Found problems: 1187
2020 China Girls Math Olympiad, 6
Let $p, q$ be integers and $p, q > 1$ , $gcd(p, \,6q)=1$. Prove that:$$\sum_{k=1}^{q-1}\left \lfloor \frac{pk}{q}\right\rfloor^2 \equiv 2p \sum_{k=1}^{q-1}k\left\lfloor \frac{pk}{q} \right\rfloor (mod \, q-1)$$
2023 Romania National Olympiad, 3
We say that a natural number $n$ is interesting if it can be written in the form
\[
n = \left\lfloor \frac{1}{a} \right\rfloor + \left\lfloor \frac{1}{b} \right\rfloor + \left\lfloor \frac{1}{c} \right\rfloor,
\] where $a,b,c$ are positive real numbers such that $a + b + c = 1.$
Determine all interesting numbers. ( $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$.)
1989 IMO Longlists, 49
Let $ t(n)$ for $ n \equal{} 3, 4, 5, \ldots,$ represent the number of distinct, incongruent, integer-sided triangles whose perimeter is $ n;$ e.g., $ t(3) \equal{} 1.$ Prove that
\[ t(2n\minus{}1) \minus{} t(2n) \equal{} \left[ \frac{6}{n} \right] \text{ or } \left[ \frac{6}{n} \plus{} 1 \right].\]
1998 Moldova Team Selection Test, 2
Determine the natural numbers that cannot be written as $\lfloor n + \sqrt{n} + \frac{1}{2} \rfloor$ for any $n \in \mathbb{N}$.
2007 Germany Team Selection Test, 1
We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by
\[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right),
\] where $\lfloor x\rfloor$ denotes the integer part of $x$.
[b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often.
[b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often.
[i]Proposed by Johan Meyer, South Africa[/i]
1991 AIME Problems, 3
Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives \begin{eqnarray*} &\ & \binom{1000}{0}(0.2)^0+\binom{1000}{1}(0.2)^1+\binom{1000}{2}(0.2)^2+\cdots+\binom{1000}{1000}(0.2)^{1000}\\ &\ & = A_0 + A_1 + A_2 + \cdots + A_{1000}, \end{eqnarray*} where $A_k = \binom{1000}{k}(0.2)^k$ for $k = 0,1,2,\ldots,1000$. For which $k$ is $A_k$ the largest?
2008 ITest, 7
Find the number of integers $n$ for which $n^2+10n<2008$.
1990 AMC 12/AHSME, 19
For how many integers $N$ between $1$ and $1990$ is the improper fraction $\frac{N^2+7}{N+4}$ not in lowest terms?
$\text{(A)} \ 0 \qquad \text{(B)} \ 86 \qquad \text{(C)} \ 90 \qquad \text{(D)} \ 104 \qquad \text{(E)} \ 105$
2004 Alexandru Myller, 4
Find the real numbers $ x>1 $ having the property that $ \sqrt[n]{\lfloor x^n \rfloor } $ is an integer for any natural number $ n\ge 2. $
[i]Mihai Piticari[/i] and [i]Dan Popescu[/i]
2017 Harvard-MIT Mathematics Tournament, 6
A positive integer $n$ is [i]magical[/i] if $\lfloor \sqrt{\lceil \sqrt{n} \rceil} \rfloor=\lceil \sqrt{\lfloor \sqrt{n} \rfloor} \rceil$. Find the number of magical integers between $1$ and $10,000$ inclusive.
2013 Brazil Team Selection Test, 2
Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.
2020 USA TSTST, 4
Find all pairs of positive integers $(a,b)$ satisfying the following conditions:
[list]
[*] $a$ divides $b^4+1$,
[*] $b$ divides $a^4+1$,
[*] $\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$.
[/list]
[i]Yang Liu[/i]
2011 China Western Mathematical Olympiad, 3
Let $n \geq 2$ be a given integer
$a)$ Prove that one can arrange all the subsets of the set $\{1,2... ,n\}$ as a sequence of subsets $A_{1}, A_{2},\cdots , A_{2^{n}}$, such that $|A_{i+1}| = |A_{i}| + 1$ or $|A_{i}| - 1$ where $i = 1,2,3,\cdots , 2^{n}$ and $A_{2^{n} + 1} = A_{1}$
$b)$ Determine all possible values of the sum $\sum \limits_{i = 1}^{2^n} (-1)^{i}S(A_{i})$ where $S(A_{i})$ denotes the sum of all elements in $A_{i}$ and $S(\emptyset) = 0$, for any subset sequence $A_{1},A_{2},\cdots ,A_{2^n}$ satisfying the condition in $a)$
PEN E Problems, 24
Let $p_{n}$ again denote the $n$th prime number. Show that the infinite series \[\sum^{\infty}_{n=1}\frac{1}{p_{n}}\] diverges.
2007 Mathematics for Its Sake, 1
Prove that the parity of each term of the sequence $ \left( \left\lfloor \left( \lfloor \sqrt q \rfloor +\sqrt{q} \right)^n \right\rfloor \right)_{n\ge 1} $ is opposite to the parity of its index, where $ q $ is a squarefree natural number.
2021 China Team Selection Test, 4
Proof that
$$ \sum_{m=1}^n5^{\omega (m)} \le \sum_{k=1}^n\lfloor \frac{n}{k} \rfloor \tau (k)^2 \le \sum_{m=1}^n5^{\Omega (m)} .$$
2018 Bosnia And Herzegovina - Regional Olympiad, 3
Solve equation $x \lfloor{x}\rfloor+\{x\}=2018$, where $x$ is real number
2020 Taiwan TST Round 3, 2
Let $H = \{ \lfloor i\sqrt{2}\rfloor : i \in \mathbb Z_{>0}\} = \{1,2,4,5,7,\dots \}$ and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A\subseteq \{1,2,\dots, n\}$ satisfies $|A| \ge C\sqrt{n}$, then there exist $a,b\in A$ such that $a-b\in H$. (Here $\mathbb Z_{>0}$ is the set of positive integers, and $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.)
2024 India National Olympiad, 6
For each positive integer $n \ge 3$, define $A_n$ and $B_n$ as
\[A_n = \sqrt{n^2 + 1} + \sqrt{n^2 + 3} + \cdots + \sqrt{n^2+2n-1}\]
\[B_n = \sqrt{n^2 + 2} + \sqrt{n^2 + 4} + \cdots + \sqrt{n^2 + 2n}.\]
Determine all positive integers $n\ge 3$ for which $\lfloor A_n \rfloor = \lfloor B_n \rfloor$.
Note. For any real number $x$, $\lfloor x\rfloor$ denotes the largest integer $N\le x$.
[i]Anant Mudgal and Navilarekallu Tejaswi[/i]
2006 National Olympiad First Round, 7
How many positive integers are there such that $\left \lfloor \frac m{11} \right \rfloor = \left \lfloor \frac m{10} \right \rfloor$? ($\left \lfloor x \right \rfloor$ denotes the greatest integer not exceeding $x$.)
$
\textbf{(A)}\ 44
\qquad\textbf{(B)}\ 48
\qquad\textbf{(C)}\ 52
\qquad\textbf{(D)}\ 54
\qquad\textbf{(E)}\ 56
$
2012 Romania Team Selection Test, 1
Prove that for any positive integer $n\geq 2$ we have that \[\sum_{k=2}^n \lfloor \sqrt[k]{n}\rfloor=\sum_{k=2}^n\lfloor\log_{k}n\rfloor.\]
2007 Indonesia TST, 3
For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.
2019 Taiwan APMO Preliminary Test, P4
We define a sequence ${a_n}$:
$$a_1=1,a_{n+1}=\sqrt{a_n+n^2},n=1,2,...$$
(1)Find $\lfloor a_{2019}\rfloor$
(2)Find $\lfloor a_{1}^2\rfloor+\lfloor a_{2}^2\rfloor+...+\lfloor a_{20}^2\rfloor$
1991 IMTS, 1
For every positive integer $n$, form the number $n/s(n)$, where $s(n)$ is the sum of digits of $n$ in base 10. Determine the minimum value of $n/s(n)$ in each of the following cases:
(i) $10 \leq n \leq 99$
(ii) $100 \leq n \leq 999$
(iii) $1000 \leq n \leq 9999$
(iv) $10000 \leq n \leq 99999$
2023 Romania National Olympiad, 4
Let $r$ and $s$ be real numbers in the interval $[1, \infty)$ such that for all positive integers $a$ and $b$ with $a \mid b \implies \left\lfloor ar \right\rfloor$ divides $\left\lfloor bs \right\rfloor$.
a) Prove that $\frac{s}{r}$ is a natural number.
b) Show that both $r$ and $s$ are natural numbers.
Here, $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to $x$.