Found problems: 4776
2021 Peru PAGMO TST, P6
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for any real numbers $x$ and $y$ the following is true:
$$x^2+y^2+2f(xy)=f(x+y)(f(x)+f(y))$$
2007 China Team Selection Test, 1
$ u,v,w > 0$,such that $ u \plus{} v \plus{} w \plus{} \sqrt {uvw} \equal{} 4$
prove that $ \sqrt {\frac {uv}{w}} \plus{} \sqrt {\frac {vw}{u}} \plus{} \sqrt {\frac {wu}{v}}\geq u \plus{} v \plus{} w$
2022 Thailand TSTST, 3
Let $S$ be the set of the positive integers greater than $1$, and let $n$ be from $S$. Does there exist a function $f$ from $S$ to itself such that for all pairwise distinct positive integers $a_1, a_2,...,a_n$ from $S$, we have $f(a_1)f(a_2)...f(a_n)=f(a_1^na_2^n...a_n^n)$?
2011 BMO TST, 1
The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.
1998 IMO Shortlist, 4
For any two nonnegative integers $n$ and $k$ satisfying $n\geq k$, we define the number $c(n,k)$ as follows:
- $c\left(n,0\right)=c\left(n,n\right)=1$ for all $n\geq 0$;
- $c\left(n+1,k\right)=2^{k}c\left(n,k\right)+c\left(n,k-1\right)$ for $n\geq k\geq 1$.
Prove that $c\left(n,k\right)=c\left(n,n-k\right)$ for all $n\geq k\geq 0$.
1995 Moldova Team Selection Test, 4
Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ satisfying the following:
$i)$ $f(1)=1$;
$ii)$ $f(m+n)(f(m)-f(n))=f(m-n)(f(m)+f(n))$ for all $m,n \in \mathbb{Z}$.
2004 Singapore Team Selection Test, 3
Find all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying
\[ f\left(\frac {x \plus{} y}{x \minus{} y}\right) \equal{} \frac {f\left(x\right) \plus{} f\left(y\right)}{f\left(x\right) \minus{} f\left(y\right)}
\]
for all $ x \neq y$.
2012 Online Math Open Problems, 28
Find the remainder when
\[\sum_{k=1}^{2^{16}}\binom{2k}{k}(3\cdot 2^{14}+1)^k (k-1)^{2^{16}-1}\]is divided by $2^{16}+1$. ([i]Note:[/i] It is well-known that $2^{16}+1=65537$ is prime.)
[i]Victor Wang.[/i]
2011 Laurențiu Duican, 3
Find the $ \mathcal{C}^1 $ class functions $ f:[0,2]\longrightarrow\mathbb{R} $ having the property that the application $ x\mapsto e^{-x} f(x) $ is nonincreasing on $ [0,1] , $ nondecreasing on $ [1,2] , $ and satisfying
$$ \int_0^2 xf(x)dx=f(0)+f(2) . $$
[i]Cristinel Mortici[/i]
2001 Putnam, 2
For each $k$, $\mathcal{C}_k$ is biased so that, when tossed, it has probability $\tfrac{1}{(2k+1)}$ of falling heads. If the $n$ coins are tossed, what is the probability that the number of heads is odd? Express the answer as a rational function $n$.
2013 Gheorghe Vranceanu, 1
Find both extrema of the function $ x\to\frac{\sin x-3}{\cos x +2} .$
1996 Korea National Olympiad, 2
Let the $f:\mathbb{N}\rightarrow\mathbb{N}$ be the function such that
(i) For all positive integers $n,$ $f(n+f(n))=f(n)$
(ii) $f(n_o)=1$ for some $n_0$
Prove that $f(n)\equiv 1.$
2007 ISI B.Stat Entrance Exam, 2
Use calculus to find the behaviour of the function
\[y=e^x\sin{x} \ \ \ \ \ \ \ -\infty <x< +\infty\]
and sketch the graph of the function for $-2\pi \le x \le 2\pi$. Show clearly the locations of the maxima, minima and points of inflection in your graph.
2006 Romania National Olympiad, 1
Find the maximal value of \[ \left( x^3+1 \right) \left( y^3 + 1\right) , \] where $x,y \in \mathbb R$, $x+y=1$.
[i]Dan Schwarz[/i]
1967 Putnam, B3
If $f$ and $g$ are continuous and periodic functions with period $1$ on the real line, then
$$\lim_{n\to \infty} \int_{0}^{1} f(x)g (nx)\; dx =\left( \int_{0}^{1} f(x)\; dx\right)\left( \int_{0}^{1} g(x)\; dx\right).$$
1981 IMO Shortlist, 7
The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.
2011 District Olympiad, 1
a) Prove that $\{x+y\}-\{y\}$ can only be equal to $\{x\}$ or $\{x\}-1$ for any $x,y\in \mathbb{R}$.
b) Let $\alpha\in \mathbb{R}\backslash \mathbb{Q}$. Denote $a_n=\{n\alpha\}$ for all $n\in \mathbb{N}^*$ and define the sequence $(x_n)_{n\ge 1}$ by
\[x_n=(a_2-a_1)(a_3-a_2)\cdot \ldots \cdot (a_{n+1}-a_n)\]
Prove that the sequence $(x_n)_{n\ge 1}$ is convergent and find it's limit.
2009 International Zhautykov Olympiad, 1
On the plane, a Cartesian coordinate system is chosen. Given points $ A_1,A_2,A_3,A_4$ on the parabola $ y \equal{} x^2$, and points $ B_1,B_2,B_3,B_4$ on the parabola $ y \equal{} 2009x^2$. Points $ A_1,A_2,A_3,A_4$ are concyclic, and points $ A_i$ and $ B_i$ have equal abscissas for each $ i \equal{} 1,2,3,4$.
Prove that points $ B_1,B_2,B_3,B_4$ are also concyclic.
2014 China National Olympiad, 2
Let $f:X\rightarrow X$, where $X=\{1,2,\ldots ,100\}$, be a function satisfying:
1) $f(x)\neq x$ for all $x=1,2,\ldots,100$;
2) for any subset $A$ of $X$ such that $|A|=40$, we have $A\cap f(A)\neq\emptyset$.
Find the minimum $k$ such that for any such function $f$, there exist a subset $B$ of $X$, where $|B|=k$, such that $B\cup f(B)=X$.
2012 Today's Calculation Of Integral, 855
Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$.
For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$
2021 CIIM, 6
Let $0 \le a < b$ be real numbers. Prove that there is no continuous function $f : [a, b] \to \mathbb{R}$ such that
\[ \int_a^b f(x)x^{2n} \mathrm dx>0 \quad \text{and} \quad \int_a^b f(x)x^{2n+1} \mathrm dx <0 \]
for every integer $n \ge 0$.
1987 IMO Longlists, 51
The function $F$ is a one-to-one transformation of the plane into itself that maps rectangles into rectangles (rectangles are closed; continuity is not assumed). Prove that $F$ maps squares into squares.
1995 Irish Math Olympiad, 5
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $ x,y$:
$ x f(x)\minus{}yf(y)\equal{}(x\minus{}y)f(x\plus{}y)$.
2010 ELMO Problems, 1
Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$.
[i]Carl Lian and Brian Hamrick.[/i]
2013 Moldova Team Selection Test, 1
Consider real numbers $x,y,z$ such that $x,y,z>0$. Prove that \[ (xy+yz+xz)\left(\frac{1}{x^2+y^2}+\frac{1}{x^2+z^2}+\frac{1}{y^2+z^2}\right) > \frac{5}{2}. \]