Found problems: 15460
2008 Singapore Junior Math Olympiad, 4
Six distinct positive integers $a,b,c.d,e, f$ are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has $10$ prime numbers while Jill claims that she has $9$ prime numbers among the sums. Who has the correct claim?
JOM 2015, 4
Given a natural number $n\ge 3$, determine all strictly increasing sequences $a_1<a_2<\cdots<a_n$ such that $\text{gcd}(a_1,a_2)=1$ and for any pair of natural numbers $(k,m)$ satisfy $n\ge m\ge 3$, $m\ge k$, $$\frac{a_1+a_2+\cdots +a_m}{a_k}$$ is a positive integer.
2015 IFYM, Sozopol, 5
If $x,y,z$ are positive integers and $z(xz+1)^2=(5z+2y)(2z+y)$, prove that $z$ is an odd perfect square.
2018 South East Mathematical Olympiad, 7
For positive integers $m,n,$ define $f(m,n)$ as the number of ordered triples $(x,y,z)$ of integers such that
$$
\begin{cases}
xyz=x+y+z+m, \\
\max\{|x|,|y|,|z|\} \leq n
\end{cases}
$$
Does there exist positive integers $m,n,$ such that $f(m,n)=2018?$ Please prove your conclusion.
2013 Math Prize for Girls Olympiad, 3
$10000$ nonzero digits are written in a $100$-by-$100$ table, one digit per cell. From left to right, each row forms a $100$-digit integer. From top to bottom, each column forms a $100$-digit integer. So the rows and columns form $200$ integers (each with $100$ digits), not necessarily distinct. Prove that if at least $199$ of these $200$ numbers are divisible by $2013$, then all of them are divisible by $2013$.
2024 Belarusian National Olympiad, 9.7
Find all pairs of positive integers $(m,n)$, for which $$(m^n-n)^m=n!+m$$
[i]D. Volkovets[/i]
2006 Taiwan National Olympiad, 1
Find all integer solutions $(x,y)$ to the equation $\displaystyle \frac{x+y}{x^2-xy+y^2}=\frac{3}{7}$.
2017 Caucasus Mathematical Olympiad, 3
Find the least positive integer $n$ satisfying the following statement: for eash pair of positive integers $a$ and $b$ such that $36$ divides $a+b$ and $n$ divides $ab$ it follows that $36$ divides both $a$ and $b$.
1997 Singapore Senior Math Olympiad, 3
Find the smallest positive integer $x$ such that $x^2$ ends with the four digits $9009$.
2018 Peru IMO TST, 10
For each positive integer $m> 1$, let $P (m)$ be the product of all prime numbers that divide $m$.
Define the sequence $a_1, a_2, a_3,...$ as followed:
$a_1> 1$ is an arbitrary positive integer,
$a_{n + 1} = a_n + P (a_n)$ for each positive integer $n$.
Prove that there exist positive integers $j$ and $k$ such that $a_j$ is the product of the first $k$ prime numbers.
2009 Canadian Mathematical Olympiad Qualification Repechage, 8
Determine an infinite family of quadruples $(a, b, c, d)$ of positive integers, each of which is a solution to $a^4+b^5+c^6=d^7$.
1997 Croatia National Olympiad, Problem 1
Find the last four digits of each of the numbers $3^{1000}$ and $3^{1997}$.
Oliforum Contest V 2017, 4
Let $p_n$ be the $n$-th prime, so that $p_1 = 2, p_2 = 3,...$ and de fine $$X_n = \{0\} \cup \{p_1,...,p_n\}$$ for each positive integer $n$. Find all $n$ for which there exist $A,B \subseteq N$ such that$ |A|,|B| \ge 2$ and
$$X_n = A + B$$, where $A + B :=\{a + b : a \in A; b \in B \}$ and $N := \{0,1, 2,...\}$.
(Salvatore Tringali)
2003 Romania Team Selection Test, 10
Let $\mathcal{P}$ be the set of all primes, and let $M$ be a subset of $\mathcal{P}$, having at least three elements, and such that for any proper subset $A$ of $M$ all of the prime factors of the number $ -1+\prod_{p\in A}p$ are found in $M$. Prove that $M= \mathcal{P}$.
[i]Valentin Vornicu[/i]
2021 Taiwan TST Round 3, N
Let $a_1$, $a_2$, $a_3$, $\ldots$ be a sequence of positive integers such that $a_1=2021$ and
$$\sqrt{a_{n+1}-a_n}=\lfloor \sqrt{a_n} \rfloor. $$
Show that there are infinitely many odd numbers and infinitely many even numbers in this sequence.
[i] Proposed by Li4, Tsung-Chen Chen, and Ming Hsiao.[/i]
2024 India National Olympiad, 3
Let $p$ be an odd prime and $a,b,c$ be integers so that the integers $$a^{2023}+b^{2023},\quad b^{2024}+c^{2024},\quad a^{2025}+c^{2025}$$ are divisible by $p$.
Prove that $p$ divides each of $a,b,c$.
$\quad$
Proposed by Navilarekallu Tejaswi
2005 China Team Selection Test, 3
Let $a_1,a_2 \dots a_n$ and $x_1, x_2 \dots x_n$ be integers and $r\geq 2$ be an integer. It is known that \[\sum_{j=0}^{n} a_j x_j^k =0 \qquad \text{for} \quad k=1,2, \dots r.\]
Prove that
\[\sum_{j=0}^{n} a_j x_j^m \equiv 0 \pmod m, \qquad \text{for all}\quad m \in \{ r+1, r+2, \cdots, 2r+1 \}.\]
2023 CMIMC Algebra/NT, 10
For a given $n$, consider the points $(x,y)\in \mathbb{N}^2$ such that $x\leq y\leq n$. An ant starts from $(0,1)$ and, every move, it goes from $(a,b)$ to point $(c,d)$ if $bc-ad=1$ and $d$ is maximized over all such points. Let $g_n$ be the number of moves made by the ant until no more moves can be made. Find $g_{2023} - g_{2022}$.
[i]Proposed by David Tang[/i]
2017 Tournament Of Towns, 5
There is a set of control weights, each of them weighs a non-integer number of grams. Any
integer weight from $1$ g to $40$ g can be balanced by some of these weights (the control
weights are on one balance pan, and the measured weight on the other pan).What is the
least possible number of the control weights?
[i](Alexandr Shapovalov)[/i]
2009 QEDMO 6th, 9
For every natural $n$ let $\phi (n)$ be the number of coprime numbers $k \in \{1,2,...,n\}$. (Example: $\phi (12) = 4$, because among the numbers $1, 2, ..., 12$ there are only the$ 4$ numbers, $1, 5, 7$ and $11$ coprime to$12.$)
If $k$ is a natural number, then one defines $\phi^k (n)=\underbrace{\strut \phi (\phi ...(\phi (n)) ...)}_{(k \, times \phi)}$ (Example: $\phi^3 (n)=\phi (\phi (\phi (n))) $)
For every whole $n> 2$ let $c(n)$ be the smallest natural number $k$ with $\phi^k (n)= 2$.
Prove that $c (ab) = c (a) + c (b)$ for odd integers $a$ and $b$, both of which are greater than $2$, .
KoMaL A Problems 2018/2019, A. 738
Consider the following sequence: $a_1 = 1$, $a_2 = 2$, $a_3 = 3$, and
\[a_{n+3} = \frac{a_{n+1}^2 + a_{n+2}^2 - 2}{a_n}\]
for all integers $n \ge 1$. Prove that every term of the sequence is a positive integer.
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]
2002 Germany Team Selection Test, 3
Determine all $(x,y) \in \mathbb{N}^2$ which satisfy $x^{2y} + (x+1)^{2y} = (x+2)^{2y}.$
2004 Czech-Polish-Slovak Match, 2
Show that for each natural number $k$ there exist only finitely many triples $(p, q, r)$ of distinct primes for which $p$ divides $qr-k$, $q$ divides $pr-k$, and $r$ divides $pq - k$.
2009 Hanoi Open Mathematics Competitions, 4
Suppose that $a=2^b$, where $b=2^{10n+1}$. Prove that $a$ is divisible by 23 for any positive integer $n$