Found problems: 15460
2024 Francophone Mathematical Olympiad, 4
Find all integers $n \ge 2$ for which there exists $n$ integers $a_1,a_2,\dots,a_n \ge 2$ such that for all indices $i \ne j$, we have $a_i \mid a_j^2+1$.
2000 Hungary-Israel Binational, 2
Prove or disprove: For any positive integer $k$ there exists an integer $n > 1$ such that the binomial coeffcient $\binom{n}{i}$ is divisible by $k$ for any $1 \leq i \leq n-1.$
2010 Greece JBMO TST, 1
Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.
OMMC POTM, 2022 4
Define a function $P(n)$ from the set of positive integers to itself, where $P(1)=1$ and if an integer $n > 1$ has prime factorization $$n = p_1^{a_1}p_2^{a_2} \dots p_k^{a_k}$$
then $$P(n) = a_1^{p_1}a_2^{p_2} \dots a_k^{p_k}.$$
Prove that $P(P(n)) \le n$ for all positive integers $n.$
[i]Proposed by Evan Chang (squareman), USA[/i]
1983 IMO Longlists, 70
Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$
2017 Macedonia JBMO TST, 5
Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.
2000 Harvard-MIT Mathematics Tournament, 7
$8712$ is an integral multiple of its reversal, $2178$, as $8712=4 * 2178$. Find another $4$-digit number which is a non-trivial integral multiple of its reversal.
2001 Poland - Second Round, 1
Find all integers $n\ge 3$ for which the following statement is true:
Any arithmetic progression $a_1,\ldots ,a_n$ with $n$ terms for which $a_1+2a_2+\ldots+na_n$ is rational contains at least one rational term.
2013 Benelux, 4
a) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers
\[g^n - n\quad\text{ and }\quad g^{n+1} - (n + 1).\]
b) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers
\[g^n - n^2\quad\text{ and }g^{n+1} - (n + 1)^2.\]
2007 Pre-Preparation Course Examination, 11
Let $p \geq 3$ be a prime and $a_1,a_2,\cdots , a_{p-2}$ be a sequence of positive integers such that for every $k \in \{1,2,\cdots,p-2\}$ neither $a_k$ nor $a_k^k-1$ is divisible by $p$. Prove that product of some of members of this sequence is equivalent to $2$ modulo $p$.
2013 USAJMO, 1
Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?
2024 Silk Road, 1
Let $n$ be a positive integer and let $p, q>n$ be odd primes. Prove that the positive integers $1, 2, \ldots, n$ can be colored in $2$ colors, such that for any $x \neq y$ of the same color, $xy-1$ is not divisible by $p$ and $q$.
2000 Rioplatense Mathematical Olympiad, Level 3, 1
Let $a$ and $b$ be positive integers such that the number $b^2 + (b +1)^2 +...+ (b + a)^2-3$ is multiple of $5$ and $a + b$ is odd. Calculate the digit of the units of the number $a + b$ written in decimal notation.
2016 Postal Coaching, 1
Show that there are infinitely many rational triples $(a, b, c)$ such that $$a + b + c = abc = 6.$$
2019 Iran MO (3rd Round), 3
Let $a,m$ be positive integers such that $Ord_m (a)$ is odd and for any integers $x,y$ so that
1.$xy \equiv a \pmod m$
2.$Ord_m(x) \le Ord_m(a)$
3.$Ord_m(y) \le Ord_m(a)$
We have either $Ord_m(x)|Ord_m(a)$ or $Ord_m(y)|Ord_m(a)$.prove that $Ord_m(a)$ contains at most one prime factor.
2002 All-Russian Olympiad, 4
From the interval $(2^{2n},2^{3n})$ are selected $2^{2n-1}+1$ odd numbers. Prove that there are two among the selected numbers, none of which divides the square of the other.
2004 Croatia National Olympiad, Problem 4
A frog jumps on the coordinate lattice, starting from the point $(1,1)$, according to the following rules:
(i) From point $(a,b)$ the frog can jump to either $(2a,b)$ or $(a,2b)$;
(ii) If $a>b$, the frog can also jump from $(a,b)$ to $(a-b,b)$, while for $a<b$ it can jump from $(a,b)$ to $(a,b-a)$.
Can the frog get to the point: (a) $(24,40)$; (b) $(40,60)$; (c) $(24,60)$; (d) $(200,4)$?
2021 Taiwan TST Round 3, 1
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]
2016 CMIMC, 9
Compute the number of positive integers $n \leq 50$ such that there exist distinct positive integers $a,b$ satisfying
\[
\frac{a}{b} +\frac{b}{a} = n \left(\frac{1}{a} + \frac{1}{b}\right).
\]
2010 IMO Shortlist, 1
Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that
\[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\]
[i]Proposed by Daniel Brown, Canada[/i]
2022 Junior Balkan Team Selection Tests - Romania, P4
Let $n$ be a positive integer with $d^2$ positive divisors. We fill a $d\times d$ board with these divisors. At a move, we can choose a row, and shift the divisor from the $i^{\text{th}}$ column to the $(i+1)^{\text{th}}$ column, for all $i=1,2,\ldots, d$ (indices reduced modulo $d$).
A configuration of the $d\times d$ board is called [i]feasible[/i] if there exists a column with elements $a_1,a_2,\ldots,a_d,$ in this order, such that $a_1\mid a_2\mid\ldots\mid a_d$ or $a_d\mid a_{d-1}\mid\ldots\mid a_1.$ Determine all values of $n$ for which ragardless of how we initially fill the board, we can reach a feasible configuration after a finite number of moves.
1999 Brazil Team Selection Test, Problem 5
(a) If $m, n$ are positive integers such that $2^n-1$ divides $m^2 + 9$, prove
that $n$ is a power of $2$;
(b) If $n$ is a power of $2$, prove that there exists a positive integer $m$ such
that $2^n-1$ divides $m^2 + 9$.
2021 CMIMC, 2.1
Find the unique 3 digit number $N=\underline{A}$ $\underline{B}$ $\underline{C},$ whose digits $(A, B, C)$ are all nonzero, with the property that the product $P=\underline{A}$ $\underline{B}$ $\underline{C}$ $\times$ $\underline{A}$ $\underline{B}$ $\times$ $\underline{A}$ is divisible by $1000$.
[i]Proposed by Kyle Lee[/i]
2003 Chile National Olympiad, 4
Juan did not like the criticism of his classmates published in his school newspaper. He found nothing better than to start ripping up the diary. First he tore it into $4$ parts and then he continued to break it in a very methodical way: namely, each piece of newspaper he found he would tear it back into $4$ or $10$ pieces randomly. Breaking this way, was he able to get exactly $2003$ pieces of the diary?
2011 China Team Selection Test, 3
A positive integer $n$ is known as an [i]interesting[/i] number if $n$ satisfies
\[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \]
for all $k=1,2,\ldots 9$.
Find the number of interesting numbers.