Olympiad problems

1996 Irish Math Olympiad, 3

Tags: function , algebra

A function $ f$ from $ [0,1]$ to $ \mathbb{R}$ has the following properties: $ (i)$ $ f(1)\equal{}1;$ $ (ii)$ $ f(x) \ge 0$ for all $ x \in [0,1]$; $ (iii)$ If $ x,y,x\plus{}y \in [0,1]$, then $ f(x\plus{}y) \ge f(x)\plus{}f(y)$. Prove that $ f(x) \le 2x$ for all $ x \in [0,1]$.

2023 China Team Selection Test, P22

Tags: function , functional equation , algebra

Find all functions $f:\mathbb {Z}\to\mathbb Z$, satisfy that for any integer ${a}$, ${b}$, ${c}$, $$2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2$$

2021 China Second Round A1, 2

Tags: number theory , function

Find a necessary and sufficient condition of $a,b,n\in\mathbb{N^*}$ such that for $S=\{a+bt\mid t=0,1,2,\cdots,n-1\}$, there exists a one-to-one mapping $f: S\to S$ such that for all $x\in S$, $\gcd(x,f(x))=1$.

2023 Israel TST, P3

Tags: algebra , functional equation , function

Find all functions $f:\mathbb{Z}\to \mathbb{Z}_{>0}$ for which \[f(x+f(y))^2+f(y+f(x))^2=f(f(x)+f(y))^2+1\] holds for any $x,y\in \mathbb{Z}$.

1984 Iran MO (2nd round), 3

Tags: function , algebra , geometry , calculus , derivative , geometric transformation

Let $f : \mathbb R \to \mathbb R$ be a function such that \[f(x+y)=f(x) \cdot f(y) \qquad \forall x,y \in \mathbb R\] Suppose that $f(0) \neq 0$ and $f(0)$ exists and it is finite $(f(0) \neq \infty)$. Prove that $f$ has derivative in each point $x \in \mathbb R.$

2005 Putnam, B1

Tags: floor function , algebra , function , polynomial , inequalities

Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a\rfloor,\lfloor 2a\rfloor)=0$ for all real numbers $a.$ (Note: $\lfloor v\rfloor$ is the greatest integer less than or equal to $v.$)

2009 India National Olympiad, 2

Tags: function , algebra

Define a a sequence $ {<{a_n}>}^{\infty}_{n\equal{}1}$ as follows $ a_n\equal{}0$, if number of positive divisors of $ n$ is [i]odd[/i] $ a_n\equal{}1$, if number of positive divisors of $ n$ is [i]even[/i] (The positive divisors of $ n$ include $ 1$ as well as $ n$.)Let $ x\equal{}0.a_1a_2a_3........$ be the real number whose decimal expansion contains $ a_n$ in the $ n$-th place,$ n\geq1$.Determine,with proof,whether $ x$ is rational or irrational.

2019 APMO, 1

Tags: number theory , functional equation , algebra , function

Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f : \mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$ for all positive integers $a,b$.

2016 Postal Coaching, 2

Tags: functional equation , function , algebra

Determine all functions $f : \mathbb R \to \mathbb R$ such that $$f(f(x)- f(y)) = f(f(x)) - 2x^2f(y) + f\left(y^2\right),$$ for all reals $x, y$.

1965 AMC 12/AHSME, 34

Tags: function , quadratic , calculus

For $ x \ge 0$ the smallest value of $ \frac {4x^2 \plus{} 8x \plus{} 13}{6(1 \plus{} x)}$ is: $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \frac {25}{12} \qquad \textbf{(D)}\ \frac {13}{6} \qquad \textbf{(E)}\ \frac {34}{5}$

1989 IMO Longlists, 39

Tags: algebra , function , polynomial , calculus , integration , combinatorics

Alice has two urns. Each urn contains four balls and on each ball a natural number is written. She draws one ball from each urn at random, notes the sum of the numbers written on them, and replaces the balls in the urns from which she took them. This she repeats a large number of times. Bill, on examining the numbers recorded, notices that the frequency with which each sum occurs is the same as if it were the sum of two natural numbers drawn at random from the range 1 to 4. What can he deduce about the numbers on the balls?

2004 India IMO Training Camp, 4

Tags: function , arithmetic sequence , number theory

Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$

2014 IMS, 4

Tags: function , limit , topology , real analysis

Let $(X,d)$ be a metric space and $f:X \to X$ be a function such that $\forall x,y\in X : d(f(x),f(y))=d(x,y)$. $\text{a})$ Prove that for all $x \in X$, $\lim_{n \rightarrow +\infty} \frac{d(x,f^n(x))}{n}$ exists, where $f^n(x)$ is $\underbrace{f(f(\cdots f(x)}_{n \text{times}} \cdots ))$. $\text{b})$ Prove that the amount of the limit does [b][u]not[/u][/b] depend on choosing $x$.

1998 Vietnam Team Selection Test, 1

Tags: function , algebra , limit , polynomial

Let $f(x)$ be a real function such that for each positive real $c$ there exist a polynomial $P(x)$ (maybe dependent on $c$) such that $| f(x) - P(x)| \leq c \cdot x^{1998}$ for all real $x$. Prove that $f$ is a real polynomial.

2011 Today's Calculation Of Integral, 704

Tags: calculus computations , function , integration , calculus , induction

A function $f_n(x)\ (n=0,\ 1,\ 2,\ 3,\ \cdots)$ satisfies the following conditions: (i) $f_0(x)=e^{2x}+1$. (ii) $f_n(x)=\int_0^x (n+2t)f_{n-1}(t)dt-\frac{2x^{n+1}}{n+1}\ (n=1,\ 2,\ 3,\ \cdots).$ Find $\sum_{n=1}^{\infty} f_n'\left(\frac 12\right).$

1999 IMC, 2

Tags: function , probability , probability and stats

We roll a regular 6-sided dice $n$ times. What is the probabilty that the total number of eyes rolled is a multiple of 5?

2008 SEEMOUS, Problem 1

Tags: function

Let $f:[1,\infty)\to(0,\infty)$ be a continuous function. Assume that for every $a>0$, the equation $f(x)=ax$ has at least one solution in the interval $[1,\infty)$. (a) Prove that for every $a>0$, the equation $f(x)=ax$ has infinitely many solutions. (b) Give an example of a strictly increasing continuous function $f$ with these properties.

1971 Miklós Schweitzer, 6

Tags: function , real analysis , limit , integration

Let $ a(x)$ and $ r(x)$ be positive continuous functions defined on the interval $ [0,\infty)$, and let \[ \liminf_{x \rightarrow \infty} (x-r(x)) >0.\] Assume that $ y(x)$ is a continuous function on the whole real line, that it is differentiable on $ [0, \infty)$, and that it satisfies \[ y'(x)=a(x)y(x-r(x))\] on $ [0, \infty)$. Prove that the limit \[ \lim_{x \rightarrow \infty}y(x) \exp \left\{ -%Error. "diaplaymath" is a bad command. \int_0^x a(u)du \right \}\] exists and is finite. [i]I. Gyori[/i]

2013 APMO, 2

Tags: floor function , algebra , function , modular arithmetic , quadratic , inequalities

Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

2025 Bulgarian Winter Tournament, 10.4

Tags: function , number theory , functional equation

The function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is such that $f(a,b) + f(b,c) = f(ac, b^2) + 1$ for any positive integers $a,b,c$. Assume there exists a positive integer $n$ such that $f(n, m) \leq f(n, m + 1)$ for all positive integers $m$. Determine all possible values of $f(2025, 2025)$.

1979 IMO Longlists, 65

Tags: function , inequalities , algebra

Given a function $f$ such that $f(x)\le x\forall x\in\mathbb{R}$ and $f(x+y)\le f(x)+f(y)\forall \{x,y\}\in\mathbb{R}$, prove that $f(x)=x\forall x\in\mathbb{R}$.

2014 Chile TST Ibero, 1

Tags: algebra , function

Consider a function $f: \mathbb{R} \to \mathbb{R}$ satisfying for all $x \in \mathbb{R}$: \[ f(x+1) = \frac{1}{2} + \sqrt{f(x) - f(x)^2}. \] Prove that there exists a $b > 0$ such that $f(x + b) = f(x)$ for all $x \in \mathbb{R}$.

2021 Saint Petersburg Mathematical Olympiad, 4

Tags: function , algebra

The following functions are written on the board, $$F(x) = x^2 + \frac{12}{x^2}, G(x) = \sin(\pi x^2), H(x) = 1.$$ If functions $f,g$ are currently on the board, we may write on the board the functions $$f(x) + g(x), f(x) - g(x), f(x)g(x), cf(x)$$ (the last for any real number $c$). Can a function $h(x)$ appear on the board such that $$|h(x) - x| < \frac{1}{3}$$ for all $x \in [1,10]$ ?

2008 Indonesia MO, 4

Tags: function , functional equation , algebra

Find all function $ f: \mathbb{N}\rightarrow\mathbb{N}$ satisfy $ f(mn)\plus{}f(m\plus{}n)\equal{}f(m)f(n)\plus{}1$ for all natural number $ n$

2011 District Olympiad, 3

Tags: function , algebra , integer part , floor

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function with the property that $ (f\circ f) (x) =[x], $ for any real number $ x. $ Show that there exist two distinct real numbers $ a,b $ so that $ |f(a)-f(b)|\ge |a-b|. $ $ [] $ denotes the integer part.