Found problems: 1187
2019 IMO Shortlist, N6
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 $z$.)
2019 LIMIT Category B, Problem 11
$$\left\lfloor\left(1\cdot2+2\cdot2^2+\ldots+100\cdot2^{100}\right)\cdot9^{-901}\right\rfloor=?$$
2013 Benelux, 1
Let $n \ge 3$ be an integer. A frog is to jump along the real axis, starting at the point $0$ and making $n$ jumps: one of length $1$, one of length $2$, $\dots$ , one of length $n$. It may perform these $n$ jumps in any order. If at some point the frog is sitting on a number $a \le 0$, its next jump must be to the right (towards the positive numbers). If at some point the frog is sitting on a number $a > 0$, its next jump must be to the left (towards the negative numbers). Find the largest positive integer $k$ for which the frog can perform its jumps in such an order that it never lands on any of the numbers $1, 2, \dots , k$.
1992 French Mathematical Olympiad, Problem 5
Determine the number of digits $1$ in the integer part of $\frac{10^{1992}}{10^{83}+7}$.
2008 Harvard-MIT Mathematics Tournament, 10
Determine the number of $ 8$-tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\le a_k\le k$, for each $ k \equal{} 1,2,3,4$, and $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$.
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?
1992 IMO Longlists, 75
A sequence $\{an\}$ of positive integers is defined by
\[a_n=\left[ n +\sqrt n + \frac 12 \right] , \qquad \forall n \in \mathbb N\]
Determine the positive integers that occur in the sequence.
2015 Hanoi Open Mathematics Competitions, 7
Solve equation $x^4 = 2x^2 + \lfloor x \rfloor$, where $ \lfloor x \rfloor$ is an integral part of $x$.
1997 Romania National Olympiad, 4
Consider the numbers $a,b, \alpha, \beta \in \mathbb{R}$ and the sets $$A=\left \{x \in \mathbb{R} : x^2+a|x|+b=0 \right \},$$ $$B=\left \{ x \in \mathbb{R} : \lfloor x \rfloor^2 + \alpha \lfloor x \rfloor + \beta = 0\right \}.$$
If $A \cap B$ has exactly three elements, prove that $a$ cannot be an integer.
2007 Balkan MO Shortlist, A7
Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which
\[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\]
is a rational number.
2000 JBMO ShortLists, 3
Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$
2007 Romania Team Selection Test, 4
Let $S$ be the set of $n$-uples $\left( x_{1}, x_{2}, \ldots, x_{n}\right)$ such that $x_{i}\in \{ 0, 1 \}$ for all $i \in \overline{1,n}$, where $n \geq 3$. Let $M(n)$ be the smallest integer with the property that any subset of $S$ with at least $M(n)$ elements contains at least three $n$-uples \[\left( x_{1}, \ldots, x_{n}\right), \, \left( y_{1}, \ldots, y_{n}\right), \, \left( z_{1}, \ldots, z_{n}\right) \] such that
\[\sum_{i=1}^{n}\left( x_{i}-y_{i}\right)^{2}= \sum_{i=1}^{n}\left( y_{i}-z_{i}\right)^{2}= \sum_{i=1}^{n}\left( z_{i}-x_{i}\right)^{2}. \]
(a) Prove that $M(n) \leq \left\lfloor \frac{2^{n+1}}{n}\right\rfloor+1$.
(b) Compute $M(3)$ and $M(4)$.
PEN I Problems, 6
Prove that for all positive integers $n$, \[\lfloor \sqrt{n}+\sqrt{n+1}+\sqrt{n+2}\rfloor =\lfloor \sqrt{9n+8}\rfloor.\]
PEN P Problems, 6
Show that every integer greater than $1$ can be written as a sum of two square-free integers.
2012 ELMO Shortlist, 4
A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles.
[i]Calvin Deng.[/i]
2010 AMC 12/AHSME, 22
What is the minimum value of $ f(x) \equal{} |x \minus{} 1| \plus{} |2x \minus{} 1| \plus{} |3x \minus{} 1| \plus{} \cdots \plus{} |119x \minus{} 1|$?
$ \textbf{(A)}\ 49 \qquad
\textbf{(B)}\ 50 \qquad
\textbf{(C)}\ 51 \qquad
\textbf{(D)}\ 52 \qquad
\textbf{(E)}\ 53$
2012 Brazil National Olympiad, 4
There exists some integers $n,a_1,a_2,\ldots,a_{2012}$ such that
\[ n^2=\sum_{1 \leq i \leq 2012}{{a_i}^{p_i}} \]
where $p_i$ is the i-th prime ($p_1=2,p_2=3,p_3=5,p_4=7,\ldots$) and $a_i>1$ for all $i$?
2022 JBMO TST - Turkey, 2
For a real number $a$, $[a]$ denotes the largest integer not exceeding $a$.
Find all positive real numbers $x$ satisfying the equation
$$x\cdot [x]+2022=[x^2]$$
1997 AIME Problems, 11
Let $x=\frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle \sum_{n=1}^{44} \sin n^\circ}.$ What is the greatest integer that does not exceed $100x$?
2005 Slovenia National Olympiad, Problem 1
Find all positive numbers $x$ such that $20\{x\}+0.5\lfloor x\rfloor = 2005$.
2013 Iran Team Selection Test, 2
Find the maximum number of subsets from $\left \{ 1,...,n \right \}$ such that for any two of them like $A,B$ if $A\subset B$ then $\left | B-A \right |\geq 3$. (Here $\left | X \right |$ is the number of elements of the set $X$.)
1976 IMO Shortlist, 4
A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where [x] denotes the smallest integer $\leq$ x)$.$
2007 National Olympiad First Round, 7
What is the sum of real numbers satisfying the equation $\left \lfloor \frac{6x+5}{8} \right \rfloor = \frac{15x-7}{5}$?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ \frac{81}{90}
\qquad\textbf{(C)}\ \frac{7}{15}
\qquad\textbf{(D)}\ \frac{4}{5}
\qquad\textbf{(E)}\ \frac{19}{15}
$
2010 Math Prize for Girls Olympiad, 1
Let $S$ be a set of 100 integers. Suppose that for all positive integers $x$ and $y$ (possibly equal) such that $x + y$ is in $S$, either $x$ or $y$ (or both) is in $S$. Prove that the sum of the numbers in $S$ is at most 10,000.
2005 QEDMO 1st, 7 (C1)
Prove:
From the set $\{1,2,...,n\}$, one can choose a subset with at most $2 \left\lfloor \sqrt n \right\rfloor +1$ elements such that the set of the pairwise differences from this subset is $\{1,2,...,n-1\}$.
($\left\lfloor x \right\rfloor$ means the greatest integer $\leq x$)