Found problems: 15460
2013 Saudi Arabia IMO TST, 3
For a positive integer $n$, we consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. (For example $n = 117$, has six divisors, namely $1,3,9,13,39,117$. Two of these divisors namely $3$ and $13$, have unit digits equal to $3$. Hence for $n = 117$, $p =33.33...$). Find, when $n$ is any positive integer, the maximum possible value of $p$.
2024 Assara - South Russian Girl's MO, 7
Find all positive integers $n$ for such the following condition holds:
"If $a$, $b$ and $c$ are positive integers such are all numbers \[ a^2+2ab+b^2,\ b^2+2bc+c^2, \ c^2+2ca+a^2 \] are divisible by $n$, then $(a+b+c)^2$ is also divisible by $n$."
[i]G.M.Sharafetdinova[/i]
2024 Singapore Junior Maths Olympiad, Q4
Suppose for some positive integer $n$, the numbers $2^n$ and $5^n$ have equal first digit. What are the possible values of this first digit?
Note: solved [url=https://artofproblemsolving.com/community/c6h312638p1685546]here[/url]
Oliforum Contest IV 2013, 7
For every positive integer $n$, define the number of non-empty subsets $\mathcal N\subseteq \{1,\ldots ,n\}$ such that $\gcd(n\in\mathcal N)=1$. Show that $f(n)$ is a perfect square if and only if $n=1$.
TNO 2008 Junior, 9
(a) Is it possible to form a prime number using all the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 exactly once?
(b) Consider the following magic square where the sum of each row, column, and diagonal is the same (in this case, 15):
\[
\begin{array}{ccc}
6 & 7 & 2 \\
1 & 5 & 9 \\
8 & 3 & 4 \\
\end{array}
\]
Is it possible to create a magic square with the same properties using the numbers 11, 12, 13, 14, 15, 16, 17, 18, and 19?
2015 EGMO, 3
Let $n, m$ be integers greater than $1$, and let $a_1, a_2, \dots, a_m$ be positive integers not greater than $n^m$. Prove that there exist positive integers $b_1, b_2, \dots, b_m$ not greater than $n$, such that \[ \gcd(a_1 + b_1, a_2 + b_2, \dots, a_m + b_m) < n, \] where $\gcd(x_1, x_2, \dots, x_m)$ denotes the greatest common divisor of $x_1, x_2, \dots, x_m$.
1991 AIME Problems, 5
Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $ 20!$ be the resulting product?
2021 Brazil Team Selection Test, 4
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
2020 HMNT (HMMO), 5
For each positive integer $n$, let an be the smallest nonnegative integer such that there is only one positive integer at most $n$ that is relatively prime to all of $n$, $n + 1$, .. , $n + a_n$. If $n < 100$, compute the largest possible value of $n - a_n$.
1962 All-Soviet Union Olympiad, 9
Given is a number with $1998$ digits which is divisible by $9$. Let $x$ be the sum of its digits, let $y$ be the sum of the digits of $x$, and $z$ the sum of the digits of $y$. Find $z$.
2018 Iran MO (1st Round), 12
How many triples $(a,b,c)$ of positive integers strictly less than $51$ are there such that $a+b+c$ is divisible by $a, b$, and $c$?
1984 Canada National Olympiad, 3
An integer is digitally divisible if both of the following conditions are fulfilled:
$(a)$ None of its digits is zero;
$(b)$ It is divisible by the sum of its digits
e.g. $322$ is digitally divisible. Show that there are infinitely many digitally divisible integers.
2010 All-Russian Olympiad Regional Round, 11.4
We call a triple of natural numbers $(a, b, c)$ [i]square [/i] if they form an arithmetic progression (in exactly this order), the number $b$ is coprime to each of the numbers $a$ and $c$, and the number $abc$ is a perfect square. Prove that for any given a square triple, there is another square triple that has at least one common number with it.
2023 CMWMC, R8
[b]p22.[/b] Find the unique ordered pair $(m, n)$ of positive integers such that $x = \sqrt[3]{m} -\sqrt[3]{n}$ satisfies $x^6 + 4x^3 - 36x^2 + 4 = 0$.
[b]p23.[/b] Jenny plays with a die by placing it flat on the ground and rolling it along any edge for each step. Initially the face with $1$ pip is face up. How many ways are there to roll the dice for $6$ steps and end with the $1$ face up again?
[b]p24.[/b] There exists a unique positive five-digit integer with all odd digits that is divisible by $5^5$. Find this integer.
PS. You should use hide for answers.
2016 Azerbaijan BMO TST, 2
Set $A$ consists of natural numbers such that these numbers can be expressed as $2x^2+3y^2,$ where $x$ and $y$ are integers. $(x^2+y^2\not=0)$
$a)$ Prove that there is no perfect square in the set $A.$
$b)$ Prove that multiple of odd number of elements of the set $A$ cannot be a perfect square.
2010 Costa Rica - Final Round, 2
Consider the sequence $x_n>0$ defined with the following recurrence relation:
\[x_1 = 0\]
and for $n>1$ \[(n+1)^2x_{n+1}^2 + (2^n+4)(n+1)x_{n+1}+ 2^{n+1}+2^{2n-2} = 9n^2x_n^2+36nx_n+32.\]
Show that if $n$ is a prime number larger or equal to $5$, then $x_n$ is an integer.
2010 AIME Problems, 1
Maya lists all the positive divisors of $ 2010^2$. She then randomly selects two distinct divisors from this list. Let $ p$ be the probability that exactly one of the selected divisors is a perfect square. The probability $ p$ can be expressed in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
2014 ELMO Shortlist, 2
Define the Fibanocci sequence recursively by $F_1=1$, $F_2=1$ and $F_{i+2} = F_i + F_{i+1}$ for all $i$. Prove that for all integers $b,c>1$, there exists an integer $n$ such that the sum of the digits of $F_n$ when written in base $b$ is greater than $c$.
[i]Proposed by Ryan Alweiss[/i]
2013 Online Math Open Problems, 47
Let $f(x,y)$ be a function from ordered pairs of positive integers to real numbers
such that
\[ f(1,x) = f(x,1) = \frac{1}{x} \quad\text{and}\quad f(x+1,y+1)f(x,y)-f(x,y+1)f(x+1,y) = 1 \]
for all ordered pairs of positive integers $(x,y)$. If $f(100,100) = \frac{m}{n}$ for two relatively prime positive integers $m,n$, compute $m+n$.
[i]David Yang[/i]
1957 Moscow Mathematical Olympiad, 357
For which integer $n$ is $N = 20^n + 16^n - 3^n - 1$ divisible by $323$?
2003 Paraguay Mathematical Olympiad, 2
With three different digits, all greater than $0$, six different three-digit numbers are formed. If we add these six numbers together the result is $4.218$. The sum of the three largest numbers minus the sum of the three smallest numbers equals $792$. Find the three digits.
2017 Peru IMO TST, 4
The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?
2008 Germany Team Selection Test, 3
Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$.
[i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]
2017 Pan-African Shortlist, N2
For which prime numbers $p$ can we find three positive integers $n$, $x$ and $y$ such that $p^n = x^3 + y^3$?
2018 Saudi Arabia IMO TST, 2
Let $n$ be an even positive integer. We fill in a number on each cell of a rectangle table of $n$ columns and multiple rows as following:
i. Each row is assigned to some positive integer $a$ and its cells are filled by $0$ or $a$ (in any order);
ii. The sum of all numbers in each row is $n$.
Note that we cannot add any more row to the table such that the conditions (i) and (ii) still hold.
Prove that if the number of $0$’s on the table is odd then the maximum odd number on the table is a perfect square.