Found problems: 4776
1954 Miklós Schweitzer, 3
[b]3.[/b] Is there a real-valued function $Af$, defined on the space of the functions, continuous on $[0,1]$, such that $f(x)\leq g(x) $ and$f(x)\not\equiv g(x) $ inply $Af< Ag$? Is this also true if the functions $f(x)$ are required to be monotonically increasing (rather than continuous) on $[0,1]$? [b](R.4)[/b]
2004 Germany Team Selection Test, 2
Find all functions $f: \Bbb{R}_{0}^{+}\rightarrow \Bbb{R}_{0}^{+}$ with the following properties:
(a) We have $f\left( xf\left( y\right) \right) \cdot f\left( y\right) =f\left( x+y\right)$ for all $x$ and $y$.
(b) We have $f\left(2\right) = 0$.
(c) For every $x$ with $0 < x < 2$, the value $f\left(x\right)$ doesn't equal $0$.
[b]NOTE.[/b] We denote by $\Bbb{R}_{0}^{+}$ the set of all non-negative real numbers.
1986 National High School Mathematics League, 8
$f(x)=|1-2x|,x\in[0,1]$. Then the number of solutions to $f(f(f(x)))=\frac{1}{2}x$ is________.
2009 China Team Selection Test, 3
Consider function $ f: R\to R$ which satisfies the conditions for any mutually distinct real numbers $ a,b,c,d$ satisfying $ \frac {a \minus{} b}{b \minus{} c} \plus{} \frac {a \minus{} d}{d \minus{} c} \equal{} 0$, $ f(a),f(b),f(c),f(d)$ are mutully different and $ \frac {f(a) \minus{} f(b)}{f(b) \minus{} f(c)} \plus{} \frac {f(a) \minus{} f(d)}{f(d) \minus{} f(c)} \equal{} 0.$ Prove that function $ f$ is linear
2011 Moldova Team Selection Test, 2
Let $x_1, x_2, \ldots, x_n$ be real positive numbers such that $x_1\cdot x_2\cdots x_n=1$. Prove the inequality
$\frac1{x_1(x_1+1)}+\frac1{x_2(x_2+1)}+\cdots+\frac1{x_n(x_n+1)}\geq\frac n2$
2010 China Team Selection Test, 1
Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings:
(1) $\sum_{v\in V} f(v)=|E|$;
(2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the number of uncolored vertices adjacent to $v$.
Let $m(G)$ be the number of good maps. Prove that if every vertex in $G$ is adjacent to at least one another vertex, then $n\leq m(G)\leq n!$.
1993 IMO Shortlist, 3
Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.
1990 IberoAmerican, 1
Let $f$ be a function defined for the non-negative integers, such that:
a) $f(n)=0$ if $n=2^{j}-1$ for some $j \geq 0$.
b) $f(n+1)=f(n)-1$ otherwise.
i) Show that for every $n \geq 0$ there exists $k \geq 0$ such that $f(n)+n=2^{k}-1$.
ii) Find $f(2^{1990})$.
1997 IMC, 6
Let $f: [0,1]\rightarrow \mathbb{R}$ continuous. We say that $f$ crosses the axis at $x$ if $f(x)=0$ but $\exists y,z \in [x-\epsilon,x+\epsilon]: f(y)<0<f(z)$ for any $\epsilon$.
(a) Give an example of a function that crosses the axis infinitely often.
(b) Can a continuous function cross the axis uncountably often?
2012 Indonesia TST, 1
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that
\[f(x+y) + f(x)f(y) = f(xy) + (y+1)f(x) + (x+1)f(y)\]
for all $x,y \in \mathbb{R}$.
2008 ISI B.Stat Entrance Exam, 3
Study the derivatives of the function
\[y=\sqrt{x^3-4x}\]
and sketch its graph on the real line.
2011 IMO Shortlist, 8
Let $k \in \mathbb{Z}^+$ and set $n=2^k+1.$ Prove that $n$ is a prime number if and only if the following holds: there is a permutation $a_{1},\ldots,a_{n-1}$ of the numbers $1,2, \ldots, n-1$ and a sequence of integers $g_{1},\ldots,g_{n-1},$ such that $n$ divides $g^{a_i}_i - a_{i+1}$ for every $i \in \{1,2,\ldots,n-1\},$ where we set $a_n = a_1.$
[i]Proposed by Vasily Astakhov, Russia[/i]
2006 ISI B.Stat Entrance Exam, 6
(a) Let $f(x)=x-xe^{-\frac1x}, \ \ x>0$. Show that $f(x)$ is an increasing function on $(0,\infty)$, and $\lim_{x\to\infty} f(x)=1$.
(b) Using part (a) or otherwise, draw graphs of $y=x-1, y=x, y=x+1$, and $y=xe^{-\frac{1}{|x|}}$ for $-\infty<x<\infty$ using the same $X$ and $Y$ axes.
2017 Junior Regional Olympiad - FBH, 1
It is given function $f(x)=3x-2$
$a)$ Find $g(x)$ if $f(2x-g(x))=-3(1+2m)x+34$
$b)$ Solve the equation: $g(x)=4(m-1)x-4(m+1)$, $m \in \mathbb{R}$
1978 Miklós Schweitzer, 4
Let $ \mathbb{Q}$ and $ \mathbb{R}$ be the set of rational numbers and the set of real numbers, respectively, and let $ f : \mathbb{Q} \rightarrow \mathbb{R}$ be a function with the following property. For every $ h \in \mathbb{Q} , \;x_0 \in \mathbb{R}$, \[ f(x\plus{}h)\minus{}f(x) \rightarrow 0\] as $ x \in \mathbb{Q}$ tends to $ x_0$. Does it follow that $ f$ is bounded on some interval?
[i]M. Laczkovich[/i]
2015 Switzerland - Final Round, 3
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, such that for arbitrary $x,y \in \mathbb{R}$: \[ (y+1)f(x)+f(xf(y)+f(x+y))=y.\]
2011 Macedonia National Olympiad, 2
Acute-angled $~$ $\triangle{ABC}$ $~$ is given. A line $~$ $l$ $~$ parallel to side $~$ $AB$ $~$ passing through vertex $~$ $C$ $~$ is drawn. Let the angle bisectors of $~$ $\angle{BAC}$ $~$ and $~$ $\angle{ABC}$ $~$ intersect the sides $~$ $BC$ and $~$ $AC$ at points $~$ $D$ $~$ and $~$ $F$, and line $~$ $l$ $~$ at points $~$ $E$ $~$ and $~$ $G$ $~$ respectively. Prove that if $~$ $\overline{DE}=\overline{GF}$ $~$ then $~$ $\overline{AC}=\overline{BC}\, .$
1993 Mexico National Olympiad, 4
$f(n,k)$ is defined by
(1) $f(n,0) = f(n,n) = 1$ and
(2) $f(n,k) = f(n-1,k-1) + f(n-1,k)$ for $0 < k < n$.
How many times do we need to use (2) to find $f(3991,1993)$?
2012 Bosnia Herzegovina Team Selection Test, 4
Define a function $f:\mathbb{N}\rightarrow\mathbb{N}$, \[f(1)=p+1,\] \[f(n+1)=f(1)\cdot f(2)\cdots f(n)+p,\] where $p$ is a prime number. Find all $p$ such that there exists a natural number $k$ such that $f(k)$ is a perfect square.
2007 USA Team Selection Test, 2
Let $n$ be a positive integer and let $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$ be two nondecreasing sequences of real numbers such that
\[ a_1 + \dots + a_i \le b_1 + \dots + b_i \text{ for every } i = 1, \dots, n \]
and
\[ a_1 + \dots + a_n = b_1 + \dots + b_n. \]
Suppose that for every real number $m$, the number of pairs $(i,j)$ with $a_i-a_j=m$ equals the numbers of pairs $(k,\ell)$ with $b_k-b_\ell = m$. Prove that $a_i = b_i$ for $i=1,\dots,n$.
2019 Durer Math Competition Finals, 3
For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$.
Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.
2006 AMC 12/AHSME, 20
Let $ x$ be chosen at random from the interval $ (0,1)$. What is the probability that
\[ \lfloor\log_{10}4x\rfloor \minus{} \lfloor\log_{10}x\rfloor \equal{} 0?
\]Here $ \lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $ x$.
$ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 3{20} \qquad \textbf{(C) } \frac 16 \qquad \textbf{(D) } \frac 15 \qquad \textbf{(E) } \frac 14$
2015 Moldova Team Selection Test, 1
Find all functions $f : \mathbb{Z}_{+} \rightarrow \mathbb{Z}_{+}$ that satisfy $f(mf(n)) = n+f(2015m)$ for all $m,n \in \mathbb{Z}_{+}$.
2012 ISI Entrance Examination, 8
Let $S = \{1,2,3,\ldots,n\}$. Consider a function $f\colon S\to S$. A subset $D$ of $S$ is said to be invariant if for all $x\in D$ we have $f(x)\in D$. The empty set and $S$ are also considered as invariant subsets. By $\deg (f)$ we define the number of invariant subsets $D$ of $S$ for the function $f$.
[b]i)[/b] Show that there exists a function $f\colon S\to S$ such that $\deg (f)=2$.
[b]ii)[/b] Show that for every $1\leq k\leq n$ there exists a function $f\colon S\to S$ such that $\deg (f)=2^{k}$.
2015 AMC 12/AHSME, 20
For every positive integer $n$, let $\operatorname{mod_5}(n)$ be the remainder obtained when $n$ is divided by $5$. Define a function $f : \{0, 1, 2, 3, \dots\} \times \{0, 1, 2, 3, 4\} \to \{0, 1, 2, 3, 4\}$ recursively as follows:
\[f(i, j) = \begin{cases}
\operatorname{mod_5}(j+1) & \text{if }i=0\text{ and }0\leq j\leq 4 \\
f(i-1, 1) & \text{if }i\geq 1\text{ and }j=0 \text{, and}\\
f(i-1, f(i, j-1)) & \text{if }i\geq 1\text{ and }1\leq j\leq 4
\end{cases}\]
What is $f(2015, 2)$?
$\textbf{(A) }0 \qquad\textbf{(B) }1 \qquad\textbf{(C) }2 \qquad\textbf{(D) }3 \qquad\textbf{(E) }4$