Olympiad problems

2024 Israel TST, P3

Find all continuous functions $f\colon \mathbb{R}_{>0}\to \mathbb{R}_{\geq 1}$ for which the following equation holds for all positive reals $x$, $y$: \[f\left(\frac{f(x)}{y}\right)-f\left(\frac{f(y)}{x}\right)=xy\left(f(x+1)-f(y+1)\right)\]

ICMC 6, 5

Tags: continuous function , real analysis , function

Let $[0, 1]$ be the set $\{x \in \mathbb{R} : 0 \leq x \leq 1\}$. Does there exist a continuous function $g : [0, 1] \to [0, 1]$ such that no line intersects the graph of $g$ infinitely many times, but for any positive integer $n$ there is a line intersecting $g$ more than $n$ times? [i]Proposed by Ethan Tan[/i]

2012 CHMMC Fall, 2

Tags: algebra , continuous function , functional

Find all continuous functions $f : R \to R$ such that $$f(x + f(y)) = f(x + y) + y,$$ for all $x, y \in R$. No proof is required for this problem.

1974 IMO Shortlist, 8

Tags: continuous function , four variables , algebra

The variables $a,b,c,d,$ traverse, independently from each other, the set of positive real values. What are the values which the expression \[ S= \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d} \] takes?

2017 Miklós Schweitzer, 10

Tags: continuous function , binary representation , random , probability , algebra

Let $X_1,X_2,\ldots$ be independent and identically distributed random variables with distribution $\mathbb{P}(X_1=0)=\mathbb{P}(X_1=1)=\frac12$. Let $Y_1$, $Y_2$, $Y_3$, and $Y_4$ be independent, identically distributed random variables, where $Y_1:=\sum_{k=1}^\infty \frac{X_k}{16^k}$. Decide whether the random variables $Y_1+2Y_2+4Y_3+8Y_4$ and $Y_1+4Y_3$ are absolutely continuous.

1969 IMO Longlists, 8

Tags: continuous function , function , functional equation , algebra

Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.

1969 IMO Shortlist, 8

Tags: continuous function , functional equation , algebra , function

Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.

Gheorghe Țițeica 2024, P1

Tags: real analysis , convergent sequence , continuous function

Find all continuous functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ such that for any sequences $(a_n)_{n\geq 1}$ and $(b_n)_{n\geq 1}$ such that the sequence $(a_n+b_n)_{n\geq 1}$ is convergent, the sequence $(f(a_n)+g(b_n))_{n\geq 1}$ is also convergent.

1985 Greece National Olympiad, 2

Tags: algebra , continuous function

Conside the continuous $ f: \mathbb{R}\to\mathbb{R}$ . It is also know that equation $f(f(f(x)))=x$ has solution in $\mathbb{R}$. Prove that equation $f(x)=x$ has solution in $\mathbb{R}$.

1974 IMO, 5

Tags: algebra , continuous function , four variables

The variables $a,b,c,d,$ traverse, independently from each other, the set of positive real values. What are the values which the expression \[ S= \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d} \] takes?

1969 IMO Longlists, 59

Tags: algebra , function , continuous function

$(SWE 2)$ For each $\lambda (0 < \lambda < 1$ and $\lambda = \frac{1}{n}$ for all $n = 1, 2, 3, \cdots)$, construct a continuous function $f$ such that there do not exist $x, y$ with $0 < \lambda < y = x + \lambda \le 1$ for which $f(x) = f(y).$

1995 Tuymaada Olympiad, 7

Tags: algebra , continuous function , functional equation

Find a continuous function $f(x)$ satisfying the identity $f(x)-f(ax)=x^n-x^m$, where $n,m\in N , 0<a<1$

1975 IMO Shortlist, 9

Tags: algebra , continuous function , translation , intersection , function

Let $f(x)$ be a continuous function defined on the closed interval $0 \leq x \leq 1$. Let $G(f)$ denote the graph of $f(x): G(f) = \{(x, y) \in \mathbb R^2 | 0 \leq$$ x \leq 1, y = f(x) \}$. Let $G_a(f)$ denote the graph of the translated function $f(x - a)$ (translated over a distance $a$), defined by $G_a(f) = \{(x, y) \in \mathbb R^2 | a \leq x \leq a + 1, y = f(x - a) \}$. Is it possible to find for every $a, \ 0 < a < 1$, a continuous function $f(x)$, defined on $0 \leq x \leq 1$, such that $f(0) = f(1) = 0$ and $G(f)$ and $G_a(f)$ are disjoint point sets ?

1969 IMO Shortlist, 59

Tags: algebra , function , continuous function

$(SWE 2)$ For each $\lambda (0 < \lambda < 1$ and $\lambda = \frac{1}{n}$ for all $n = 1, 2, 3, \cdots)$, construct a continuous function $f$ such that there do not exist $x, y$ with $0 < \lambda < y = x + \lambda \le 1$ for which $f(x) = f(y).$

1983 IMO Shortlist, 11

Tags: functional equation , continuous function , algebra

Let $f : [0, 1] \to \mathbb R$ be continuous and satisfy: \[ \begin{cases}bf(2x) = f(x), &\mbox{ if } 0 \leq x \leq 1/2,\\ f(x) = b + (1 - b)f(2x - 1), &\mbox{ if } 1/2 \leq x \leq 1,\end{cases}\] where $b = \frac{1+c}{2+c}$, $c > 0$. Show that $0 < f(x)-x < c$ for every $x, 0 < x < 1.$

1993 Poland - Second Round, 6

Tags: function , functional equation , continuous function , algebra

A continuous function $f : R \to R$ satisfies the conditions $f(1000) = 999$ and $f(x)f(f(x)) = 1$ for all real $x$. Determine $f(500)$.

2025 All-Russian Olympiad, 11.8

Tags: algebra , continuous function , function

Let $ f: \mathbb{R} \to \mathbb{R} $ be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of $ f $. It is known that the graph of $ f $ contains exactly $ N $ chords, one of which has length 2025. Find the minimum possible value of $ N $.

1974 IMO Longlists, 22

Tags: algebra , continuous function , four variables

The variables $a,b,c,d,$ traverse, independently from each other, the set of positive real values. What are the values which the expression \[ S= \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d} \] takes?

2019 District Olympiad, 4

Tags: calculus , continuous function , convergence , sequence

Let $f: [0, \infty) \to [0, \infty)$ be a continuous function with $f(0)>0$ and having the property $$x-y<f(y)-f(x) \le 0~\forall~0 \le x<y.$$ Prove that: $a)$ There exists a unique $\alpha \in (0, \infty)$ such that $(f \circ f)(\alpha)=\alpha.$ $b)$ The sequence $(x_n)_{n \ge 1},$ defined by $x_1 \ge 0$ and $x_{n+1}=f(x_n)~\forall~n \in \mathbb{N}$ is convergent.

2023 Romania National Olympiad, 4

Tags: continuous function , real analysis , differentiable function

We consider a function $f:\mathbb{R} \rightarrow \mathbb{R}$ for which there exist a differentiable function $g : \mathbb{R} \rightarrow \mathbb{R}$ and exist a sequence $(a_n)_{n \geq 1}$ of real positive numbers, convergent to $0,$ such that \[ g'(x) = \lim_{n \to \infty} \frac{f(x + a_n) - f(x)}{a_n}, \forall x \in \mathbb{R}. \] a) Give an example of such a function f that is not differentiable at any point $x \in \mathbb{R}.$ b) Show that if $f$ is continuous on $\mathbb{R}$, then $f$ is differentiable on $\mathbb{R}.$

1998 Romania National Olympiad, 4

Tags: real analysis , continuous function , increasing

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function with the property that for any $a,b \in \mathbb{R},$ $a<b,$ there are $c_1,c_2 \in [a,b],$ $c_1 \le c_2$ such that $f(c_1)= \min_{x \in [a,b]} f(x)$ and $f(c_2)= \max_{x \in [a,b]} f(x).$ Prove that $f$ is increasing.

2010 Miklós Schweitzer, 6

Tags: function , continuous function

Is there a continuous function $ f: \mathbb {R} ^ {2} \rightarrow \mathbb {R} $ for every $ d \in \mathbb {R} $ we have $ g_{m,d}(x) = f (x, m x + d) $ is strictly monotonic on $ \mathbb {R} $ if $ m \ge 0, $ and not monotone on any non-empty open interval if $ m <0? $