Found problems: 15460
2017 Canada National Olympiad, 2
Define a function $f(n)$ from the positive integers to the positive integers such that $f(f(n))$ is the number of positive integer divisors of $n$. Prove that if $p$ is a prime, then $f(p)$ is prime.
2024 Belarus - Iran Friendly Competition, 2.1
Prove that the equation $2+x^3y+y^2+z^2=0$ has no solutions in integers.
2017 ISI Entrance Examination, 6
Let $p_1,p_2,p_3$ be primes with $p_2\neq p_3$ such that $4+p_1p_2$ and $4+p_1p_3$ are perfect squares. Find all possible values of $p_1,p_2,p_3$.
2016 Greece National Olympiad, 1
Find all triplets of nonnegative integers $(x,y,z)$ and $x\leq y$ such that
$x^2+y^2=3 \cdot 2016^z+77$
2007 Estonia National Olympiad, 4
Let $a, b,c$ be positive integers such that $gcd(a, b, c) = 1$ and each product of two is divided by the third.
a) Prove that each of these numbers is equal to the least two remaining numbers the quotient of the coefficient and the highest coefficient.
b) Give an example of one of these larger numbers $a, b$ and $c$
2014 Balkan MO, 2
A [i]special number[/i] is a positive integer $n$ for which there exists positive integers $a$, $b$, $c$, and $d$ with \[ n = \frac {a^3 + 2b^3} {c^3 + 2d^3}. \] Prove that
i) there are infinitely many special numbers;
ii) $2014$ is not a special number.
[i]Romania[/i]
2011 IFYM, Sozopol, 5
Does there exist a strictly increasing sequence $\{a_n\}_{n=1}^\infty$ of natural numbers with the following property: for $\forall$ $c\in \mathbb{Z}$ the sequence $c+a_1,c+a_2,...,c+a_n...$ has finite number of primes? Explain your answer.
2022 LMT Spring, 10
In a room, there are $100$ light switches, labeled with the positive integers ${1,2, . . . ,100}$. They’re all initially turned off. On the $i$ th day for $1 \le i \le 100$, Bob flips every light switch with label number $k$ divisible by $i$ a total of $\frac{k}{i}$ times. Find the sum of the labels of the light switches that are turned on at the end of the $100$th day.
2022 BMT, 1
What is the sum of all positive $2$-digit integers whose sum of digits is $16$?
2017 Iberoamerican, 1
For every positive integer $n$ let $S(n)$ be the sum of its digits. We say $n$ has a property $P$ if all terms in the infinite secuence $n, S(n), S(S(n)),...$ are even numbers, and we say $n$ has a property $I$ if all terms in this secuence are odd. Show that for, $1 \le n \le 2017$ there are more $n$ that have property $I$ than those who have $P$.
2024 Australian Mathematical Olympiad, P1
Determine all triples $(k, m, n) $ of positive integers satisfying $$k!+m!=k!n!.$$
2013 Portugal MO, 4
Which is the leastest natural number $n$ such that $n!$ has, at least, $2013$ divisors?
2013 IberoAmerican, 3
Let $A = \{1,...,n\}$ with $n \textgreater 5$. Prove that one can find $B$ a finite set of positive integers such that $A$ is a subset of $B$ and
$\displaystyle\sum_{x \in B} x^2 = \displaystyle\prod_{x \in B} x$
1990 IMO Longlists, 95
Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.
2002 Switzerland Team Selection Test, 3
Let $d_1,d_2,d_3,d_4$ be the four smallest divisors of a positive integer $n$ (having at least four divisors). Find all $n$ such that $d_1^2+d_2^2+d_3^2+d_4^2 = n$.
2000 Irish Math Olympiad, 4
Show that in each set of ten consecutive integers there is one that is coprime with each of the other integers. (For example, in the set $ \{ 114,115,...,123 \}$ there are two such numbers: $ 119$ and $ 121.)$
2006 APMO, 3
Let $p\ge5$ be a prime and let $r$ be the number of ways of placing $p$ checkers on a $p\times p$ checkerboard so that not all checkers are in the same row (but they may all be in the same column). Show that $r$ is divisible by $p^5$. Here, we assume that all the checkers are identical.
2018 Turkey Team Selection Test, 8
For integers $m\geq 3$, $n$ and $x_1,x_2, \ldots , x_m$ if $x_{i+1}-x_i \equiv x_i-x_{i-1} (mod n) $ for every $2\leq i \leq m-1$, we say that the $m$-tuple $(x_1,x_2,\ldots , x_m)$ is an arithmetic sequence in $(mod n)$. Let $p\geq 5$ be a prime number and $1<a<p-1$ be an integer. Let ${a_1,a_2,\ldots , a_k}$ be the set of all possible remainders when positive powers of $a$ are divided by $p$. Show that if a permutation of ${a_1,a_2,\ldots , a_k}$ is an arithmetic sequence in $(mod p)$, then $k=p-1$.
2001 Tournament Of Towns, 5
Let $a$ and $d$ be positive integers. For any positive integer $n$, the number $a+nd$ contains a block of consecutive digits which constitute the number $n$. Prove that $d$ is a power of 10.
2024 Princeton University Math Competition, A1 / B3
A quadratic polynomial $f(x) = Ax^2 + Bx + C$ is [I]small[/I] if $A, B, C$ are single-digit positive integers. It is [I]full[/I] if there are only finitely many positive integers that cannot be expressed as $f(x) + 3y$ for some positive integers $x$ and $y.$ Find the number of quadratic polynomials that are both small and full.
2021 Girls in Mathematics Tournament, 3
A natural number is called [i]chaotigal [/i] if it and its successor both have the sum of their digits divisible by $2021$. How many digits are in the smallest chaotigal number?
2003 Estonia Team Selection Test, 2
Let $n$ be a positive integer. Prove that if the number overbrace $\underbrace{\hbox{99...9}}_{\hbox{n}}$ is divisible by $n$, then the number $\underbrace{\hbox{11...1}}_{\hbox{n}}$ is also divisible by $n$.
(H. Nestra)
1983 Canada National Olympiad, 1
Find all positive integers $w$, $x$, $y$ and $z$ which satisfy $w! = x! + y! + z!$.
1989 AMC 8, 22
The letters $\text{A}$, $\text{J}$, $\text{H}$, $\text{S}$, $\text{M}$, $\text{E}$ and the digits $1$, $9$, $8$, $9$ are "cycled" separately as follows and put together in a numbered list:
\[\begin{tabular}[t]{lccc}
& & AJHSME & 1989 \\
& & & \\
1. & & JHSMEA & 9891 \\
2. & & HSMEAJ & 8919 \\
3. & & SMEAJH & 9198 \\
& & ........ &
\end{tabular}\]
What is the number of the line on which $\text{AJHSME 1989}$ will appear for the first time?
$\text{(A)}\ 6 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 24$
2012 Saint Petersburg Mathematical Olympiad, 4
Some notzero reals numbers are placed around circle. For every two neighbour numbers $a,b$ it true, that $a+b$ and $\frac{1}{a}+\frac{1}{b}$ are integer. Prove that there are not more than $4$ different numbers.