Found problems: 4776
Determine all continuous functions $f: \mathbb R \to \mathbb R$ such that
\[f(x + y)f(x - y) = (f(x)f(y))^2, \quad \forall(x, y) \in\mathbb{R}^2.\]
Denote by $C,\ l$ the graphs of the cubic function $C: y=x^3-3x^2+2x$, the line $l: y=ax$.
(1) Find the range of $a$ such that $C$ and $l$ have intersection point other than the origin.
(2) Denote $S(a)$ by the area bounded by $C$ and $l$. If $a$ move in the range found in (1), then find the value of $a$ for which $S(a)$ is minimized.
50 points
Determine all functions $f:\mathbb R\to\mathbb R$ such that for all $x, y \in \mathbb R$
$$f(xf(y) - yf(x)) = f(xy) - xy.$$
Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$
$$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$
$$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$
[i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]
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 $n$.
Find the total number of different integer values the function \[ f(x) = [x] + [2x] + [\frac{5x}{3}] + [3x] + [4x] \] takes for real numbers $x$ with $0 \leq x \leq 100$.
Let $p = 2^{16}+1$ be a prime, and let $S$ be the set of positive integers not divisible by $p$.
Let $f: S \to \{0, 1, 2, ..., p-1\}$ be a function satisfying
\[ f(x)f(y) \equiv f(xy)+f(xy^{p-2}) \pmod{p} \quad\text{and}\quad f(x+p) = f(x) \]
for all $x,y \in S$.
Let $N$ be the product of all possible nonzero values of $f(81)$.
Find the remainder when when $N$ is divided by $p$.
[i]Proposed by Yang Liu and Ryan Alweiss[/i]
Let $a, b$ be two nonnegative real numbers and $n$ a positive integer. Prove that \[\left(1-2^{-n}\right)\left|a^{2^n}-b^{2^n}\right|\ge\sqrt{ab}\left|a^{2^n-1}-b^{2^n-1}\right|.\]
Prove that these three statements are equivalent:
(a) For every continuous function $f:S^n \to \mathbb R^n$, there exists an $x\in S^n$ such that $f(x)=f(-x)$.
(b) There is no antipodal mapping $f:S^n \to S^{n-1}$.
(c) For every covering of $S^n$ with closed sets $A_0,\dots,A_n$, there exists an index $i$ such that $A_i\cap -A_i\neq \emptyset$.
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties:
(i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$;
(ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite.
Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic.
[i]Proposed by Ang Jie Jun, Singapore[/i]
Find all polynomial functions with real coefficients for which $$(x-2)P(x+2)+(x+2)P(x-2)=2xP(x)$$ for all real $x$
Let $f(x)$ be an odd function on $\mathbb{R}$, such that $f(x)=x^2$ when $x\ge 0$. Knowing that for all $x\in [a,a+2]$, the inequality $f(x+a)\ge 2f(x)$ holds, find the range of real number $a$.
Al has the cards $1,2,\dots,10$ in a row in increasing order. He first chooses the cards labeled $1$, $2$, and $3$, and rearranges them among their positions in the row in one of six ways (he can leave the positions unchanged). He then chooses the cards labeled $2$, $3$, and $4$, and rearranges them among their positions in the row in one of six ways. (For example, his first move could have made the sequence $3,2,1,4,5,\dots,$ and his second move could have rearranged that to $2,4,1,3,5,\dots$.) He continues this process until he has rearranged the cards with labels $8$, $9$, $10$. Determine the number of possible orderings of cards he can end up with.
[i]Proposed by Ray Li[/i]
Find all real numbers $x,y$ greater than $1$, satisfying the condition that the numbers $\sqrt{x-1}+\sqrt{y-1}$ and $\sqrt{x+1}+\sqrt{y+1}$ are nonconsecutive integers.
Find all functions $f: \mathbb{N} \rightarrow [0, +\infty)$ such that $f(1000)=10$ and $f(n+1)= \sum_{k=1}^n \frac{1}{f^2(k) + f(k)f(k+1) + f^2(k+1)}$ for all $n \in \mathbb{N}$. (Here, $f^2(i)$ means $(f(i))^2$.)
A function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the conditions
\[\begin{cases}f(x+24)\le f(x)+24\\ f(x+77)\ge f(x)+77\end{cases}\quad\text{for all}\ x\in\mathbb{R}\]
Prove that $f(x+1)=f(x)+1$ for all real $x$.
Let $p_0(x),p_1(x),p_2(x),\ldots$ be polynomials such that $p_0(x)=x$ and for all positive integers $n$, $\dfrac{d}{dx}p_n(x)=p_{n-1}(x)$. Define the function $p(x):[0,\infty)\to\mathbb{R}$ by $p(x)=p_n(x)$ for all $x\in [n,n+1)$. Given that $p(x)$ is continuous on $[0,\infty)$, compute \[\sum_{n=0}^\infty p_n(2009).\]
Let $n$ be a positive integer, and $x$ be a positive real number. Prove that $$\sum_{k=1}^{n} \left( x \left[\frac{k}{x}\right] - (x+1)\left[\frac{k}{x+1}\right]\right) \leq n,$$ where $[x]$ denotes the largest integer not exceeding $x$.
The function $f: \mathbb{R}\rightarrow\mathbb{R}$ is twice differentiable and satisfies $f(0)=2,f'(0)=-2,f(1)=1$.
Prove that there is a $\xi \in ]0,1[$ for which we have $f(\xi)\cdot f'(\xi)+f''(\xi)=0$.
You know that the binary function $\diamond$ takes in two non-negative integers and has the following properties:
\begin{align*}0\diamond a&=1\\ a\diamond a&=0\end{align*}
$\text{If } a<b, \text{ then } a\diamond b\&=(b-a)[(a-1)\diamond (b-1)].$
Find a general formula for $x\diamond y$, assuming that $y\gex>0$.
A multi-digit number is written on the blackboard. Susan puts in a number of plus signs between some pairs of adjacent digits. The addition is performed and the process is repeated with the sum. Prove that regardless of what number was initially on the blackboard, Susan can always obtain a single-digit number in at most ten steps.
Let $n$ be a positive integer and define $A_n = \{1, 2, ..., n\}$. How many functions $f : A_n \to A_n$ are there such that for all $x, y \in A_n$, if $x < y$ then $f(x) \ge f(y)$?
Find an integral formula for the solution of the differential equation
$$\delta (\delta-1)(\delta-2) \cdots(\delta -n +1) y= f(x), \;\;\, x\geq 1,$$
for $y$ as a function of $f$ satisfying the initial conditions $y(1)=y'(1)=\ldots= y^{(n-1)}(1)=0$, where $f$ is continuous and $\delta$ is the differential operator $ x \frac{d}{dx}.$
Let $F_n$ be the set of all bijective functions $f:\left\{1,2,\ldots,n\right\}\rightarrow\left\{1,2,\ldots,n\right\}$ that satisfy the conditions:
1. $f(k)\leq k+1$ for all $k\in\left\{1,2,\ldots,n\right\}$
2. $f(k)\neq k$ for all $k\in\left\{2,3,\ldots,n\right\}$
Find the probability that $f(1)\neq1$ for $f$ randomly chosen from $F_n$.
Find all functions $f : \mathbb N \to \mathbb N$ such that:
i) $f^{2000}(m)=f(m)$ for all $m \in \mathbb N$,
ii) $f(mn)=\dfrac{f(m)f(n)}{f(\gcd(m,n))}$, for all $m,n\in \mathbb N$, and
iii) $f(m)=1$ if and only if $m=1$.