Found problems: 1187
Solve equation $x \lfloor{x}\rfloor+\{x\}=2018$, where $x$ is real number
Find all nonzero subsets $A$ of the set $\left\{2,3,4,5,\cdots\right\}$ such that $\forall n\in A$, we have that $n^{2}+4$ and $\left\lfloor{\sqrt{n}\right\rfloor}+1$ are both in $A$.
Find all postitive integers n such that
$$\left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor=n^2$$
where $\lfloor x \rfloor$ represents the largest integer less than the real number $x$.
Consider $ n$ points in a plane which are vertices of a convex polygon. Prove that the set of the lengths of the sides and the diagonals of the polygon has at least $ \lfloor n/2\rfloor$ elements.
suppose that $\mathcal F\subseteq X^{(K)}$ and $|X|=n$. we know that for every three distinct elements of $\mathcal F$ like $A,B$ and $C$ we have $A\cap B \not\subset C$.
a)(10 points) Prove that :
\[|\mathcal F|\le \dbinom{k}{\lfloor\frac{k}{2}\rfloor}+1\]
b)(15 points) if elements of $\mathcal F$ do not necessarily have $k$ elements, with the above conditions show that:
\[|\mathcal F|\le \dbinom{n}{\lceil\frac{n-2}{3}\rceil}+2\]
Evaluate the sum \[ \dfrac {1}{2 \lfloor \sqrt {1} \rfloor + 1} + \dfrac {1}{2 \lfloor \sqrt {2} \rfloor + 1} + \dfrac {1}{2 \lfloor \sqrt {3} \rfloor + 1} + \cdots + \dfrac {1}{2 \lfloor \sqrt {100} \rfloor + 1} \]
Let $\lfloor \bullet \rfloor$ denote the floor function. For nonnegative integers $a$ and $b$, their [i]bitwise xor[/i], denoted $a \oplus b$, is the unique nonnegative integer such that $$ \left \lfloor \frac{a}{2^k} \right \rfloor+ \left\lfloor\frac{b}{2^k} \right\rfloor - \left\lfloor \frac{a\oplus b}{2^k}\right\rfloor$$ is even for every $k \ge 0$. Find all positive integers $a$ such that for any integers $x>y\ge 0$, we have \[ x\oplus ax \neq y \oplus ay. \]
[i]Carl Schildkraut[/i]
For a positive integer $a$, define a sequence of integers $x_1,x_2,\ldots$ by letting $x_1=a$ and $x_{n+1}=2x_n+1$ for $n\geq 1$. Let $y_n=2^{x_n}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_1,\ldots,y_k$ are all prime.
For $2k$ real numbers $a_1, a_2, ..., a_k$, $b_1, b_2, ..., b_k$ define a sequence of numbers $X_n$ by \[
X_n = \sum_{i=1}^k [a_in + b_i] \quad (n=1,2,...).
\] If the sequence $X_N$ forms an arithmetic progression, show that $\textstyle\sum_{i=1}^k a_i$ must be an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.
For a real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$ and let $\{x\} = x - \lfloor x\rfloor$. If $a, b, c$ are distinct real numbers, prove that
\[\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)}\]
is an integer if and only if $\{a\} + \{b\} + \{c\}$ is an integer.
Let $k$ be a given positive integer. The sequence $x_n$ is defined as follows: $x_1 =1$ and $x_{n+1}$ is the least positive integer which is not in $\{x_{1}, x_{2},..., x_{n}, x_{1}+k, x_{2}+2k,..., x_{n}+nk \}$. Show that there exist real number $a$ such that $x_n = \lfloor an\rfloor$ for all positive integer $n$.
Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10?$
For a given positive integer $n >2$, let $C_{1},C_{2},C_{3}$ be the boundaries of three convex $n-$ gons in the plane , such that
$C_{1}\cap C_{2}, C_{2}\cap C_{3},C_{1}\cap C_{3}$ are finite. Find the maximum number of points of the sets $C_{1}\cap C_{2}\cap C_{3}$.
Which natural numbers cannot be presented in that way: $[n+\sqrt{n}+\frac{1}{2}]$, $n\in\mathbb{N}$
$[y]$ is the greatest integer function.
Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$.
A prison has $2004$ cells, numbered $1$ through $2004$. A jailer, carrying out the terms of a partial amnesty, unlocked every cell. Next he locked every second cell. Then he turned the key in every third cell, locking the opened cells, and unlocking the locked ones. He continued this way, on $n^{\text{th}}$ trip, turning the key in every $n^{\text{th}}$ cell, and he finished his mission after $2004$ trips. How many prisoners were released?
How many positive integers $n$, with $n\le 2007$, yield a solution for $x$ (where $x$ is real) in the equation $\lfloor x \rfloor+\lfloor 2x\rfloor+\lfloor 3x\rfloor=n$?
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]