Found problems: 1340
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
Given a nonnegative real number $x,$ let $\langle x\rangle$ denote the fractional part of $x;$ that is, $\langle x\rangle=x-\lfloor x\rfloor,$ where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x.$ Suppose that $a$ is positive, $\langle a^{-1}\rangle=\langle a^2\rangle,$ and $2<a^2<3.$ Find the value of $a^{12}-144a^{-1}.$
Assume that $a$ is a given irrational number.
(a) Prove that for each positive real number $\epsilon$ there exists at least one integer $q\ge0$ such that $aq-\lfloor aq\rfloor<\epsilon$.
(b) Prove that for given $\epsilon>0$ there exist infinitely many rational numbers $\frac pq$ such that $q>0$ and $\left|a-\frac pq\right|<\frac\epsilon q$.
Let $a_0, a_1, a_2,\ldots $ be a sequence of real numbers satisfying $a_0=1$ and $a_n=a_{\lfloor 7n/9\rfloor}+a_{\lfloor n/9\rfloor}$ for $n=1, 2,\ldots $
Prove that there exists a positive integer $k$ with $a_k<\frac{k}{2001!}$.
Let $n$ be a positive integer and $x$ positive real number such that none of numbers $x,2x,\dots,nx$ and none of $\frac{1}{x},\frac{2}{x},\dots,\frac{\left\lfloor nx\right\rfloor }{x}$ is an integer. Prove that \[
\left\lfloor x\right\rfloor +\left\lfloor 2x\right\rfloor +\dots+\left\lfloor nx\right\rfloor +\left\lfloor \frac{1}{x}\right\rfloor +\left\lfloor \frac{2}{x}\right\rfloor +\dots+\left\lfloor \frac{\left\lfloor nx\right\rfloor }{x}\right\rfloor =n\left\lfloor nx\right\rfloor \]
Show that for all positive integers $m$ and $n$, \[\gcd(m, n) = m+n-mn+2\sum^{m-1}_{k=0}\left \lfloor \frac{kn}{m}\right \rfloor.\]
Sofía colours $46$ cells of a $9 \times 9$ board red. If Pedro can find a $2 \times 2$ square from the board that has $3$ or more red cells, he wins; otherwise, Sofía wins. Determine the player with the winning strategy.
Let $m$ and $n$ be integers greater than 1. Prove that $\left\lfloor \dfrac{mn}{6} \right\rfloor$ non-overlapping 2-by-3 rectangles can be placed in an $m$-by-$n$ rectangle. Note: $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x$.
The following facts are known in a mathematical contest:
[list]
(a) The number of problems tested was $n\ge 4$
(b) Each problem was solved by exactly four contestants.
(c) For each pair of problems, there is exactly one contestant who solved both problems
[/list]
Assuming the number of contestants is greater than or equal to $4n$, find the minimum value of $n$ for which there always exists a contestant who solved all the problems.
Let $a_1,a_2 \dots a_n$ and $x_1, x_2 \dots x_n$ be integers and $r\geq 2$ be an integer. It is known that \[\sum_{j=0}^{n} a_j x_j^k =0 \qquad \text{for} \quad k=1,2, \dots r.\]
Prove that
\[\sum_{j=0}^{n} a_j x_j^m \equiv 0 \pmod m, \qquad \text{for all}\quad m \in \{ r+1, r+2, \cdots, 2r+1 \}.\]
A polynomial $P$ of degree $2015$ satisfies the equation $P(n)=\frac{1}{n^2}$ for $n=1, 2, \dots, 2016$. Find $\lfloor 2017P(2017)\rfloor$.
How many integers $100 \le x \le 999$ have the property that, among the six digits in $\lfloor 280 + \frac{x}{100} \rfloor$ and $x$, exactly two are identical?
For a matrix $(p_{ij})$ of the format $m\times n$ with real entries, set
\[a_i =\displaystyle\sum_{j=1}^n p_{ij}\text{ for }i = 1,\cdots,m\text{ and }b_j =\displaystyle\sum_{i=1}^m p_{ij}\text{ for }j = 1, . . . , n\longrightarrow(1)\]
By integering a real number, we mean replacing the number with the integer closest to it. Prove that integering the numbers $a_i, b_j, p_{ij}$ can be done in such a way that $(1)$ still holds.
Solve the equation $x\cdot [x\cdot [x \cdot [x]]] = 88$ in the set of real numbers.
Find all numbers $a$ such that
(1) all numbers $[a], [2a], . . . , [Na]$ are distinct and
(2) all numbers $\left[ \frac{1}{a}\right], \left[ \frac{2}{a}\right], ..., \left[ \frac{M}{a}\right]$ are distinct.
Call a subset $ S$ of $ \{1,2,\dots,n\}$ [i]mediocre[/i] if it has the following property: Whenever $ a$ and $ b$ are elements of $ S$ whose average is an integer, that average is also an element of $ S.$ Let $ A(n)$ be the number of mediocre subsets of $ \{1,2,\dots,n\}.$ [For instance, every subset of $ \{1,2,3\}$ except $ \{1,3\}$ is mediocre, so $ A(3)\equal{}7.$] Find all positive integers $ n$ such that $ A(n\plus{}2)\minus{}2A(n\plus{}1)\plus{}A(n)\equal{}1.$
A finite sequence of integers $a_1, a_2, \dots, a_n$ is called [i]regular[/i] if there exists a real number $x$ satisfying \[ \left\lfloor kx \right\rfloor = a_k \quad \text{for } 1 \le k \le n. \] Given a regular sequence $a_1, a_2, \dots, a_n$, for $1 \le k \le n$ we say that the term $a_k$ is [i]forced[/i] if the following condition is satisfied: the sequence \[ a_1, a_2, \dots, a_{k-1}, b \] is regular if and only if $b = a_k$. Find the maximum possible number of forced terms in a regular sequence with $1000$ terms.
Given a positive integer $n,$ what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same?
[When $n=8,$ the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is [i]at least[/i] 3.]
Twenty-five points are given on the plane. Among any three of them, one can choose two less than one inch apart. Prove that there are 13 points among them which lie in a circle of radius 1.
Let $j$ and $k$ be integers. Prove that the inequality
$$\lfloor(j+k)\alpha\rfloor+\lfloor(j+k)\beta\rfloor\ge\lfloor j\alpha\rfloor+\lfloor j\beta\rfloor+\lfloor k(\alpha+\beta)\rfloor$$holds for all real numbers $\alpha,\beta$ if and only if $j=k$.
Let $n$ and $b$ be positive integers. We say $n$ is $b$-discerning if there exists a set consisting of $n$ different positive integers less than $b$ that has no two different subsets $U$ and $V$ such that the sum of all elements in $U$ equals the sum of all elements in $V$.
(a) Prove that $8$ is $100$-discerning.
(b) Prove that $9$ is not $100$-discerning.
[i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]
Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$.
[i]Evan O' Dorney.[/i]
Find all solutions to the equation:
\[ \left(\left\lfloor x+\frac{7}{3} \right\rfloor \right)^2-\left\lfloor x-\frac{9}{4} \right\rfloor = 16 \]
How many of the first 1000 positive integers can be expressed in the form
\[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \]
where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?
Find $\lim_{n\to\infty} \int_0^1 |\sin nx|^3dx\ (n=1,\ 2,\ \cdots).$
[i]2010 Kyoto Institute of Technology entrance exam/Textile, 2nd exam[/i]