Olympiad problems

1968 Spain Mathematical Olympiad, 2

Justify if continuity can be affirmed, denied or cannot be decided in the point$ x = 0$ of a real function $f(x)$ of real variable, in each of the three (independent) cases . a) It is known only that for all natural $n$: $f\left( \frac{1}{2n}\right)= 1$ and $f\left( \frac{1}{2n+1}\right)= -1$. b) It is known that for all nonnegative real $x$ is $f(x) = x^2$ and for negative real $x$ is $f(x) = 0$. c) It is only known that for all natural $n$ it is $f\left( \frac{1}{n}\right)= 1$.

1986 Poland - Second Round, 1

Tags: algebra , continuous , functional , functional equation

Determine all functions $ f : \mathbb{R} \to \mathbb{R} $ continuous at zero and such that for every real number $ x $ the equality holds $$ 2f(2x) = f(x) + x.$$

1998 Belarus Team Selection Test, 3

Tags: continuous , functional , functional equation , algebra

Find all continuous functions $f: R \to R$ such that $g(g(x)) = g(x)+2x$ for all real $x$.

2007 Estonia Team Selection Test, 5

Tags: continuous , algebra , functional , functional equation

Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.

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 $x$.

2007 Estonia Team Selection Test, 5

Tags: continuous , functional , algebra , functional equation

Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.

1986 Austrian-Polish Competition, 9

Tags: functional , algebra , continuous , composition , functional equation

Find all continuous monotonic functions $f : R \to R$ that satisfy $f (1) = 1$ and $f(f (x)) = f (x)^2$ for all $x \in R$.

1999 All-Russian Olympiad Regional Round, 11.1

Tags: algebra , continuous , function

The function $f(x)$, defined on the entire real line, is known but that for any $a > 1 $ the function $f(x)+f(ax)$ is continuous on the entire line. Prove that $f(x)$ is also continuous along the entire line.

1995 VJIMC, Problem 3

Tags: real analysis , function , continuous

Let $f:\mathbb R\to\mathbb R$ be a continuous function. Do there exist continuous functions $g:\mathbb R\to\mathbb R$ and $h:\mathbb R\to\mathbb R$ such that $f(x)=g(x)\sin x+h(x)\cos x$ holds for every $x\in\mathbb R$?

1989 Swedish Mathematical Competition, 2

Tags: functional equation , algebra , continuous , functional

Find all continuous functions $f$ such that $f(x)+ f(x^2) = 0$ for all real numbers $x$.

1981 Poland - Second Round, 3

Tags: functional , algebra , continuous

Prove that there is no continuous function $ f: \mathbb{R} \to \mathbb{R} $ satisfying the condition $ f(f(x)) = - x $ for every $ x $.

1986 Tournament Of Towns, (116) 4

Tags: functional equation , functional , algebra , function , continuous , analysis

The function $F$ , defined on the entire real line, satisfies the following relation (for all $x$ ) : $F(x +1 )F(x) + F(x + 1 ) + 1 = 0$ . Prove that $F$ is not continuous. (A.I. Plotkin, Leningrad)

1974 Putnam, B4

Tags: function , continuous

A function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is said to be [i]continuous in each variable separately [/i] if, for each fixed value $y_0$ of $y$, the function $f(x, y_0)$ is contnuous in the usual sense as a function in $x,$ and similarly $f(x_0 , y)$ is continuous as a function of $y$ for each fixed $x_0$. Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be continuous in each variable separately. Show that there exists a sequence of continuous functions $g_n: \mathbb{R}^{2} \rightarrow \mathbb{R}$ such that $$f(x,y) =\lim_{n\to \infty}g_{n}(x,y)$$ for all $(x,y)\in \mathbb{R}^{2}.$

2018 ISI Entrance Examination, 3

Tags: function , calculus , continuous

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that for all $x\in\mathbb{R}$ and for all $t\geqslant 0$, $$f(x)=f(e^tx)$$ Show that $f$ is a constant function.