Olympiad problems

1998 Poland - Second Round, 1

Let $A_n = \{1,2,...,n\}$. Prove or disprove: For all integers $n \ge 2$ there exist functions $f,g : A_n \to A_n$ which satisfy $f(f(k)) = g(g(k)) = k$ for $1 \le k \le n$, and $g(f(k)) = k +1$ for $1 \le k \le n -1$.

2019 Taiwan TST Round 1, 1

Tags: functional equation , algebra , function

Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that $$ f\left(xf\left(y\right)-f\left(x\right)-y\right) = yf\left(x\right)-f\left(y\right)-x $$ holds for all $ x,y \in \mathbb{R} $

2014 Abels Math Contest (Norwegian MO) Final, 1b

Tags: functional , functional equation , function , algebra

Find all functions $f : R-\{0\} \to R$ which satisfy $(1 + y)f(x) - (1 + x)f(y) = yf(x/y) - xf(y/x)$ for all real $x, y \ne 0$, and which take the values $f(1) = 32$ and $f(-1) = -4$.

2004 Romania National Olympiad, 2

Tags: superior algebra , algebra , polynomial , function

Let $f \in \mathbb Z[X]$. For an $n \in \mathbb N$, $n \geq 2$, we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$, for all $x \in \mathbb Z$. (a) Prove that $f_n$ is well defined. (b) Find all polynomials $f \in \mathbb Z[X]$ such that for all $n \in \mathbb N$, $n \geq 2$, the function $f_n$ is surjective. [i]Bogdan Enescu[/i]

2012 Brazil Team Selection Test, 1

Tags: function , golden ratio , ratio , algebra , floor function

Let $\phi = \frac{1+\sqrt5}{2}$. Prove that a positive integer appears in the list $$\lfloor \phi \rfloor , \lfloor 2 \phi \rfloor, \lfloor 3\phi \rfloor ,... , \lfloor n\phi \rfloor , ... $$ if and only if it appears exactly twice in the list $$\lfloor 1/ \phi \rfloor , \lfloor 2/ \phi \rfloor, \lfloor 3/\phi \rfloor , ... ,\lfloor n/\phi \rfloor , ... $$

2018 CMIMC Number Theory, 9

Tags: function , euler

Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$. Compute \[\sum_{n=1}^{\infty}\frac{\phi(n)}{5^n+1}.\]

2007 Today's Calculation Of Integral, 244

Tags: calculus computations , calculus , integration , function , geometry

A quartic funtion $ y \equal{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx\plus{}e\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta ).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.

2014 Taiwan TST Round 1, 2

Tags: induction , strong induction , function , modular arithmetic , number theory , algebra , polynomial

For a fixed integer $k$, determine all polynomials $f(x)$ with integer coefficients such that $f(n)$ divides $(n!)^k$ for every positive integer $n$.

2022-IMOC, A3

Tags: functional equation , function , algebra

Find all functions $f:\mathbb R\to \mathbb R$ such that $$xy(f(x+y)-f(x)-f(y))=2f(xy)$$ for all $x,y\in \mathbb R.$ [i]Proposed by USJL[/i]

2019 IFYM, Sozopol, 6

Tags: function , algebra

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that: $xf(y)+yf(x)=(x+y)f(x^2+y^2), \forall x,y \in \mathbb{N}$

1993 National High School Mathematics League, 2

Tags: function

$f(x)=a\sin x+b\sqrt[3]{x}+4$. If $f(\lg\log_{3}10)=5$, then the value of $f(\lg\lg 3)$ is $\text{(A)}-5\qquad\text{(B)}-3\qquad\text{(C)}3\qquad\text{(D)}$ not sure

1995 IMC, 2

Tags: calculus , function , integral , real analysis , inequalities

Let $f$ be a continuous function on $[0,1]$ such that for every $x\in [0,1]$, we have $\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}$. Show that $\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}$.

2021 Moldova Team Selection Test, 5

Tags: algebra , function , geometry

Let $ABC$ be an equilateral triangle. Find all positive integers $n$, for which the function $f$, defined on all points $M$ from the circle $S$ circumscribed to triangle $ABC$, defined by the formula $f:S \rightarrow R, f(M)=MA^n+MB^n+MC^n$, is a constant function.

2012 International Zhautykov Olympiad, 1

Tags: induction , number theory , function

Do there exist integers $m, n$ and a function $f\colon \mathbb R \to \mathbb R$ satisfying simultaneously the following two conditions? $\bullet$ i) $f(f(x))=2f(x)-x-2$ for any $x \in \mathbb R$; $\bullet$ ii) $m \leq n$ and $f(m)=n$.

2019 AMC 10, 9

Tags: function

The function $f$ is defined by $$f(x) = \Big\lfloor \lvert x \rvert \Big\rfloor - \Big\lvert \lfloor x \rfloor \Big\rvert$$for all real numbers $x$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number $r$. What is the range of $f$? $\textbf{(A) } \{-1, 0\} \qquad\textbf{(B) } \text{The set of nonpositive integers} \qquad\textbf{(C) } \{-1, 0, 1\}$ $\textbf{(D) } \{0\} \qquad\textbf{(E) } \text{The set of nonnegative integers} $

2009 Germany Team Selection Test, 3

Tags: algebra , polynomial , function , calculus

Find all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that $ \forall x,y,z \in \mathbb{R}$ we have: If \[ x^3 \plus{} f(y) \cdot x \plus{} f(z) \equal{} 0,\] then \[ f(x)^3 \plus{} y \cdot f(x) \plus{} z \equal{} 0.\]

2019 Turkey MO (2nd round), 5

Tags: algebra , function , combinatorics

Let $f:\{1,2,\dots,2019\}\to\{-1,1\}$ be a function, such that for every $k\in\{1,2,\dots,2019\}$, there exists an $\ell\in\{1,2,\dots,2019\}$ such that $$ \sum_{i\in\mathbb{Z}:(\ell-i)(i-k)\geqslant 0} f(i)\leqslant 0. $$ Determine the maximum possible value of $$ \sum_{i\in\mathbb{Z}:1\leqslant i\leqslant 2019} f(i). $$

2011 Harvard-MIT Mathematics Tournament, 7

Tags: function , hmmt

Let $A = \{1,2,\ldots,2011\}$. Find the number of functions $f$ from $A$ to $A$ that satisfy $f(n) \le n$ for all $n$ in $A$ and attain exactly $2010$ distinct values.

1999 AMC 12/AHSME, 18

Tags: function , trigonometry

How many zeros does $ f(x) \equal{} \cos(\log(x)))$ have on the interval $ 0 < x < 1$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ \text{infinitely many}$

1985 Polish MO Finals, 3

Tags: continuous , function , inequalities

The function $f : R \to R$ satisfies $f(3x) = 3f(x) - 4f(x)^3$ for all real $x$ and is continuous at $x = 0$. Show that $|f(x)| \le 1$ for all $x$.

2019 IMC, 3

Tags: real analysis , function

Let $f:(-1,1)\to \mathbb{R}$ be a twice differentiable function such that $$2f’(x)+xf''(x)\geqslant 1 \quad \text{ for } x\in (-1,1).$$ Prove that $$\int_{-1}^{1}xf(x)dx\geqslant \frac{1}{3}.$$ [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karim Rakhimov, Scuola Normale Superiore and National University of Uzbekistan[/i]