Found problems: 4776
[i](a)[/i] Show that there exists exactly one function $ f : \mathbb Q^+ \to \mathbb Q^+$ satisfying the following conditions:
[b](i)[/b] if $0 < q < \frac 12$, then $f(q)=1+f \left( \frac{q}{1-2q} \right);$
[b](ii)[/b] if $1 < q \leq 2$, then $f(q) = 1+f(q + 1);$
[b](iii)[/b] $f(q)f(1/q) = 1$ for all $q \in \mathbb Q^+.$
[i](b)[/i] Find the smallest rational number $q \in \mathbb Q^+$ such that $f(q) = \frac{19}{92}.$
How many one-to-one functions $f : \{1, 2, \cdots, 9\} \rightarrow \{1, 2, \cdots, 9\}$ satisfy (i) and (ii)?
(i) $f(1)>f(2)$, $f(9)<9$.
(ii) For each $i=3, 4, \cdots, 8$, if $f(1), \cdots, f(i-1)$ are smaller than $f(i)$, then $f(i+1)$ is also smaller than $f(i)$.
Show that for any integer $n\geq2$ and all integers $a_{1},a_{2},...,a_{n}$ the product $\prod_{i<j}{(a_{j}-a_{i})}$ is divisible by $\prod_{i<j}{(j-i)}$ .
For all positive integers $ n$, let $ f(n) \equal{} \log_{2002} n^2$. Let
\[ N \equal{} f(11) \plus{} f(13) \plus{} f(14)
\]
Which of the following relations is true?
$ \textbf{(A)}\ N < 1 \qquad \textbf{(B)}\ N \equal{} 1 \qquad \textbf{(C)}\ 1 < N < 2 \qquad \textbf{(D)}\ N \equal{} 2 \qquad \textbf{(E)}\ N > 2$
Suppose that the function $f(x)=a x^2 +bx+c$, where $a,b,c$ are real, satisfies the condition $|f(x)|\leq 1$ for $|x|\leq1$. Prove that $|f'(x)|\leq 4$ for $|x|\leq1$.
Let $x$ and $y$ be real numbers such that $\frac{\sin{x}}{\sin{y}} = 3$ and $\frac{\cos{x}}{\cos{y}} = \frac{1}{2}$. The value of $\frac{\sin{2x}}{\sin{2y}} + \frac{\cos{2x}}{\cos{2y}}$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
Let $n \ge 2$ be an integer. There are $n$ beads numbered $1, 2, \ldots, n$. Two necklaces made out of some of these beads are considered the same if we can get one by rotating the other (with no flipping allowed). For example, with $n \ge 5$, the necklace with four beads $1, 5, 3, 2$ in the clockwise order is same as the one with $5, 3, 2, 1$ in the clockwise order, but is different from the one with $1, 2, 3, 5$ in the clockwise order.
We denote by $D_0(n)$ (respectively $D_1(n)$) the number of ways in which we can use all the beads to make an even number (resp. an odd number) of necklaces each of length at least $3$. Prove that $n - 1$ divides $D_1(n) - D_0(n)$.
Let $ \mathbb{R}$ be the set of real numbers. Does there exist a function $ f: \mathbb{R} \mapsto \mathbb{R}$ which simultaneously satisfies the following three conditions?
[b](a)[/b] There is a positive number $ M$ such that $ \forall x:$ $ \minus{} M \leq f(x) \leq M.$
[b](b)[/b] The value of $f(1)$ is $1$.
[b](c)[/b] If $ x \neq 0,$ then
\[ f \left(x \plus{} \frac {1}{x^2} \right) \equal{} f(x) \plus{} \left[ f \left(\frac {1}{x} \right) \right]^2
\]
Let $ n\ge 2 $ be a positive integer and $ S= \{1,2,\cdots ,n\} $. Let two functions $ f:S \rightarrow \{1,-1\} $ and $ g:S \rightarrow S $ satisfy:
i) $ f(x)f(y)=f(x+y) , \forall x,y \in S $ \\
ii) $ f(g(x))=f(x) , \forall x \in S $\\
iii) $f(x+n)=f(x) ,\forall x \in S$\\
iv) $ g $ is bijective.\\
Find the number of pair of such functions $ (f,g)$ for every $n$.
Let $f:\mathbb{N}\rightarrow \mathbb{N}$ be a strictly increasing function such that $f(2)=2$ and $f(mn)=f(m)f(n)$
for every pair of relatively prime positive integers $m$ and $n$. Prove that $f(n)=n$ for every positive integer $n$.
$\lim _{x \rightarrow 0^{+}} \frac{[x]}{\tan x}$ where $[x]$ is the greatest integer function
[list=1]
[*] -1
[*] 0
[*] 1
[*] Does not exists
[/list]
Construct a function $ f:[0,1]\longrightarrow\mathbb{R} $ that is primitivable, bounded, and doesn't touch its bounds.
[i]Dorian Popa[/i]
Let f:N→N be a function satisfying $ kf(n)\le f(kn)\le kf(n) \plus{} k \minus{} 1$ for all $ k, n\in N$.
(a)Prove that $ f(a) \plus{} f(b)\le f(a \plus{} b)\le f(a) \plus{} f(b) \plus{} 1$ for all $ a, b\in N$.
(b)If $ f$ satisfies $ f(2007n)\le 2007f(n) \plus{} 200$ for every $ n\in N$, show that there exists $ c\in N$ such that $ f(2007c) \equal{} 2007f(c)$.
The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ \[ f(m+n)-f(m)-f(n)=0 \text{ or } 1. \] Determine $f(1982)$.
Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.
A network is a simple directed graph such that each edge $ e$ has two intger lower and upper capacities $ 0\leq c_l(e)\leq c_u(e)$. A circular flow on this graph is a function such that:
1) For each edge $ e$, $ c_l(e)\leq f(e)\leq c_u(e)$.
2) For each vertex $ v$: \[ \sum_{e\in v^\plus{}}f(e)\equal{}\sum_{e\in v^\minus{}}f(e)\]
a) Prove that this graph has a circular flow, if and only if for each partition $ X,Y$ of vertices of the network we have:
\[ \sum_{\begin{array}{c}{e\equal{}xy}\\{x\in X,y\in Y}\end{array}} c_l(e)\leq \sum_{\begin{array}{c}{e\equal{}yx}\\{y\in Y,x\in X}\end{array}} c_l(e)\]
b) Suppose that $ f$ is a circular flow in this network. Prove that there exists a circular flow $ g$ in this network such that $ g(e)\equal{}\lfloor f(e)\rfloor$ or $ g(e)\equal{}\lceil f(e)\rceil$ for each edge $ e$.
Show that for each integer $a$, there is a unique decomposition
\[ a = \sum_{j=0}^{n} d_j 2^j , d_j \in (-1,0,1) \] such that no two consecutive $d_j$'s are nonzero. Show further that if $f$ is nondecreasing function from the set of all non-negative integers in to the set of all non-negative real numbers, and if $a = \sum_{j=0}^{n} c_j 2^j$ is any other decomposition of $a$ with $c_j \in (-1,0,1)$ , then
\[ \sum_{j=0}^{n} |d_j| f(j) \leq \sum_{j=0}^{n} |c_j| f(j) \]
Determine $ m > 0$ so that $ x^4 \minus{} (3m\plus{}2)x^2 \plus{} m^2 \equal{} 0$ has four real solutions forming an arithmetic series: i.e., that the solutions may be written $ a, a\plus{}b, a\plus{}2b,$ and $ a\plus{}3b$ for suitable $ a$ and $ b$.
A. 1
B. 3
C. 7
D. 12
E. None of these
Does there exist a function $f : \mathbb N \to \mathbb N$, such that $f(f(n)) =n + 1987$ for every natural number $n$? [i](IMO Problem 4)[/i]
[i]Proposed by Vietnam.[/i]
Find $n$ such that $\dfrac1{2!9!}+\dfrac1{3!8!}+\dfrac1{4!7!}+\dfrac1{5!6!}=\dfrac n{10!}$.
Prove that for each $ n$:
\[ \sum_{k\equal{}1}^n\binom{n\plus{}k\minus{}1}{2k\minus{}1}\equal{}F_{2n}\]
If $f$ and $g$ are real-valued functions of one real variable, show that there exist $x$ and $y$ in $[0,1]$ such that $$|xy-f(x)-g(y)|\geq \frac{1}{4}.$$
The sequence of functions $f_n:[0,1]\to\mathbb R$ $(n\ge2)$ is given by $f_n=1+x^{n^2-1}+x^{n^2+2n}$. Let $S_n$ denote the area of the figure bounded by the graph of the function $f_n$ and the lines $x=0$, $x=1$, and $y=0$. Compute
$$\lim_{n\to\infty}\left(\frac{\sqrt{S_1}+\sqrt{S_2}+\ldots+\sqrt{S_n}}n\right)^n.$$
A function $ f$ is defined by $ f(z) \equal{} (4 \plus{} i) z^2 \plus{} \alpha z \plus{} \gamma$ for all complex numbers $ z$, where $ \alpha$ and $ \gamma$ are complex numbers and $ i^2 \equal{} \minus{} 1$. Suppose that $ f(1)$ and $ f(i)$ are both real. What is the smallest possible value of $ | \alpha | \plus{} |\gamma |$?
$ \textbf{(A)} \; 1 \qquad \textbf{(B)} \; \sqrt {2} \qquad \textbf{(C)} \; 2 \qquad \textbf{(D)} \; 2 \sqrt {2} \qquad \textbf{(E)} \; 4 \qquad$
We consider the following triangular array
\[ \begin{array}{cccccccc}
0 & 1 & 1 & 2 & 3 & 5 & 8 & \ldots \\
\ & 0 & 1 & 1 & 2 & 3 & 5 & \ldots \\
\ & \ & 2 & 3 & 5 & 8 & 13 & \ldots \\
\ & \ & \ & 4 & 7 & 11 & 18 & \ldots \\
\ & \ & \ & \ & 12 & 19 & 31 & \ldots \\
\end{array} \]
which is defined by the conditions
i) on the first two lines, each element, starting with the third one, is the sum of the preceding two elements;
ii) on the other lines each element is the sum of the two numbers found on the same column above it.
a) Prove that all the lines satisfy the first condition i);
b) Let $a,b,c,d$ be the first elements of 4 consecutive lines in the array. Find $d$ as a function of $a,b,c$.