Found problems: 247
2013 Moldova Team Selection Test, 2
Consider a board on $2013 \times 2013$ squares, what is the maximum number of chess knights that can be placed so that no $2$ attack each other?
2006 Iran Team Selection Test, 2
Let $n$ be a fixed natural number.
[b]a)[/b] Find all solutions to the following equation :
\[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \]
[b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) :
\[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]
2014 Math Prize For Girls Problems, 20
How many complex numbers $z$ such that $\left| z \right| < 30$ satisfy the equation
\[
e^z = \frac{z - 1}{z + 1} \, ?
\]
2007 Iran MO (3rd Round), 8
In this question you must make all numbers of a clock, each with using 2, exactly 3 times and Mathematical symbols. You are not allowed to use English alphabets and words like $ \sin$ or $ \lim$ or $ a,b$ and no other digits.
[img]http://i2.tinypic.com/5x73dza.png[/img]
2013 Harvard-MIT Mathematics Tournament, 6
Find the number of integers $n$ such that \[1+\left\lfloor\dfrac{100n}{101}\right\rfloor=\left\lceil\dfrac{99n}{100}\right\rceil.\]
2000 Junior Balkan Team Selection Tests - Romania, 3
Let be a real number $ a. $ For any real number $ p $ and natural number $ k, $ let be the set
$$ A_k(p)=\{ px\in\mathbb{Z}\mid k=\lceil x \rceil \} . $$
Find all real numbers $ b $ such that $ \# A_n(a)=\# A_n(b) , $ for any natural number $ n. $
$ \# $ [i]denotes the cardinal.[/i]
[i]Eugen Păltânea[/i]
2001 Brazil National Olympiad, 2
Given $a_0 > 1$, the sequence $a_0, a_1, a_2, ...$ is such that for all $k > 0$, $a_k$ is the smallest integer greater than $a_{k-1}$ which is relatively prime to all the earlier terms in the sequence.
Find all $a_0$ for which all terms of the sequence are primes or prime powers.
2012 China Team Selection Test, 2
Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that
\[0<|xy-zw|<C\alpha ^{-4}\]
where $\alpha =\frac{|X|}{n}$.
2013 Math Prize For Girls Problems, 14
How many positive integers $n$ satisfy the inequality
\[
\left\lceil \frac{n}{101} \right\rceil + 1 > \frac{n}{100} \, ?
\]
Recall that $\lceil a \rceil$ is the least integer that is greater than or equal to $a$.
2010 Polish MO Finals, 1
The integer number $n > 1$ is given and a set $S \subset \{0, 1, 2, \ldots, n-1\}$ with $|S| > \frac{3}{4} n$. Prove that there exist integer numbers $a, b, c$ such that the remainders after the division by $n$ of the numbers:
\[a, b, c, a+b, b+c, c+a, a+b+c\]
belong to $S$.
2020 Germany Team Selection Test, 3
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
2008 Bulgaria National Olympiad, 3
Let $n\in\mathbb{N}$ and $0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi$ and $b_1,b_2,\ldots ,b_n$ are real numbers for which the following inequality is satisfied :
\[\left|\sum_{i\equal{}1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}\]
for all $ k\in\mathbb{N}$. Prove that $ b_1\equal{}b_2\equal{}\ldots \equal{}b_n\equal{}0$.
2014 AMC 10, 2
Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing $6$ cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?
${ \textbf{(A)}\ \text{Tuesday}\qquad\textbf{(B)}\ \text{Wednesday}\qquad\textbf{(C)}\ \text{Thursday}\qquad\textbf{(D)}}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}$
2023 Iran MO (3rd Round), 3
There's infinity of the following blocks on the table:$1*1 , 1*2 , 1*3 ,.., 1*n$. We have a $n*n$ table and Ali chooses some of these blocks so that the sum of their area is at least $n^2$. Then , Amir tries to cover the $n*n$ table so that none of blocks go out of the table and they don't overlap and he wanna maximize the covered area in the $n*n$ table with those blocks chosen by Ali. Let $k$ be the maximum coverable area independent of Ali's choice. Prove that:
$$n^2 - \lceil \frac{n^2}{4} \rceil \leq k \leq n^2 - \lfloor \frac{n^2}{8} \rfloor$$
*Note : the blocks can be placed only vertically or horizontally.
2006 Iran Team Selection Test, 2
Let $n$ be a fixed natural number.
[b]a)[/b] Find all solutions to the following equation :
\[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \]
[b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) :
\[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]
2007 IberoAmerican, 1
Given an integer $ m$, define the sequence $ \left\{a_{n}\right\}$ as follows:
\[ a_{1}\equal{}\frac{m}{2},\ a_{n\plus{}1}\equal{}a_{n}\left\lceil a_{n}\right\rceil,\textnormal{ if }n\geq 1\]
Find all values of $ m$ for which $ a_{2007}$ is the first integer appearing in the sequence.
Note: For a real number $ x$, $ \left\lceil x\right\rceil$ is defined as the smallest integer greater or equal to $ x$. For example, $ \left\lceil\pi\right\rceil\equal{}4$, $ \left\lceil 2007\right\rceil\equal{}2007$.
2017 Princeton University Math Competition, 9
The set $\{(x, y) \in R^2| \lfloor x + y\rfloor \cdot \lceil x + y\rceil = (\lfloor x\rfloor + \lceil y \rceil ) (\lceil x \rceil + \lfloor y\rfloor), 0 \le x, y \le 100\}$ can be thought of as a collection of line segments in the plane. If the total length of those line segments is $a + b\sqrt{c}$ for $c$ squarefree, find $a + b + c$.
($\lfloor z\rfloor$ is the greatest integer less than or equal to $z$, and $\lceil z \rceil$ is the least integer greater than or equal to $z$, for $z \in R$.)
2024 Brazil Cono Sur TST, 3
Given a positive integer $n$, define $\tau(n)$ as the number of positive divisors of $n$ and $\sigma(n)$ as the sum of those divisors. For example, $\tau(12) = 6$ and $\sigma(12) = 28$. Find all positive integers $n$ that satisfy:
\[
\sigma(n) = \tau(n) \cdot \lceil \sqrt{n} \rceil
\]
2013 NIMO Problems, 8
A person flips $2010$ coins at a time. He gains one penny every time he flips a prime number of heads, but must stop once he flips a non-prime number. If his expected amount of money gained in dollars is $\frac{a}{b}$, where $a$ and $b$ are relatively prime, compute $\lceil\log_{2}(100a+b)\rceil$.
[i]Proposed by Lewis Chen[/i]
2013 National Olympiad First Round, 28
In the beginning, there is a pair of positive integers $(m,n)$ written on the board. Alice and Bob are playing a turn-based game with the following move. At each turn, a player erases one of the numbers written on the board, and writes a different positive number not less than the half of the erased one. If a player cannot write a new number at some turn, he/she loses the game. For how many starting pairs $(m,n)$ from the pairs $(7,79)$, $(17,71)$, $(10,101)$, $(21,251)$, $(50,405)$, can Alice guarantee to win when she makes the first move?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ \text{None of above}
$
2010 Contests, A1
Given a positive integer $n,$ what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same?
[When $n=8,$ the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is [i]at least[/i] 3.]
2000 All-Russian Olympiad, 2
Tanya chose a natural number $X\le100$, and Sasha is trying to guess this number. He can select two natural numbers $M$ and $N$ less than $100$ and ask about $\gcd(X+M,N)$. Show that Sasha can determine Tanya's number with at most seven questions.
2009 Indonesia TST, 3
Let $ n \ge 2009$ be an integer and define the set:
\[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}.
\]
Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that
\[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}.
\]
2012 ELMO Shortlist, 8
Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$.
[i]Victor Wang.[/i]
MOAA Team Rounds, 2019.6
Let $f(x, y) = \left\lfloor \frac{5x}{2y} \right\rfloor + \left\lceil \frac{5y}{2x} \right\rceil$. Suppose $x, y$ are chosen independently uniformly at random from the interval $(0, 1]$. Let $p$ be the probability that $f(x, y) < 6$. If $p$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$, compute $m + n$.
(Note: $\lfloor x\rfloor $ is defined as the greatest integer less than or equal to $x$ and $\lceil x \rceil$ is defined as the least integer greater than or equal to$ x$.)