Found problems: 4776
Consider a graph with $n$ vertices and $\frac{7n}{4}$ edges.
(a) Prove that there are two cycles of equal length.
(25 points)
(b) Can you give a smaller function than $\frac{7n}{4}$ that still fits in part (a)? Prove your claim.
We say function $a(n)$ is smaller than $b(n)$ if there exists an $N$ such that for each $n>N$ ,$a(n)<b(n)$
(At most 5 points)
[i]Proposed by Afrooz Jabal'ameli[/i]
Show that there is no continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for every real number $x$
\[f(x-f(x)) = \dfrac x2.\]
There is a unique ordered triple of real numbers $(a, b, c)$ that makes the piecewise function
\begin{align*}
f(x) = \begin{cases}
(x - a)^2 + b & \text{if } x \geq c \\
x^3 - x & \text{if } x < c
\end{cases}
\end{align*}
twice continuously differentiable for all real $x.$ The value of $a + b + c$ can be expressed as a common fraction $p/q.$ Compute $p + q.$
Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$.
Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.
Given a positive integer $n \geq 2$, whose canonical prime factorization is $n = p_1^{\alpha_1}p_2^{\alpha_2} \ldots p_k^{\alpha_k}$, we define the following functions:
$$\varphi(n) = n\bigg(1 -\frac{1}{p_1}\bigg) \bigg(1 -\frac{1}{p_2}\bigg) \ldots \bigg(1 -\frac {1}{p_k}\bigg) ; \overline{\varphi}(n) = n\bigg(1 +\frac{1}{p_1}\bigg) \bigg(1 +\frac{1}{p_2}\bigg) \ldots \bigg(1 + \frac{1}{p_k}\bigg)$$
Consider all positive integers $n$ such that $\overline{\varphi}(n)$ is a multiple of $n + \varphi(n) $.
(a) Prove that $n$ is even.
(b) Determine all positive integers $n$ that satisfy this property.
Find $\lim_{n\to\infty} \int_0^{\pi} e^{x}|\sin nx|dx.$
For real number $r$ let $f(r)$ denote the integer that is the closest to $r$ (if the fractional part of $r$ is $1/2$, let $f(r)$ be $r-1/2$). Let $a>b>c$ rational numbers such that for all integers $n$ the following is true: $f(na)+f(nb)+f(nc)=n$. What can be the values of $a$, $b$ and $c$?
[i]Submitted by Gábor Damásdi, Budapest[/i]
Let $f : N \to N$ be a function such that $f(f(1995)) = 95, f(xy) = f(x)f(y)$ and $f(x) \le x$ for all $x,y$.
Find all possible values of $f(1995)$.
There are 2002 towns in a kingdom. Some of the towns are connected by roads in such a manner that, if all roads from one city closed, one can still travel between any two cities. Every year, the kingdom chooses a non-self-intersecting cycle of roads, founds a new town, connects it by roads with each city from the chosen cycle, and closes all the roads from the original cycle. After several years, no non-self-intersecting cycles remained. Prove that at that moment there are at least 2002 towns, exactly one road going out from each of them.
If $a>1$ and $b>2$ are positive integers, show that $a^{b}+1 \geq b(a+1)$, and determine when equality holds.
Find all functions f : R-->R such that
f (f (x) + y^2) = x −1 + (y + 1)f (y)
holds for all real numbers x, y
Let $N_0=\{0, 1, 2 \cdots \}$. Find all functions: $N_0 \to N_0$ such that:
(1) $f(n) < f(n+1)$, all $n \in N_0$;
(2) $f(2)=2$;
(3) $f(mn)=f(m)f(n)$, all $m, n \in N_0$.
Let $f:[0,1]\to\mathbb{R}$ be a function for which there exists a constant $K>0$ such that $|f(x)-f(y)|\le K|x-y|$ for all $x,y\in [0,1].$ Suppose also that for each rational number $r\in [0,1],$ there exist integers $a$ and $b$ such that $f(r)=a+br.$ Prove that there exist finitely many intervals $I_1,\dots,I_n$ such that $f$ is a linear function on each $I_i$ and $[0,1]=\bigcup_{i=1}^nI_i.$
Prove that every bijective function $ f: \mathbb{Z}\rightarrow\mathbb{Z}$ can be written in the way $ f\equal{}u\plus{}v$ where $ u,v: \mathbb{Z}\rightarrow\mathbb{Z}$ are bijective functions.
Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$.
Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.
Suppose two convex quadrangles in the plane $P$ and $P'$, share a point $O$ such that, for every line $l$ trough $O$, the segment along which $l$ and $P$ meet is longer then the segment along which $l$ and $P'$ meet. Is it possible that the ratio of the area of $P'$ to the area of $P$ is greater then $1.9$?
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(n+1) > f(f(n)).\]
Let $\mathbb{N}$ be the set of all natural numbers. Let $f:\mathbb{N} \to \mathbb{N}$ be a bijective function. Show that there exists three numbers $a$, $b$, $c$ in arithmatic progression such that $f(a)<f(b)<f(c)$
For a unit circle $O$, arrange points $A,B,C,D$ and $E$ in that order evenly along $O$'s circumference. For each of those points, draw the arc centered at that point inside O from the point to its left to the point to its right. Denote the outermost intersections of these arcs as $A', B', C', D'$ and $E'$, where the prime of any point is opposite the point. The length of $AC'$ can be written as an expression $f(x)$, where $f$ is a trigonometric function. Find this expression.
Let $d$ be a positive integer.
The seqeunce $a_1, a_2, a_3,...$ of positive integers is defined by $a_1 = 1$ and $a_{n + 1} = n\left \lfloor \frac{a_n}{n} \right \rfloor+ d$ for $n = 1,2,3, ...$ .
Prove that there exists a positive integer $N$ so that the terms $a_N,a_{N + 1}, a_{N + 2},...$ form an arithmetic progression.
Note: If $x$ is a real number, $\left \lfloor x \right \rfloor $ denotes the largest integer that is less than or equal to $x$.
Prove that $\ln n \geq k\ln 2$, where $n$ is a natural number and $k$ is the number of distinct primes that divide $n$.
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions:
$(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$
$(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.
Let $ {\mathbb Q}^ \plus{}$ be the set of positive rational numbers. Construct a function $ f : {\mathbb Q}^ \plus{} \rightarrow {\mathbb Q}^ \plus{}$ such that
\[ f(xf(y)) \equal{} \frac {f(x)}{y}
\]
for all $ x$, $ y$ in $ {\mathbb Q}^ \plus{}$.
Let $f(x)=\cos(a_1+x)+{1\over2}\cos(a_2+x)+{1\over4}\cos(a_3+x)+\ldots+{1\over2^{n-1}}\cos(a_n+x)$, where $a_i$ are real constants and $x$ is a real variable. If $f(x_1)=f(x_2)=0$, prove that $x_1-x_2$ is a multiple of $\pi$.
Determine the number of functions $f : Z^+ \to Z^+$ so that for all positive integers $x$ we have $f(f(x)) = f(x + 1)$, and $\max (f(2), . . . , f(14)) \le f(1) - 2 = 12$.