Found problems: 1187
2017 AMC 12/AHSME, 20
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,1)$. What is the probability that $\lfloor \log_2{x} \rfloor=\lfloor \log_2{y} \rfloor$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number $r$?
$\textbf{(A)}\ \frac{1}{8}\qquad\textbf{(B)}\ \frac{1}{6}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2}$
2013 China Western Mathematical Olympiad, 4
There are $n$ coins in a row, $n\geq 2$. If one of the coins is head, select an odd number of consecutive coins (or even 1 coin) with the one in head on the leftmost, and then flip all the selected coins upside down simultaneously. This is a $move$. No move is allowed if all $n$ coins are tails.
Suppose $m-1$ coins are heads at the initial stage, determine if there is a way to carry out $ \lfloor\frac {2^m}{3}\rfloor $ moves
2004 India IMO Training Camp, 2
Find all primes $p \geq 3$ with the following property: for any prime $q<p$, the number
\[
p - \Big\lfloor \frac{p}{q} \Big\rfloor q
\]
is squarefree (i.e. is not divisible by the square of a prime).
2010 Iran MO (3rd Round), 3
suppose that $\mathcal F\subseteq p(X)$ and $|X|=n$. we know that for every $A_i,A_j\in \mathcal F$ that $A_i\supseteq A_j$ we have $3\le |A_i|-|A_j|$. prove that:
$|\mathcal F|\le \lfloor\frac{2^n}{3}+\frac{1}{2}\dbinom{n}{\lfloor\frac{n}{2}\rfloor}\rfloor$
(20 points)
1987 IMO Longlists, 42
Find the integer solutions of the equation
\[ \left[ \sqrt{2m} \right] = \left[ n(2+\sqrt 2) \right] \]
2000 Taiwan National Olympiad, 2
Let $n$ be a positive integer and $A=\{ 1,2,\ldots ,n\}$. A subset of $A$ is said to be connected if it consists of one element or several consecutive elements. Determine the maximum $k$ for which there exist $k$ distinct subsets of $A$ such that the intersection of any two of them is connected.
2009 Moldova Team Selection Test, 3
[color=darkblue]Weightlifter Ruslan has just finished the exercise with a weight, which has $ n$ small weights on one side and $ n$ on the another. At each stage he takes some weights from one of the sides, such that at any moment the difference of the numbers of weights on the sides does not exceed $ k$. What is the minimal number of stages (in function if $ n$ and $ k$), which Ruslan need to take off all weights..[/color]
1998 Romania Team Selection Test, 1
Let $n\ge 2$ be an integer. Show that there exists a subset $A\in \{1,2,\ldots ,n\}$ such that:
i) The number of elements of $A$ is at most $2\lfloor\sqrt{n}\rfloor+1$
ii) $\{ |x-y| \mid x,y\in A, x\not= y\} = \{ 1,2,\ldots n-1 \}$
[i]Radu Todor[/i]
2010 Contests, 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 Baltic Way, 7
A rectangular array has $ n$ rows and $ 6$ columns, where $ n \geq 2$. In each cell there is written either $ 0$ or $ 1$. All rows in the array are different from each other. For each two rows $ (x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})$ and $ (y_{1},y_{2},y_{3},y_{4},y_{5},y_{6})$, the row $ (x_{1}y_{1},x_{2}y_{2},x_{3}y_{3},x_{4}y_{4},x_{5}y_{5},x_{6}y_{6})$ can be found in the array as well. Prove that there is a column in which at least half of the entries are zeros.
2008 Kazakhstan National Olympiad, 1
Let $ F_n$ be a set of all possible connected figures, that consist of $ n$ unit cells. For each element $ f_n$ of this set, let $ S(f_n)$ be the area of that minimal rectangle that covers $ f_n$ and each side of the rectangle is parallel to the corresponding side of the cell. Find $ max(S(f_n))$,where $ f_n\in F_n$?
Remark: Two cells are called connected if they have a common edge.
1975 Canada National Olympiad, 4
For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.
1985 AIME Problems, 10
How many of the first 1000 positive integers can be expressed in the form
\[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \]
where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?
1998 Greece National Olympiad, 4
Let a function $g:\mathbb{N}_0\to\mathbb{N}_0$ satisfy $g(0)=0$ and $g(n)=n-g(g(n-1))$ for all $n\ge 1$. Prove that:
a) $g(k)\ge g(k-1)$ for any positive integer $k$.
b) There is no $k$ such that $g(k-1)=g(k)=g(k+1)$.
2021 Thailand TST, 2
Prove that, for all positive integers $m$ and $n$, we have $$\left\lfloor m\sqrt{2} \right\rfloor\cdot\left\lfloor n\sqrt{7} \right\rfloor<\left\lfloor mn\sqrt{14} \right\rfloor.$$
1998 AMC 12/AHSME, 30
For each positive integer $n$, let
\[a_n = \frac {(n + 9)!}{(n - 1)!}.\]
Let $k$ denote the smallest positive integer for which the rightmost nonzero digit of $a_k$ is odd. The rightmost nonzero digit of $a_k$ is
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 7\qquad
\textbf{(E)}\ 9$
2008 Moldova Team Selection Test, 4
A non-empty set $ S$ of positive integers is said to be [i]good[/i] if there is a coloring with $ 2008$ colors of all positive integers so that no number in $ S$ is the sum of two different positive integers (not necessarily in $ S$) of the same color. Find the largest value $ t$ can take so that the set $ S\equal{}\{a\plus{}1,a\plus{}2,a\plus{}3,\ldots,a\plus{}t\}$ is good, for any positive integer $ a$.
[hide="P.S."]I have the feeling that I've seen this problem before, so if I'm right, maybe someone can post some links...[/hide]
PEN D Problems, 1
If $p$ is an odd prime, prove that \[{k \choose p}\equiv \left\lfloor \frac{k}{p}\right\rfloor \pmod{p}.\]
PEN I Problems, 11
Let $p$ be a prime number of the form $4k+1$. Show that \[\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right \rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) = \frac{p-1}{2}.\]
2011 Uzbekistan National Olympiad, 2
Prove that $ \forall n\in\mathbb{N}$,$ \exists a,b,c\in$$\bigcup_{k\in\mathbb{N}}(k^{2},k^{2}+k+3\sqrt 3) $ such that $n=\frac{ab}{c}$.
2020 BMT Fall, 10
For $k\ge 1$, define $a_k=2^k$. Let $$S=\sum_{k=1}^{\infty}\cos^{-1}\left(\frac{2a_k^2-6a_k+5}{\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\right).$$ Compute $\lfloor 100S\rfloor$.
1984 Iran MO (2nd round), 2
Consider the function
\[f(x)= \sin \biggl( \frac{\pi}{2} \lfloor x \rfloor \biggr).\]
Find the period of $f$ and sketch diagram of $f$ in one period. Also prove that $\lim_{x \to 1} f(x)$ does not exist.
2014 Harvard-MIT Mathematics Tournament, 4
Compute \[\sum_{k=0}^{100}\left\lfloor\dfrac{2^{100}}{2^{50}+2^k}\right\rfloor.\] (Here, if $x$ is a real number, then $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.)
2010 ELMO Shortlist, 4
Let $-2 < x_1 < 2$ be a real number and define $x_2, x_3, \ldots$ by $x_{n+1} = x_n^2-2$ for $n \geq 1$. Assume that no $x_n$ is $0$ and define a number $A$, $0 \leq A \leq 1$ in the following way: The $n^{\text{th}}$ digit after the decimal point in the binary representation of $A$ is a $0$ if $x_1x_2\cdots x_n$ is positive and $1$ otherwise. Prove that $A = \frac{1}{\pi}\cos^{-1}\left(\frac{x_1}{2}\right)$.
[i]Evan O' Dorney.[/i]
2013 Romanian Masters In Mathematics, 2
Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?