Found problems: 22
2007 Grigore Moisil Intercounty, 2
[b]a)[/b] Show that there is no function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the property that
$$ f(f(x))=\left\{ \begin{matrix} \sqrt{2007} ,& \quad x\in\mathbb{Q} \\ 2007, & \quad x\not\in \mathbb{Q} \end{matrix} \right. , $$
for any real number $ x. $
[b]b)[/b] Prove that there is an infinite number of functions $ g:\mathbb{R}\longrightarrow\mathbb{R} $ having the property that
$$ g(g(x))=\left\{ \begin{matrix} 2007 ,& \quad x\in\mathbb{Q} \\ \sqrt{2007}, & \quad x\not\in \mathbb{Q} \end{matrix} \right. , $$
for any real number $ x. $
2025 ISI Entrance UGB, 4
Let $S^1 = \{ z \in \mathbb{C} \mid |z| =1 \}$ be the unit circle in the complex plane. Let $f \colon S^1 \longrightarrow S^2$ be the map given by $f(z) = z^2$. We define $f^{(1)} \colon = f$ and $f^{(k+1)} \colon = f \circ f^{(k)}$ for $k \geq 1$. The smallest positive integer $n$ such that $f^{(n)}(z) = z$ is called the [i]period[/i] of $z$. Determine the total number of points in $S^1$ of period $2025$.
(Hint : $2025 = 3^4 \times 5^2$)
1984 Polish MO Finals, 1
Find the number of all real functions $f$ which map the sum of $n$ elements into the sum of their images, such that $f^{n-1}$ is a constant function and $f^{n-2}$ is not. Here $f^0(x) = x$ and $f^k = f \circ f^{k-1}$ for $k \ge 1$.
2017 NZMOC Camp Selection Problems, 7
Let $a, b, c, d, e$ be distinct positive integers such that $$a^4 + b^4 = c^4 + d^4 = e^5.$$
Show that $ac + bd$ is composite.
2020 Brazil National Olympiad, 6
Let $f (x) = 2x^2 + x - 1$, $f^0(x) = x$ and $f^{n + 1}(x) = f (f^n(x))$ for all real $x$ and $n \ge 0$ integer .
(a) Determine the number of real distinct solutions of the equation of $f^3(x) = x$.
(b) Determine, for each integer $n \ge 0$, the number of real distinct solutions of the equation $f^n(x) = 0$.
2007 Grigore Moisil Intercounty, 3
Let be two functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ such that $ g $ has infinite limit at $ \infty . $
[b]a)[/b] Prove that if $ g $ continuous then $ \lim_{x\to\infty } f(x) =\lim_{x\to\infty } f(g(x)) $
[b]b)[/b] Provide an example of what $ f,g $ could be if $ f $ has no limit at $ \infty $ and $ \lim_{x\to\infty } f(g(x)) =0. $
1981 Austrian-Polish Competition, 9
For a function $f : [0,1] \to [0,1] $ we define $f^1 = f $ and $f^{n+1} (x) = f (f^n(x))$ for $0 \le x \le 1$ and $n \in N$. Given that there is a $n$ such that $|f^n(x) - f^n(y)| < |x - y| $ for all distinct $x, y \in [0,1]$, prove that there is a unique $x_0 \in [0,1]$ such that $f (x_0) = x_0$.
2023 Grosman Mathematical Olympiad, 3
Find all pairs of polynomials $p$, $q$ with complex coefficients so that
\[p(x)\cdot q(x)=p(q(x)).\]
1998 Tournament Of Towns, 6
In a function $f (x) = (x^2 + ax + b )/ (x^2 + cx + d)$ , the quadratics $x^2 + ax + b$ and $x^2 + cx + d$ have no common roots. Prove that the next two statements are equivalent:
(i) there is a numerical interval without any values of $f(x)$ ,
(ii) $f(x)$ can be represented in the form $f (x) = f_1 (f_2( ... f_{n-1} (f_n (x))... ))$ where each of the functions $f_j$ is o f one of the three forms $k_j x + b_j, 1/x, x^2$ .
(A Kanel)
1989 Tournament Of Towns, (219) 3
Given $1000$ linear functions $f_k(x)=p_k x + q_k$ where $k = 1 , 2 ,... , 1000$, it is necessary to evaluate their composite $f(x) =f_1 (f_2(f_3 ... f_{1000}(x)...))$ at the point $x_0$ . Prove that this can be done in no more than $30$ steps, where at each step one may execute simultaneously any number of arithmetic operations on pairs of numbers obtained from the previous step (at the first step one may use the numbers $p_1 , p_2 ,... ,p_{1000}, q_l , q_2 ,... ,q_{1000} , x_o$).
{S. Fomin, Leningrad)
2006 Victor Vâlcovici, 1
Let be a nondegenerate and closed interval $ I $ of real numbers, a short map $ m:I\longrightarrow I, $ and a sequence of functions $ \left( x_n \right)_{n\ge 1} :I\longrightarrow\mathbb{R} $ such that $ x_1 $ is the identity map and
$$ 2x_{n+1}=x_n+m\circ x_n , $$
for any natural numbers $ n. $ Prove that:
[b]a)[/b] there exists a nondegenerate interval having the property that any point of it is a fixed point for $ m. $
[b]b)[/b] $ \left( x_n \right)_{n\ge 1} $ is pointwise convergent and that its limit function is a short map.
1986 Austrian-Polish Competition, 9
Find all continuous monotonic functions $f : R \to R$ that satisfy $f (1) = 1$ and $f(f (x)) = f (x)^2$ for all $x \in R$.
2024 Irish Math Olympiad, P10
Let $\mathbb{Z}_+=\{1,2,3,4...\}$ be the set of all positive integers. Find, with proof, all functions $f : \mathbb{Z}_+ \mapsto \mathbb{Z}_+$ with the property that $$f(x+f(y)+f(f(z)))=z+f(y)+f(f(x))$$ for all positive integers $x,y,z$.
2016 Puerto Rico Team Selection Test, 6
$N$ denotes the set of all natural numbers. Define a function $T: N \to N$ such that $T (2k) = k$ and $T (2k + 1) = 2k + 2$. We write $T^2 (n) = T (T (n))$ and in general $T^k (n) = T^{k-1} (T (n))$ for all $k> 1$.
(a) Prove that for every $n \in N$, there exists $k$ such that $T^k (n) = 1$.
(b) For $k \in N$, $c_k$ denotes the number of elements in the set $\{n: T^k (n) = 1\}$.
Prove that $c_{k + 2} = c_{k + 1} + c_k$, for $1 \le k$.
2013 Balkan MO Shortlist, A7
Suppose that $k$ is a positive integer. A bijective map $f : Z \to Z$ is said to be $k$-[i]jumpy [/i] if $|f(z) - z| \le k$ for all integers $z$.
Is it that case that for every $k$, each $k$-jumpy map is a composition of $1$-jumpy maps?
[i]It is well known that this is the case when the support of the map is finite.[/i]
2013 India Regional Mathematical Olympiad, 1
Find the number of eight-digit numbers the sum of whose digits is $4$
1998 Belarus Team Selection Test, 1
Let $n\ge 2$ be positive integer. Find the least possible number of elements of tile set $A =\{1,2,...,2n-1,2n\}$ that should be deleted in order to the sum of any two different elements remained be a composite number.
1995 Singapore Team Selection Test, 1
Let $f(x) = \frac{1}{1+x}$ where $x$ is a positive real number, and for any positive integer $n$,
let $g_n(x) = x + f(x) + f(f(x)) + ... + f(f(... f(x)))$, the last term being $f$ composed with itself $n$ times. Prove that
(i) $g_n(x) > g_n(y)$ if $x > y > 0$.
(ii) $g_n(1) = \frac{F_1}{F_2}+\frac{F_2}{F_3}+...+\frac{F_{n+1}}{F_{n+2}}$ , where $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} +F_n$ for $n \ge 1$.
1991 Austrian-Polish Competition, 9
For a positive integer $n$ denote $A = \{1,2,..., n\}$. Suppose that $g : A\to A$ is a fixed function with $g(k) \ne k$ and $g(g(k)) = k$ for $k \in A$. How many functions $f: A \to A$ are there such that $f(k)\ne g(k)$ and $f(f(f(k))= g(k)$ for $k \in A$?
1966 Swedish Mathematical Competition, 4
Let $f(x) = 1 + \frac{2}{x}$. Put $f_1(x) = f(x)$, $f_2(x) = f(f_1(x))$, $f_3(x) = f(f_2(x))$, $... $. Find the solutions to $x = f_n(x)$ for $n > 0$.
1995 All-Russian Olympiad Regional Round, 10.1
Given function $f(x) = \dfrac{1}{\sqrt[3]{1-x^3}}$, find $\underbrace{f(... f(f(19))...)}_{95}$.
.
2023 Israel TST, P3
Given a polynomial $P$ and a positive integer $k$, we denote the $k$-fold composition of $P$ by $P^{\circ k}$. A polynomial $P$ with real coefficients is called [b]perfect[/b] if for each integer $n$ there is a positive integer $k$ so that $P^{\circ k}(n)$ is an integer. Is it true that for each perfect polynomial $P$, there exists a positive $m$ so that for each integer $n$ there is $0<k\leq m$ for which $P^{\circ k}(n)$ is an integer?