Found problems: 1340
A function $f$ is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real $x$. If $x = 0$ is a root of $f(x) = 0$, what is the least number of roots $f(x) = 0$ must have in the interval $-1000 \le x \le 1000$?
A necklace consists of 100 blue and several red beads. It is known that every segment of the necklace containing 8 blue beads contain also at least 5 red beads. What minimum number of red beads can be in the necklace?
[i]Proposed by A. Golovanov[/i]
A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not inclued the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed $1000x.$
Find all positive integers $m$ and $n$ such that the inequality:
\[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \]
is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.
Let $n$ be a positive integer. Let $a$ be an integer such that $\gcd (a,n)=1$. Prove that
\[\frac{a^{\phi (n)}-1}{n}=\sum_{i\in R}\frac{1}{ai}\left[\frac{ai}{n}\right]\pmod{n}\]
where $R$ is the reduced residue system of $n$ with each element a positive integer at most $n$.
Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.
Let $P_1$ be a regular $n$-gon, where $n\in\mathbb{N}$. We construct $P_2$ as the regular $n$-gon whose vertices are the midpoints of the edges of $P_1$. Continuing analogously, we obtain regular $n$-gons $P_3,P_4,\ldots ,P_m$. For $m\ge n^2-n+1$, find the maximum number $k$ such that for any colouring of vertices of $P_1,\ldots ,P_m$ in $k$ colours there exists an isosceles trapezium $ABCD$ whose vertices $A,B,C,D$ have the same colour.
[i]Radu Ignat[/i]
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
[i]Proposed by Hong Kong[/i]
Let $\lambda$ the positive root of the equation $t^2-1998t-1=0$. It is defined the sequence $x_0,x_1,x_2,\ldots,x_n,\ldots$ by $x_0=1,\ x_{n+1}=\lfloor\lambda{x_n}\rfloor\mbox{ for }n=1,2\ldots$ Find the remainder of the division of $x_{1998}$ by $1998$.
Note: $\lfloor{x}\rfloor$ is the greatest integer less than or equal to $x$.
Prove that the sum:
\[ S_n=\binom{n}{1}+\binom{n}{3}\cdot 2005+\binom{n}{5}\cdot 2005^2+...=\sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor}\binom{n}{2k+1}\cdot 2005^k \]
is divisible by $2^{n-1}$ for any positive integer $n$.
Let $f(0, 0) = 5^{2003}, f(0, n) = 0$ for every integer $n \neq 0$ and
\[\begin{array}{c}\ f(m, n) = f(m-1, n) - 2 \cdot \Bigg\lfloor \frac{f(m-1, n)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n-1)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n+1)}{2}\Bigg\rfloor \end{array}\]
for every natural number $m > 0$ and for every integer $n$.
Prove that there exists a positive integer $M$ such that $f(M, n) = 1$ for all integers $n$ such that $|n| \leq \frac{(5^{2003}-1)}{2}$ and $f(M, n) = 0$ for all integers n such that $|n| > \frac{5^{2003}-1}{2}.$
Find all positive integers $m$ and $n$ such that the inequality:
\[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \]
is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.
[i]Proposed by Harmel Nestra, Estionia[/i]
Let $R$ be the set of points $(x, y)$ such that $\lfloor x^2 \rfloor = \lfloor y \rfloor$ and $\lfloor y^2 \rfloor = \lfloor x \rfloor$. Compute the area of region $R$. Recall that $\lfloor z \rfloor$ is the greatest integer that is less than or equal to $z$.
Let $n$ be a positive integer. Determine the size of the largest subset of $\{ -n, -n+1, \dots, n-1, n\}$ which does not contain three elements $a$, $b$, $c$ (not necessarily distinct) satisfying $a+b+c=0$.
Given $ n$ points in a line so that any distance occurs at most twice, show that the number of distance occurring exactly once is at least $ \lfloor n/2 \rfloor$.
[i]V. T. Sos, L. Szekely[/i]
A real number $a$ is given. The sequence $n_{1}< n_{2}< n_{3}< ...$ consists of all the positive integral $n$ such that $\{na\}< \frac{1}{10}$. Prove that there are at most three different numbers among the numbers $n_{2}-n_{1}$, $n_{3}-n_{2}$, $n_{4}-n_{3}$, $\ldots$.
[i]A corollary of a theorem from ergodic theory[/i]
An infinite sequence of natural number $\{x_n\}_{n\ge 1}$ is such that $x_{n+1}$ is obtained by adding one of the non-zero digits of $x_n$ to itself. Show this sequence contains an even number.
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 $x$.
Find the minimum real $x$ that satisfies
$$\lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots$$
Find all possible pairs of real numbers $ (x, y) $ that satisfy the equalities $ y ^ 2- [x] ^ 2 = 2001 $ and $ x ^ 2 + [y] ^ 2 = 2001 $.
Jeremy has a magic scale, each side of which holds a positive integer. He plays the following game: each turn, he chooses a positive integer $n$. He then adds $n$ to the number on the left side of the scale, and multiplies by $n$ the number on the right side of the scale. (For example, if the turn starts with $4$ on the left and $6$ on the right, and Jeremy chooses $n = 3$, then the turn ends with $7$ on the left and $18$ on the right.) Jeremy wins if he can make both sides of the scale equal.
(a) Show that if the game starts with the left scale holding $17$ and the right scale holding $5$, then Jeremy can win the game in $4$ or fewer turns.
(b) Prove that if the game starts with the right scale holding $b$, where $b\geq 2$, then Jeremy can win the game in $b-1$ or fewer turns.
Let $r_1,r_2,\ldots,r_m$ be positive rational numbers with a sum of $1$. Find the maximum values of the function $f:\mathbb N\to\mathbb Z$ defined by
$$f(n)=n-\lfloor r_1n\rfloor-\lfloor r_2n\rfloor-\ldots-\lfloor r_mn\rfloor$$
Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?
Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$-element set $X$. Prove that from $\mathcal{F}$ we can select $\lceil \log_2 n \rceil + 1$ sets that form a separating family on $X$, i.e., for any two distinct elements of $X$ there is a selected set containing exactly one of the two elements.
Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=614827&hilit=Schweitzer+2014+separating