Found problems: 1187
2006 Thailand Mathematical Olympiad, 9
Compute the largest integer not exceeding $$\frac{2549^3}{2547\cdot 2548} -\frac{2547^3}{2548\cdot 2549}$$
2010 Contests, 2
Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as
\[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \]
where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.
2015 International Zhautykov Olympiad, 2
Let $ A_n $ be the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that every two neighbouring terms of each subsequence have different parity,and $ B_n $ the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that all the terms of each subsequence have the same parity ( for example,the partition $ {(1,4,5,8),(2,3),(6,9),(7)} $ is an element of $ A_9 $,and the partition $ {(1,3,5),(2,4),(6)} $ is an element of $ B_6 $ ).
Prove that for every positive integer $ n $ the sets $ A_n $ and $ B_{n+1} $ contain the same number of elements.
2000 Croatia National Olympiad, Problem 3
Let $m>1$ be an integer. Determine the number of positive integer solutions of the equation $\left\lfloor\frac xm\right\rfloor=\left\lfloor\frac x{m-1}\right\rfloor$.
1987 IMO Longlists, 42
Find the integer solutions of the equation
\[ \left[ \sqrt{2m} \right] = \left[ n(2+\sqrt 2) \right] \]
PEN F Problems, 5
Prove that there is no positive rational number $x$ such that \[x^{\lfloor x\rfloor }=\frac{9}{2}.\]
1982 IMO Longlists, 9
Given any two real numbers $\alpha$ and $\beta , 0 \leq \alpha < \beta \leq 1$, prove that there exists a natural number $m$ such that
\[\alpha < \frac{\phi(m)}{m} < \beta.\]
2010 China Team Selection Test, 2
Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$,
and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.
1993 Balkan MO, 2
A positive integer given in decimal representation $\overline{ a_na_{n-1} \ldots a_1a_0 }$ is called [i]monotone[/i] if $a_n\leq a_{n-1} \leq \cdots \leq a_0$. Determine the number of monotone positive integers with at most 1993 digits.
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]
2012 Junior Balkan Team Selection Tests - Moldova, 2
Let $ a,b,c,d$ be positive real numbers and $cd=1$. Prove that there exists a positive integer $n$ such that
$ab\leq n^2\leq (a+c)(b+d)$
VMEO IV 2015, 11.3
How many natural number $n$ less than $2015$ that is divisible by $\lfloor\sqrt[3]{n}\rfloor$ ?
2012 Princeton University Math Competition, B5
Considering all numbers of the form $n = \lfloor \frac{k^3}{2012} \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct?
2004 National Olympiad First Round, 10
Let $a_1 = \sqrt 7$ and $b_i = \lfloor a_i \rfloor$, $a_{i+1} = \dfrac{1}{b_i - \lfloor b_i \rfloor}$ for each $i\geq i$. What is the smallest integer $n$ greater than $2004$ such that $b_n$ is divisible by $4$? ($\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$)
$
\textbf{(A)}\ 2005
\qquad\textbf{(B)}\ 2006
\qquad\textbf{(C)}\ 2007
\qquad\textbf{(D)}\ 2008
\qquad\textbf{(E)}\ \text{None of above}
$
1997 Singapore Team Selection Test, 2
For any positive integer n, evaluate $$\sum_{i=0}^{\lfloor \frac{n+1}{2} \rfloor}
{n-i+1 \choose i}$$
, where $\lfloor n \rfloor$ is the greatest integer less than or equal to $n$ .
2019 PUMaC Team Round, 5
Let $f(x) = x^3 + 3x^2 + 1$. There is a unique line of the form $y = mx + b$ such that $m > 0$ and this line intersects $f(x)$ at three points, $A, B, C$ such that $AB = BC = 2$. Find $\lfloor 100m \rfloor$.
2008 ITest, 44
Now Wendy wanders over and joins Dr. Lisi and her younger siblings. Thinking she knows everything there is about how to work with arithmetic series, she nearly turns right around to walk back home when Dr. Lisi poses a more challenging problem. "Suppose I select two distinct terms at random from the $2008$ term sequence. What's the probability that their product is positive?" If $a$ and $b$ are relatively prime positive integers such that $a/b$ is the probability that the product of the two terms is positive, find the value of $a+b$.
2005 Danube Mathematical Olympiad, 2
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 $n$.
2004 Turkey Team Selection Test, 1
Find all possible values of $x-\lfloor x\rfloor$ if $\sin \alpha = 3/5$ and $x=5^{2003}\sin {(2004\alpha)}$.
2008 Harvard-MIT Mathematics Tournament, 7
Compute $ \sum_{n \equal{} 1}^\infty\sum_{k \equal{} 1}^{n \minus{} 1}\frac {k}{2^{n \plus{} k}}$.
1995 Cono Sur Olympiad, 3
Let $n$ be a natural number and $f(n) = 2n - 1995 \lfloor \frac{n}{1000} \rfloor$($\lfloor$ $\rfloor$ denotes the floor function).
1. Show that if for some integer $r$: $f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times), then $n$ is multiple of $1995$.
2. Show that if $n$ is multiple of 1995, then there exists r such that:$f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times). Determine $r$ if $n=1995.500=997500$
2014 Portugal MO, 6
One hundred musicians are planning to organize a festival with several concerts. In each concert, while some of the one hundred musicians play on stage, the others remain in the audience assisting to the players. What is the least number of concerts so that each of the musicians has the chance to listen to each and every one of the other musicians on stage?
2015 Saudi Arabia GMO TST, 1
Let be given the sequence $(x_n)$ defined by $x_1 = 1$ and $x_{n+1} = 3x_n + \lfloor x_n \sqrt5 \rfloor$ for all $n = 1,2,3,...,$ where $\lfloor x \rfloor$ denotes the greatest integer that does not exceed $x$. Prove that for any positive integer $n$ we have $$x_nx_{n+2} - x^2_{n+1} = 4^{n-1}$$
Trần Nam Dũng
2001 Baltic Way, 4
Let $p$ and $q$ be two different primes. Prove that
\[\left\lfloor\frac{p}{q}\right\rfloor+\left\lfloor\frac{2p}{q}\right\rfloor+\left\lfloor\frac{3p}{q}\right\rfloor+\ldots +\left\lfloor\frac{(q-1)p}{q}\right\rfloor=\frac{1}{2}(p-1)(q-1) \]
2006 Pan African, 3
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.