Found problems: 1340
Tanya chose a natural number $X\le100$, and Sasha is trying to guess this number. He can select two natural numbers $M$ and $N$ less than $100$ and ask about $\gcd(X+M,N)$. Show that Sasha can determine Tanya's number with at most seven questions.
Consider a board on $2013 \times 2013$ squares, what is the maximum number of chess knights that can be placed so that no $2$ attack each other?
Let $p=4k+1$ be a prime. Show that \[\sum^{k}_{i=1}\left \lfloor \sqrt{ ip }\right \rfloor = \frac{p^{2}-1}{12}.\]
Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$,
\[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]
Prove that for any integer $n$ one can find integers $a$ and $b$ such that
\[n=\left[ a\sqrt{2}\right]+\left[ b\sqrt{3}\right] \]
Let $A$ be the set of positive integers of the form $a^2 +2b^2$, where $a$ and $b$ are integers and $b \neq 0$. Show that if $p$ is a prime number and $p^2 \in A$, then $p \in A$.
Two players $A$ and $B$ have one stone each on a $100 \times 100$ chessboard. They move their stones one after the other, and a move means moving one's stone to a neighbouring field (horizontally or vertically, not diagonally). At the beginning of the game, the stone of $A$ lies in the lower left corner, and the one of $B$ in the lower right corner. Player $A$ starts.
Prove: Player $A$ is, independently from that what $B$ does, able to reach, after finitely many steps, the field $B$'s stone is lying on at that moment.
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]
Show that the sequence $\{a_{n}\}_{n \ge 1}$ defined by $a_{n}=\lfloor n\sqrt{2}\rfloor$ contains an infinite number of integer powers of $2$.
Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
(You may use the fact that $\sin{n^\circ}$ is irrational for positive integers $n$ not divisible by $30$.)
[i]Ray Li[/i]
Let $T=\text{TNFTPP}$, and let $R=T-914$. Let $x$ be the smallest real solution of \[3x^2+Rx+R=90x\sqrt{x+1}.\] Find the value of $\lfloor x\rfloor$.
Let $f(n)=n+\lfloor \sqrt{n}\rfloor$. Prove that, for every positive integer $m$, the sequence \[m, f(m), f(f(m)), f(f(f(m))), \cdots\] contains at least one square of an integer.
Find the maximum possible number of kings on a $12\times 12$ chess table so that each king attacks exactly one of the other kings (a king attacks only the squares that have a common point with the square he sits on).
Find the smallest positive integer $c$ for which the following statement holds: Let $k$ and $n$ be positive integers. Suppose there exist pairwise distinct subsets $S_1$, $S_2$, $\dots$, $S_{2k}$ of $\{1, 2, \dots, n\}$, such that $S_i \cap S_j \neq \varnothing$ and $S_i \cap S_{j+k} \neq \varnothing$ for all $1 \le i,j \le k$. Then $1000k \le c \cdot 2^n$.
[i]Proposed by Yang Liu[/i]
Compute \[\left\lfloor \dfrac{2007!+2004!}{2006!+2005!}\right\rfloor.\] (Note that $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)
For each real number $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. For example $\lfloor 2.8 \rfloor = 2$. Let $r \ge 0$ be a real number such that for all integers $m, n, m|n$ implies $\lfloor mr \rfloor| \lfloor nr \rfloor$. Prove that $r$ is an integer.
Prove that there is no positive rational number $x$ such that \[x^{\lfloor x\rfloor }=\frac{9}{2}.\]
Ten women sit in $ 10$ seats in a line. All of the $ 10$ get up and then reseat themselves using all $ 10$ seats, each sitting in the seat she was in before or a seat next to the one she occupied before. In how many ways can the women be reseated?
$ \textbf{(A)}\ 89\qquad
\textbf{(B)}\ 90\qquad
\textbf{(C)}\ 120\qquad
\textbf{(D)}\ 2^{10}\qquad
\textbf{(E)}\ 2^2 3^8$
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose the Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$?
$\textbf{(A) } 9
\qquad\textbf{(B) } 11
\qquad\textbf{(C) } 12
\qquad\textbf{(D) } 13
\qquad\textbf{(E) } 15
$
Let two natural number $n > 1$ and $m$ be given. Find the least positive integer $k$ which has the following property: Among $k$ arbitrary integers $a_1, a_2, \ldots, a_k$ satisfying the condition $a_i - a_j$ ( $1 \leq i < j \leq k$) is not divided by $n$, there exist two numbers $a_p, a_s$ ($p \neq s$) such that $m + a_p - a_s$ is divided by $n$.
Let $r\geq1$ be areal number that holds with the property that for each pair of positive integer numbers $m$ and $n$, with $n$ a multiple of $m$, it is true that $\lfloor{nr}\rfloor$ is multiple of $\lfloor{mr}\rfloor$. Show that $r$ has to be an integer number.
[b]Note: [/b][i]If $x$ is a real number, $\lfloor{x}\rfloor$ is the greatest integer lower than or equal to $x$}.[/i]
At what index the harmonic series has a fractional part of $ 1/12? $
Given an integer $n \geq 2$ determine the integral part of the number
$ \sum_{k=1}^{n-1} \frac {1} {({1+\frac{1} {n}}) \dots ({1+\frac {k} {n})}}$ - $\sum_{k=1}^{n-1} (1-\frac {1} {n}) \dots(1-\frac{k}{n})$