Found problems: 1187
2014 District Olympiad, 2
Let $M$ be the set of palindromic integers of the form $5n+4$ where $n\ge 0$ is an integer.
[list=a]
[*]If we write the elements of $M$ in increasing order, what is the $50^{\text{th}}$ number?
[*]Among all numbers in $M$ with nonzero digits which sum up to $2014$ which is the largest and smallest one?[/list]
2007 Croatia Team Selection Test, 2
Prove that the sequence $a_{n}=\lfloor n\sqrt 2 \rfloor+\lfloor n\sqrt 3 \rfloor$ contains infintely many even and infinitely many odd numbers.
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.
1995 Poland - Second Round, 3
Let $a,b,c,d$ be positive irrational numbers with $a+b = 1$.
Show that $c+d = 1$ if and only if $[na]+[nb] = [nc]+[nd]$ for all positive integers $n$.
2006 Bulgaria Team Selection Test, 3
[b]Problem 3.[/b] Let $n\geq 3$ is given natural number, and $M$ is the set of the first $n$ primes. For any nonempty subset $X$ of $M$ with $P(X)$ denote the product of its elements. Let $N$ be a set of the kind $\ds\frac{P(A)}{P(B)}$, $A\subset M, B\subset M, A\cap B=\emptyset$ such that the product of any 7 elements of $N$ is integer. What is the maximal number of elements of $N$?
[i]Alexandar Ivanov[/i]
2002 Romania Team Selection Test, 4
For any positive integer $n$, let $f(n)$ be the number of possible choices of signs $+\ \text{or}\ - $ in the algebraic expression $\pm 1\pm 2\ldots \pm n$, such that the obtained sum is zero. Show that $f(n)$ satisfies the following conditions:
a) $f(n)=0$ for $n=1\pmod{4}$ or $n=2\pmod{4}$.
b) $2^{\frac{n}{2}-1}\le f(n)\le 2^n-2^{\lfloor\frac{n}{2}\rfloor+1}$, for $n=0\pmod{4}$ or $n=3\pmod{4}$.
[i]Ioan Tomsecu[/i]
Bangladesh Mathematical Olympiad 2020 Final, #5
For a positive real number $ [x] $ be its integer part. For example, $[2.711] = 2, [7] = 7, [6.9] = 6$. $z$ is the maximum real number such that [$\frac{5}{z}$] + [$\frac{6}{z}$] = 7. Find the value of$ 20z$.
2014 Online Math Open Problems, 9
Let $N = 2014! + 2015! + 2016! + \dots + 9999!$. How many zeros are at the end of the decimal representation of $N$?
[i]Proposed by Evan Chen[/i]
1997 IberoAmerican, 1
Let $r\geq1$ be areal number that holds with the property that for each pair of positive integer numbers $m$ and $n$, with $n$ a multiple of $m$, it is true that $\lfloor{nr}\rfloor$ is multiple of $\lfloor{mr}\rfloor$. Show that $r$ has to be an integer number.
[b]Note: [/b][i]If $x$ is a real number, $\lfloor{x}\rfloor$ is the greatest integer lower than or equal to $x$}.[/i]
2004 Purple Comet Problems, 7
How many positive integers less that $200$ are relatively prime to either $15$ or $24$?
OMMC POTM, 2024 6
Find the remainder modulo $101$ of
$$\left\lfloor \dfrac{1}{(2 \cos \left(\frac{4\pi}{7} \right))^{103}}\right\rfloor$$
2005 AMC 12/AHSME, 23
Two distinct numbers $ a$ and $ b$ are chosen randomly from the set $ \{ 2, 2^2, 2^3, \ldots, 2^{25} \}$. What is the probability that $ \log_{a} b$ is an integer?
$ \textbf{(A)}\ \frac {2}{25} \qquad \textbf{(B)}\ \frac {31}{300} \qquad \textbf{(C)}\ \frac {13}{100} \qquad \textbf{(D)}\ \frac {7}{50} \qquad \textbf{(E)}\ \frac {1}{2}$
2010 AMC 12/AHSME, 21
Let $ a>0$, and let $ P(x)$ be a polynomial with integer coefficients such that
\[ P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and}\]
\[ P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.}\]
What is the smallest possible value of $ a$?
$ \textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$
1988 China Team Selection Test, 4
Let $k \in \mathbb{N},$ $S_k = \{(a, b) | a, b = 1, 2, \ldots, k \}.$ Any two elements $(a, b)$, $(c, d)$ $\in S_k$ are called "undistinguishing" in $S_k$ if $a - c \equiv 0$ or $\pm 1 \pmod{k}$ and $b - d \equiv 0$ or $\pm 1 \pmod{k}$; otherwise, we call them "distinguishing". For example, $(1, 1)$ and $(2, 5)$ are undistinguishing in $S_5$. Considering the subset $A$ of $S_k$ such that the elements of $A$ are pairwise distinguishing. Let $r_k$ be the maximum possible number of elements of $A$.
(i) Find $r_5$.
(ii) Find $r_7$.
(iii) Find $r_k$ for $k \in \mathbb{N}$.
2009 Olympic Revenge, 4
Let $d_i(k)$ the number of divisors of $k$ greater than $i$.
Let $f(n)=\sum_{i=1}^{\lfloor \frac{n^2}{2} \rfloor}d_i(n^2-i)-2\sum_{i=1}^{\lfloor \frac{n}{2} \rfloor}d_i(n-i)$.
Find all $n \in N$ such that $f(n)$ is a perfect square.
2005 Singapore MO Open, 1
An integer is square-free if it is not divisible by $a^2$ for any integer $a>1$. Let $S$ be the set of positive square-free integers. Determine, with justification, the value of\[\sum_{k\epsilon S}\left[\sqrt{\frac{10^{10}}{k}}\right]\]where $[x]$ denote the greatest integer less than or equal to $x$
2006 Brazil National Olympiad, 6
Professor Piraldo takes part in soccer matches with a lot of goals and judges a match in his own peculiar way. A match with score of $m$ goals to $n$ goals, $m\geq n$, is [i]tough[/i] when $m\leq f(n)$, where $f(n)$ is defined by $f(0) = 0$ and, for $n \geq 1$, $f(n) = 2n-f(r)+r$, where $r$ is the largest integer such that $r < n$ and $f(r) \leq n$.
Let $\phi ={1+\sqrt 5\over 2}$. Prove that a match with score of $m$ goals to $n$, $m\geq n$, is tough if $m\leq \phi n$ and is not tough if $m \geq \phi n+1$.
2009 Romanian Master of Mathematics, 1
For $ a_i \in \mathbb{Z}^ \plus{}$, $ i \equal{} 1, \ldots, k$, and $ n \equal{} \sum^k_{i \equal{} 1} a_i$, let $ d \equal{} \gcd(a_1, \ldots, a_k)$ denote the greatest common divisor of $ a_1, \ldots, a_k$.
Prove that $ \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i \equal{} 1} (a_i!)}$ is an integer.
[i]Dan Schwarz, Romania[/i]
2011 All-Russian Olympiad, 3
A convex 2011-gon is drawn on the board. Peter keeps drawing its diagonals in such a way, that each newly drawn diagonal intersected no more than one of the already drawn diagonals. What is the greatest number of diagonals that Peter can draw?
2021 Latvia Baltic Way TST, P14
Prove that there exist infinitely many triples of positive integers $(a,b,c)$ so that $a,b,c$ are pairwise coprime and
$$\bigg \lfloor \frac{a^2}{2021} \bigg \rfloor + \bigg \lfloor \frac{b^2}{2021} \bigg \rfloor = \bigg \lfloor \frac{c^2}{2021} \bigg \rfloor.$$
PEN I Problems, 13
Suppose that $n \ge 2$. Prove that \[\sum_{k=2}^{n}\left\lfloor \frac{n^{2}}{k}\right\rfloor = \sum_{k=n+1}^{n^{2}}\left\lfloor \frac{n^{2}}{k}\right\rfloor.\]
2012 Serbia Team Selection Test, 2
Let $\sigma(x)$ denote the sum of divisors of natural number $x$, including $1$ and $x$. For every $n\in \mathbb{N}$ define $f(n)$ as number of natural numbers $m, m\leq n$, for which $\sigma(m)$ is odd number. Prove that there are infinitely many natural numbers $n$, such that $f(n)|n$.
2014 PUMaC Number Theory B, 8
Find the number of positive integers $n \le 2014$ such that there exists integer $x$ that satisfies the condition that $\frac{x+n}{x-n}$ is an odd perfect square.
1999 Romania Team Selection Test, 16
Let $X$ be a set with $n$ elements, and let $A_{1}$, $A_{2}$, ..., $A_{m}$ be subsets of $X$ such that:
1) $|A_{i}|=3$ for every $i\in\left\{1,2,...,m\right\}$;
2) $|A_{i}\cap A_{j}|\leq 1$ for all $i,j\in\left\{1,2,...,m\right\}$ such that $i \neq j$.
Prove that there exists a subset $A$ of $X$ such that $A$ has at least $\left[\sqrt{2n}\right]$ elements, and for every $i\in\left\{1,2,...,m\right\}$, the set $A$ does not contain $A_{i}$.
[i]Alternative formulation.[/i] Let $X$ be a finite set with $n$ elements and $A_{1},A_{2},\ldots, A_{m}$ be three-elements subsets of $X$, such that $|A_{i}\cap A_{j}|\leq 1$, for every $i\neq j$. Prove that there exists $A\subseteq X$ with $|A|\geq \lfloor \sqrt{2n}\rfloor$, such that none of $A_{i}$'s is a subset of $A$.
2001 Brazil Team Selection Test, Problem 2
A set $S$ consists of $k$ sequences of $0,1,2$ of length $n$. For any two sequences $(a_i),(b_i)\in S$ we can construct a new sequence $(c_i)$ such that $c_i=\left\lfloor\frac{a_i+b_i+1}2\right\rfloor$ and include it in $S$. Assume that after performing finitely many such operations we obtain all the $3n$ sequences of $0,1,2$ of length $n$. Find the least possible value of $k$.