Found problems: 4776
1959 Putnam, A4
If $f$ and $g$ are real-valued functions of one real variable, show that there exist $x$ and $y$ in $[0,1]$ such that $$|xy-f(x)-g(y)|\geq \frac{1}{4}.$$
2007 South East Mathematical Olympiad, 1
Let $f(x)$ be a function satisfying $f(x+1)-f(x)=2x+1 (x \in \mathbb{R})$.In addition, $|f(x)|\le 1$ holds for $x\in [0,1]$. Prove that $|f(x)|\le 2+x^2$ holds for $x \in \mathbb{R}$.
2008 Stars Of Mathematics, 1
Prove that for any positive integer $m$, the equation
\[ \frac{n}{m}\equal{}\lfloor\sqrt[3]{n^2}\rfloor\plus{}\lfloor\sqrt{n}\rfloor\plus{}1\]
has (at least) a positive integer solution $n_{m}$.
[i]Cezar Lupu & Dan Schwarz[/i]
2000 Spain Mathematical Olympiad, 3
Show that there is no function $f : \mathbb N \to \mathbb N$ satisfying $f(f(n)) = n + 1$ for each positive integer $n.$
2013 Romania National Olympiad, 2
Given $f:\mathbb{R}\to \mathbb{R}$ an arbitrary function and $g:\mathbb{R}\to \mathbb{R}$ a function of the second degree, with the property:
for any real numbers m and n equation $f\left( x \right)=mx+n$ has solutions if and only if the equation $g\left( x \right)=mx+n$ has solutions
Show that the functions $f$ and $g$ are equal.
1988 USAMO, 3
A function $f(S)$ assigns to each nine-element subset of $S$ of the set $\{1,2,\ldots, 20\}$ a whole number from $1$ to $20$. Prove that regardless of how the function $f$ is chosen, there will be a ten-element subset $T\subset\{1,2,\ldots, 20\}$ such that $f(T - \{k\})\neq k$ for all $k\in T$.
1990 Balkan MO, 4
Find the least number of elements of a finite set $A$ such that there exists a function $f : \left\{1,2,3,\ldots \right\}\rightarrow A$ with the property: if $i$ and $j$ are positive integers and $i-j$ is a prime number, then $f(i)$ and $f(j)$ are distinct elements of $A$.
2006 Iran MO (3rd Round), 6
$P,Q,R$ are non-zero polynomials that for each $z\in\mathbb C$, $P(z)Q(\bar z)=R(z)$.
a) If $P,Q,R\in\mathbb R[x]$, prove that $Q$ is constant polynomial.
b) Is the above statement correct for $P,Q,R\in\mathbb C[x]$?
2014 Tuymaada Olympiad, 5
For two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\ell(x)$ such that $P(x)=Q(\ell(x))$ for all real $x$. How many such linear functions $\ell(x)$ can exist?
[i](A. Golovanov)[/i]
1989 AIME Problems, 8
Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that
\[ \begin{array}{r} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1\,\,\,\,\,\,\,\, \\ 4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12\,\,\,\,\, \\ 9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123. \\ \end{array} \] Find the value of \[16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7.\]
1969 IMO Longlists, 61
$(SWE 4)$ Let $a_0, a_1, a_2, \cdots$ be determined with $a_0 = 0, a_{n+1} = 2a_n + 2^n$. Prove that if $n$ is power of $2$, then so is $a_n$
2008 Moldova Team Selection Test, 2
Let $ a_1,\ldots,a_n$ be positive reals so that $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le\frac n2$. Find the minimal value of
$ \sqrt{a_1^2\plus{}\frac1{a_2^2}}\plus{}\sqrt{a_2^2\plus{}\frac1{a_3^2}}\plus{}\ldots\plus{}\sqrt{a_n^2\plus{}\frac1{a_1^2}}$.
2007 Nicolae Coculescu, 1
Let $w\in \mathbb{C}\setminus \mathbb{R}$, $|w|\neq 1$. Prove that $f\colon \mathbb{C} \to \mathbb{C}$, given by $f(z)= z+w\overline{z}$, is a bijection, and find its inverse.
2004 Romania Team Selection Test, 3
Find all one-to-one mappings $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $n$ the following relation holds:
\[ f(f(n)) \leq \frac {n+f(n)} 2 . \]
1998 IMC, 6
Let $f: [0,1]\rightarrow\mathbb{R}$ be a continuous function satisfying $xf(y)+yf(x)\le 1$ for every $x,y\in[0,1]$.
(a) Show that $\int^1_0 f(x)dx \le \frac{\pi}4$.
(b) Find such a funtion for which equality occurs.
2012 ELMO Shortlist, 9
Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that
\[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\]
and
\[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\]
[i]Calvin Deng.[/i]
1969 AMC 12/AHSME, 32
Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relation $u_{n+1}-u_n=3+4(n-1)$, $n=1,2,3,\cdots$. If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is:
$\textbf{(A) }3\qquad
\textbf{(B) }4\qquad
\textbf{(C) }5\qquad
\textbf{(D) }6\qquad
\textbf{(E) }11$
2020-2021 OMMC, 2
The function $f(x)$ is defined on the reals such that
$$f\left(\frac{1-4x}{4-x}\right) = 4-xf(x)$$
for all $x \ne 4$. There exists two distinct real numbers $a, b \ne 4$ such that $f(a) = f(b) = \frac{5}{2}$. $a+b$ can be represented as $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $10p + q$.
2001 Nordic, 2
Let ${f}$ be a bounded real function defined for all real numbers and satisfying for all real numbers ${x}$ the condition ${ f \Big(x+\frac{1}{3}\Big) + f \Big(x+\frac{1}{2}\Big)=f(x)+ f \Big(x+\frac{5}{6}\Big)}$ . Show that ${f}$ is periodic.
2014 Chile TST Ibero, 1
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 $x \in \mathbb{R}$.
1996 Moldova Team Selection Test, 11
Let $A{}$ be a set with $n{}$ $(n\geq3)$ elements. Iterations $f^2,f^2,\ldots$ of the function $f:A\rightarrow A$ are defined as $f^2(x)=f(f(x)), f^{i+1}=f(f^i(x)), \forall i\geq2$. Find the number of functions $f:A\rightarrow A$ with the property: the function $f^{n-2}$ is constant, but $f^{n-3}$ is not.
1964 Miklós Schweitzer, 8
Let $ F$ be a closed set in the $ n$-dimensional Euclidean space. Construct a function that is $ 0$ on $ F$, positive outside $ F$ , and whose partial derivatives all exist.
MIPT student olimpiad spring 2024, 3
Is it true that if a function $f: R \to R$ is continuous and takes rational values at rational points,
then at least at one point it is differentiable?
2018 IMO Shortlist, A5
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
1995 South africa National Olympiad, 2
Find all pairs $(m,n)$ of natural numbers with $m<n$ such that $m^2+1$ is a multiple of $n$ and $n^2+1$ is a multiple of $m$.