Found problems: 15460
2022 Rioplatense Mathematical Olympiad, 1
Prove that there exists infinitely many positive integers $n$ for which the equation$$x^2+y^{11}-z^{2022!}=n$$has no solution $(x,y,z)$ over the integers.
2003 Croatia Team Selection Test, 1
Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.
2010 South East Mathematical Olympiad, 2
For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.
1981 Bundeswettbewerb Mathematik, 4
Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.
2002 Croatia National Olympiad, Problem 3
Let $f(x)=x^{2002}-x^{2001}+1$. Prove that for every positive integer $m$, the numbers $m,f(m),f(f(m)),\ldots$ are pairwise coprime.
2022 JHMT HS, 4
For an integer $a$ and positive integers $n$ and $k$, let $f_k(a, n)$ be the remainder when $a^k$ is divided by $n$. Find the largest composite integer $n\leq 100$ that guarantees the infinite sequence
\[ f_1(a,n),f_2(a,n),f_3(a,n),\ldots,f_i(a,n),\ldots \]
to be periodic for all integers $a$ (i.e., for each choice of $a$, there is some positive integer $T$ such that $f_k(a,n) = f_{k+T}(a,n)$ for all $k$).
2002 France Team Selection Test, 2
Consider the set $S$ of integers $k$ which are products of four distinct primes. Such an integer $k=p_1p_2p_3p_4$ has $16$ positive divisors $1=d_1<d_2<\ldots <d_{15}<d_{16}=k$. Find all elements of $S$ less than $2002$ such that $d_9-d_8=22$.
2015 IFYM, Sozopol, 5
A positive interger number $k$ is called “$t-m$”-property if forall positive interger number $a$, there exists a positive integer number $n$ such that
${{1}^{k}}+{{2}^{k}}+{{3}^{k}}+...+{{n}^{k}} \equiv a (\bmod m).$
a) Find all positive integer numbers $k$ which has $t-20$-property.
b) Find smallest positive integer number $k$ which has $t-{{20}^{15}}$-property.
2010 Hanoi Open Mathematics Competitions, 8
If $n$ and $n^3+2n^2+2n+4$ are both perfect squares, fi nd $n$.
2005 MOP Homework, 6
Let $p$ be a prime number, and let $0 \le a_1<a_2<...<a_m<p$ and $0 \le b_1<b_2<...<b_n<p$ be arbitrary integers. Denote by $k$ the number of different remainders of $a_i+b_j$, $1 \le i \le m$ and $1 \le j \le n$, modulo $p$. Prove that
(i) if $m+n>p$, then $k=p$
(ii) if $m+n \le p$, then $k \ge m+n-1$
2023 Brazil EGMO Team Selection Test, 2
Let $A$ be a finite set made up of prime numbers. Determine if there exists an infinite set $B$ that satisfies the following conditions:
$(i)$ the prime factors of any element of $B$ are in $A$;
$(ii)$ no term of $B$ divides another element of this set.
2024 Israel National Olympiad (Gillis), P5
For positive integral $k>1$, we let $p(k)$ be its smallest prime divisor. Given an integer $a_1>2$, we define an infinite sequence $a_n$ by $a_{n+1}=a_n^n-1$ for each $n\geq 1$. For which values of $a_1$ is the sequence $p(a_n)$ bounded?
2023 All-Russian Olympiad, 6
Consider all $100$-digit numbers divisible by $19$. Prove that the number of such numbers not containing the digits $4, 5$, and $6$ is the number of such numbers that do not contain the digits $1, 4$ and $7$.
1979 IMO Longlists, 15
Let $n \geq 2$ be an integer. Find the maximal cardinality of a set $M$ of pairs $(j, k)$ of integers, $1 \leq j < k \leq n$, with the following property: If $(j, k) \in M$, then $(k,m) \not \in M$ for any $m.$
2004 Finnish National High School Mathematics Competition, 5
Finland is going to change the monetary system again and replace the Euro by the Finnish Mark.
The Mark is divided into $100$ pennies.
There shall be coins of three denominations only, and the number of coins a person has to carry in order to be able
to pay for any purchase less than one mark should be minimal.
Determine the coin denominations.
2010 Macedonia National Olympiad, 1
Solve the equation
\[ x^3+2y^3-4x-5y+z^2=2012, \]
in the set of integers.
2019 Durer Math Competition Finals, 8
Let $N$ be a positive integer such that $N$ and $N^2$ both end in the same four digits $\overline{abcd}$, where $a \ne 0$. What is the four-digit number $\overline{abcd}$?
2014 Junior Balkan Team Selection Tests - Romania, 1
We call a composite positive integer $n$ nice if it is possible to arrange its factors that are larger than $1$ on a circle such that two neighboring numbers are not coprime. How many of the elements of the set $\{1, 2, 3, ..., 100\}$ are nice?
2018 IMO, 5
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number
$$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$
is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$.
[i]Proposed by Bayarmagnai Gombodorj, Mongolia[/i]
2013 IFYM, Sozopol, 4
Let $k<<n$ denote that $k<n$ and $k\mid n$. Let $f:\{1,2,...,2013\}\rightarrow \{1,2,...,M\}$ be such that, if $n\leq 2013$ and $k<<n$, then $f(k)<<f(n)$. What’s the least possible value of $M$?
2015 May Olympiad, 1
The teacher secretly thought of a three-digit $S$ number. Students $A, B, C$ and $D$ tried to guess, saying, respectively, $541$, $837$, $291$ and $846$. The teacher told them, “Each of you got it right exactly one digit of $S$ and in the correct position ”. What is the number $S$?
2001 National Olympiad First Round, 23
Which of the followings is false for the sequence $9,99,999,\dots$?
$\textbf{(A)}$ The primes which do not divide any term of the sequence are finite.
$\textbf{(B)}$ Infinitely many primes divide infinitely many terms of the sequence.
$\textbf{(C)}$ For every positive integer $n$, there is a term which is divisible by at least $n$ distinct prime numbers.
$\textbf{(D)}$ There is an inteter $n$ such that every prime number greater than $n$ divides infinitely many terms of the sequence.
$\textbf{(E)}$ None of above
2003 Junior Tuymaada Olympiad, 2
Find all natural $ x $ for which $ 3x+1 $ and $ 6x-2 $ are perfect squares, and the number $ 6x^2-1 $ is prime.
2025 Belarusian National Olympiad, 10.8
Given a set $S$ that consists of $n \geq 3$ positive integers. It is known that if for some (not necessarily distinct) numbers $a,b,c,d$ from $S$ the equality $a-b=2(c-d)$ holds, then $a=b$ and $c=d$. Let $M$ be the biggest element in $S$.
a) Prove that $M > \frac{n^2}{3}$.
b) For $n=1024$ find the biggest possible value of $M$.
[i]M. Zorka, Y. Sheshukou[/i]
2009 Postal Coaching, 6
Find all pairs $(m, n)$ of positive integers $m$ and $n$ for which one has $$\sqrt{ m^2 - 4} < 2\sqrt{n} - m < \sqrt{ m^2 - 2}$$