iota (g) =g^n ,\eta (g) =g^{2n} $ are endomorphisms. Prove that $ G $ is commutative if $ \iota $ is injective or surjective. [i]Gh. Andrei[/i]