Found problems: 15460
2002 China Team Selection Test, 2
Find all non-negative integers $m$ and $n$, such that $(2^n-1) \cdot (3^n-1)=m^2$.
1984 AIME Problems, 2
The integer $n$ is the smallest positive multiple of 15 such that every digit of $n$ is either 8 or 0. Compute $\frac{n}{15}$.
2013 Junior Balkan Team Selection Tests - Romania, 2
Call the number $\overline{a_1a_2... a_m}$ ($a_1 \ne 0,a_m \ne 0$) the reverse of the number $\overline{a_m...a_2a_1}$. Prove that the sum between a number $n$ and its reverse is a multiple of $81$ if and only if the sum of the digits of $n$ is a multiple of $81$.
2018 Mexico National Olympiad, 3
A sequence $a_2, a_3, \dots, a_n$ of positive integers is said to be [i]campechana[/i], if for each $i$ such that $2 \leq i \leq n$ it holds that exactly $a_i$ terms of the sequence are relatively prime to $i$. We say that the [i]size[/i] of such a sequence is $n - 1$. Let $m = p_1p_2 \dots p_k$, where $p_1, p_2, \dots, p_k$ are pairwise distinct primes and $k \geq 2$. Show that there exist at least two different campechana sequences of size $m$.
2013 Romania National Olympiad, 4
Let $n$ be a positive integer and $M = {1, 2, . . . , 2n + 1}$. Find out in how many ways we can split the set $M$ into three mutually disjoint nonempty sets $A,B,C$ so that both the following are true:
(i) for each $a \in A$ and $b \in B$, the remainder of the division of $a$ by $b$ belongs to $C$
(ii) for each $c \in C$ there exists $a \in A$ and $b \in B$ such that $c$ is the remainder of the division of $a$ by $b$.
2018 Middle European Mathematical Olympiad, 4
(a) Prove that for every positive integer $m$ there exists an integer $n\ge m$ such that
$$\left \lfloor \frac{n}{1} \right \rfloor \cdot \left \lfloor \frac{n}{2} \right \rfloor \cdots \left \lfloor \frac{n}{m} \right \rfloor =\binom{n}{m} \\\\\\\\\\\\\\\ (*)$$
(b) Denote by $p(m)$ the smallest integer $n \geq m$ such that the equation $ (*)$ holds. Prove that
$p(2018) = p(2019).$
Remark: For a real number $x,$ we denote by $\left \lfloor x \right \rfloor$ the largest integer not larger than $x.$
2024 Austrian MO National Competition, 6
For each prime number $p$, determine the number of residue classes modulo $p$ which can
be represented as $a^2+b^2$ modulo $p$, where $a$ and $b$ are arbitrary integers.
[i](Daniel Holmes)[/i]
1966 IMO Longlists, 34
Find all pairs of positive integers $\left( x;\;y\right) $ satisfying the equation $2^{x}=3^{y}+5.$
1988 USAMO, 1
By a [i]pure repeating decimal[/i] (in base $10$), we mean a decimal $0.\overline{a_1\cdots a_k}$ which repeats in blocks of $k$ digits beginning at the decimal point. An example is $.243243243\cdots = \tfrac{9}{37}$. By a [i]mixed repeating decimal[/i] we mean a decimal $0.b_1\cdots b_m\overline{a_1\cdots a_k}$ which eventually repeats, but which cannot be reduced to a pure repeating decimal. An example is $.011363636\cdots = \tfrac{1}{88}$.
Prove that if a mixed repeating decimal is written as a fraction $\tfrac pq$ in lowest terms, then the denominator $q$ is divisible by $2$ or $5$ or both.
1979 IMO Longlists, 34
Notice that in the fraction $\frac{16}{64}$ we can perform a simplification as $\cancel{\frac{16}{64}}=\frac 14$ obtaining a correct equality. Find all fractions whose numerators and denominators are two-digit positive integers for which such a simplification is correct.
2009 Harvard-MIT Mathematics Tournament, 2
Suppose N is a $6$-digit number having base-$10$ representation $\underline{a}\text{ }\underline{b}\text{ }\underline{c}\text{ }\underline{d}\text{ }\underline{e}\text{ }\underline{f}$. If $N$ is $6/7$ of the number having base-$10$ representation $\underline{d}\text{ }\underline{e}\text{ }\underline{f}\text{ }\underline{a}\text{ }\underline{b}\text{ }\underline{c}$, find $N$.
2011 Hanoi Open Mathematics Competitions, 1
An integer is called "octal" if it is divisible by $8$ or if at least one of its digits is $8$.
How many integers between $1$ and $100$ are octal?
(A): $22$, (B): $24$, (C): $27$, (D): $30$, (E): $33$
1995 Iran MO (2nd round), 2
Let $n \geq 0$ be an integer. Prove that
\[ \lceil \sqrt n +\sqrt{n+1}+\sqrt{n+2} \rceil = \lceil \sqrt{9n+8} \rceil\]
Where $\lceil x \rceil $ is the smallest integer which is greater or equal to $x.$
2022 Argentina National Olympiad Level 2, 4
Determine the smallest positive integer $n$ that is equal to the sum of $11$ consecutive positive integers, the sum of $12$ consecutive positive integers and the sum of $13$ consecutive positive integers.
2020 Israel Olympic Revenge, P3
For each positive integer $n$, define $f(n)$ to be the least positive integer for which the following holds:
For any partition of $\{1,2,\dots, n\}$ into $k>1$ disjoint subsets $A_1, \dots, A_k$, [u]all of the same size[/u], let $P_i(x)=\prod_{a\in A_i}(x-a)$. Then there exist $i\neq j$ for which
\[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\]
a) Prove that there is a constant $c$ so that $f(n)\le c\cdot \sqrt{n}$ for all $n$.
b) Prove that for infinitely many $n$, one has $f(n)\ge \ln(n)$.
Russian TST 2016, P1
Let $a{}$ and $b{}$ be natural numbers greater than one. Let $n{}$ be a natural number for which $a\mid 2^n-1$ and $b\mid 2^n+1$. Prove that there is no natural $k{}$ such that $a\mid 2^k+1$ and $b\mid 2^k-1$.
2013 Dutch IMO TST, 4
Determine all positive integers $n\ge 2$ satisfying $i+j\equiv\binom ni +\binom nj \pmod{2}$ for all $i$ and $j$ with $0\le i\le j\le n$.
2013 Brazil Team Selection Test, 1
Call admissible a set $A$ of integers that has the following property:
If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$.
Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers.
[i]Proposed by Warut Suksompong, Thailand[/i]
2011 Greece Junior Math Olympiad, 2
We consider the set of four-digit positive integers $x =\overline{abcd}$ with digits different than zero and pairwise different. We also consider the integers $y = \overline{dcba}$ and we suppose that $x > y$. Find the greatest and the lowest value of the difference $x-y$, as well as the corresponding four-digit integers $x,y$ for which these values are obtained.
2010 Contests, 1
Find all primes $p,q$ such that $p^3-q^7=p-q$.
1998 Brazil Team Selection Test, Problem 2
There are $n\ge3$ integers around a circle. We know that for each of these numbers the ratio between the sum of its two neighbors and the number is a positive integer. Prove that the sum of the $n$ ratios is not greater than $3n$.
2004 China Team Selection Test, 3
$ S$ is a non-empty subset of the set $ \{ 1, 2, \cdots, 108 \}$, satisfying:
(1) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c \in S$, such that $ \gcd(a,c)\equal{}\gcd(b,c)\equal{}1$.
(2) For any two numbers $ a,b \in S$ ( may not distinct), there exists $ c' \in S$, $ c' \neq a$, $ c' \neq b$, such that $ \gcd(a, c') > 1$, $ \gcd(b,c') >1$.
Find the largest possible value of $ |S|$.
1995 AMC 12/AHSME, 29
For how many three-element sets of positive integers $\{a,b,c\}$ is it true that $a \times b \times c = 2310$?
$\textbf{(A)}\ 32 \qquad
\textbf{(B)}\ 36 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 43 \qquad
\textbf{(E)}\ 45$
2015 Saudi Arabia BMO TST, 4
Let $n \ge 2$ be an integer and $p_1 < p_2 < ... < p_n$ prime numbers. Prove that there exists an integer $k$ relatively prime with $p_1p_2... p_n$ and such that $gcd (k + p_1p_2...p_i, p_1p_2...p_n) = 1$ for all $i = 1, 2,..., n - 1$.
Malik Talbi
2001 Taiwan National Olympiad, 5
Let $f(n)=\sum_{k=0}^{n-1}x^ky^{n-1-k}$ with, $x$, $y$ real numbers. If $f(n)$, $f(n+1)$, $f(n+2)$, $f(n+3)$, are integers for some $n$, prove $f(n)$ is integer for all $n$.