Found problems: 15460
2010 China Team Selection Test, 3
Given positive integer $k$, prove that there exists a positive integer $N$ depending only on $k$ such that for any integer $n\geq N$, $\binom{n}{k}$ has at least $k$ different prime divisors.
2024 Thailand October Camp, 3
Recall that for an arbitrary prime $p$, we define a [b]primitive root[/b] modulo $p$ as an integer $r$ for which the least positive integer $v$ such that $r^{v}\equiv 1\pmod{p}$ is $p-1$.\\
Prove or disprove the following statement:
[center]
For every prime $p>2023$, there exists positive integers $1\leqslant a<b<c<p$\\ such that $a,b$ and $c$ are primitive roots modulo $p$ but $abc$ is not a primitive root modulo $p$.
[/center]
1996 Korea National Olympiad, 6
Find the minimum value of $k$ such that there exists two sequence ${a_i},{b_i}$ for $i=1,2,\cdots ,k$ that satisfies the following conditions.
(i) For all $i=1,2,\cdots ,k,$ $a_i,b_i$ is the element of $S=\{1996^n|n=0,1,2,\cdots\}.$
(ii) For all $i=1,2,\cdots, k, a_i\ne b_i.$
(iii) For all $i=1,2,\cdots, k, a_i\le a_{i+1}$ and $b_i\le b_{i+1}.$
(iv) $\sum_{i=1}^{k} a_i=\sum_{i=1}^{k} b_i.$
2021 Durer Math Competition (First Round), 1
Albrecht is travelling in his car on the motorway at a constant speed. The journey is very long so Marvin who is sitting next to Albrecht gets bored and decides to calculate the speed of the car. He was a bit careless but he noted that at noon they passed milestone $XY$ (where $X$ and $Y$ are digits), at $12:42$ milestone $YX$ and at $1$pm they arrived at milestone $X0Y$. What did Marvin deduce, what is the speed of the car?
2008 Postal Coaching, 6
Consider the set $A = \{1, 2, 3, ..., 2008\}$. We say that a set is of [i]type[/i] $r, r \in \{0, 1, 2\}$, if that set is a nonempty subset of $A$ and the sum of its elements gives the remainder $r$ when divided by $3$. Denote by $X_r, r \in \{0, 1, 2\}$ the class of sets of type $r$. Determine which of the classes $X_r, r \in \{0, 1, 2\}$, is the largest.
2013 Costa Rica - Final Round, N1
Find all triples $(a, b, p)$ of positive integers, where $p$ is a prime number, such that $a^p - b^p = 2013$.
2018 IMAR Test, 4
Prove that every non-negative integer $n$ is expressible in the form $n=t^2+u^2+v^2+w^2$, where $t,u,v,w$ are integers such that $t+u+v+w$ is a perfect square.
[i]* * *[/i]
2024 Germany Team Selection Test, 1
For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.
2010 Junior Balkan Team Selection Tests - Romania, 1
Determine the prime numbers $p, q, r$ with the property $\frac {1} {p} + \frac {1} {q} + \frac {1} {r} \ge 1$
2000 Korea - Final Round, 1
Prove that for any prime $p$, there exist integers $x,y,z,$ and $w$ such that $x^2+y^2+z^2-wp=0$ and $0<w<p$
2004 Croatia Team Selection Test, 1
Find all pairs $(x,y)$ of positive integers such that $x(x+y)=y^2+1.$
2009 Federal Competition For Advanced Students, P1, 2
For a positive integers $n,k$ we define k-multifactorial of n as $Fk(n)$ = $(n)$ . $(n-k)$
$(n-2k)$...$(r)$, where $r$ is the reminder when $n$ is divided by $k$ that satisfy $1<=r<=k$
Determine all non-negative integers $n$ such that $F20(n)+2009$ is a perfect square.
2000 Junior Balkan Team Selection Tests - Moldova, 2
The number $665$ is represented as a sum of $18$ natural numbers nenule $a_1, a_2, ..., a_{18}$.
Determine the smallest possible value of the smallest common multiple of the numbers $a_1, a_2, ..., a_{18}$.
1969 IMO Shortlist, 7
$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.
2024 Baltic Way, 18
An infinite sequence $a_1, a_2,\ldots$ of positive integers is such that $a_n \geq 2$ and $a_{n+2}$ divides $a_{n+1} + a_n$ for all $n \geq 1$. Prove that there exists a prime which divides infinitely many terms of the sequence.
the 12th XMO, Problem 3
Let $a_0=0,a_1\in\mathbb Z_+.$ For integer $n\geq 2,a_n$ is the smallest positive integer satisfy that for $\forall 0\leq i\neq j\leq n-1,a_n\nmid (a_i-a_j).$
(1) If $a_1=2023,$ calculate $a_{10000}.$
(2) If $a_t\leq\frac{a_1}2,$ find the maximum value of $\frac t{a_1}.$
2003 Tuymaada Olympiad, 4
Given are polynomial $f(x)$ with non-negative integral coefficients and positive integer $a.$ The sequence $\{a_{n}\}$ is defined by $a_{1}=a,$ $a_{n+1}=f(a_{n}).$ It is known that the set of primes dividing at least one of the terms of this sequence is finite.
Prove that $f(x)=cx^{k}$ for some non-negative integral $c$ and $k.$
[i]Proposed by F. Petrov[/i]
[hide="For those of you who liked this problem."]
Check [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62259]this thread[/url] out.[/hide]
Gheorghe Țițeica 2025, P4
[list=a]
[*] Prove that for any positive integers $a,b,c$, there exists a positive integer $N$ such that $$(N+a^2)(N+b^2)(N+c^2)$$ is a perfect square.
[*] Prove that there exist five distinct positive integers $a,b,c,d,e$ for which there exists a positive integer $N$ such that $$(N+a^2)(N+b^2)(N+c^2)(N+d^2)(N+e^2)$$ is a perfect square.
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[i]Luminița Popescu[/i]
2006 USAMO, 1
Let $p$ be a prime number and let $s$ be an integer with $0 < s < p.$ Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and
\[ \left \{\frac{sm}{p} \right\} < \left \{\frac{sn}{p} \right \} < \frac{s}{p} \]
if and only if $s$ is not a divisor of $p-1$.
Note: For $x$ a real number, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of x.
2021 Kyiv City MO Round 1, 9.2
Roma wrote on the board each of the numbers $2018, 2019, 2020$, $100$ times each. Let us denote by $S(n)$ the sum of digits of positive integer $n$. In one action, Roma can choose any positive integer $k$ and instead of any three numbers $a, b, c$ written on the board write the numbers $2S(a + b) + k, 2S(b + c) + k$ and $2S(c + a) + k$. Can Roma after several such actions make $299$ numbers on the board equal, and the last one differing from them by $1$?
[i]Proposed by Oleksii Masalitin[/i]
2019 LIMIT Category A, Problem 7
The digit in unit place of $1!+2!+\ldots+99!$ is
$\textbf{(A)}~3$
$\textbf{(B)}~0$
$\textbf{(C)}~1$
$\textbf{(D)}~7$
2006 Romania Team Selection Test, 3
For which pairs of positive integers $(m,n)$ there exists a set $A$ such that for all positive integers $x,y$, if $|x-y|=m$, then at least one of the numbers $x,y$ belongs to the set $A$, and if $|x-y|=n$, then at least one of the numbers $x,y$ does not belong to the set $A$?
[i]Adapted by Dan Schwarz from A.M.M.[/i]
2009 Puerto Rico Team Selection Test, 5
The [i]weird [/i] mean of two numbers $ a$ and $ b$ is defined as $ \sqrt {\frac {2a^2 + 3b^2}{5}}$. $ 2009$ positive integers are placed around a circle such that each number is equal to the the weird mean of the two numbers beside it. Show that these $ 2009$ numbers must be equal.
1983 Putnam, A1
How many positive integers $n$ are there such that $n$ is an exact divisors of at least one of the numbers $10^{40}$ and $20^{30}$?
2019 Hong Kong TST, 2
Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.