Found problems: 15460
2008 Princeton University Math Competition, A8/B9
Find all sets of three primes $p, q$, and $r$ such that $p + q = r$ and $(r -p)(q - p) - 27p$ is a perfect square.
2023 ELMO Shortlist, N1
Let \(m\) be a positive integer. Find, in terms of \(m\), all polynomials \(P(x)\) with integer coefficients such that for every integer \(n\), there exists an integer \(k\) such that \(P(k)=n^m\).
[i]Proposed by Raymond Feng[/i]
IV Soros Olympiad 1997 - 98 (Russia), 11.4
There is a set of $1998$ different natural numbers. It is known that none of them can be represented as the sum of several other numbers in this set. What is the smallest value that the largest of these numbers can take?
2009 Germany Team Selection Test, 2
For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$.
[*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list]
[i]Proposed by Bruno Le Floch, France[/i]
PEN E Problems, 40
Prove that there do not exist eleven primes, all less than $20000$, which form an arithmetic progression.
2013 China Team Selection Test, 1
For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$, we define
\[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.\]
Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$, $\Omega(P(k))$ is even. Show that $n$ is an even number.
2006 Swedish Mathematical Competition, 1
If positive integers $a$ and $b$ have 99 and 101 different positive divisors respectively (including 1 and the number itself), can the product $ab$ have exactly 150 positive divisors?
2003 Iran MO (3rd Round), 10
let p be a prime and a and n be natural numbers such that (p^a -1 )/ (p-1) = 2 ^n
find the number of natural divisors of na. :)
2001 IMO Shortlist, 3
Let $ a_1 \equal{} 11^{11}, \, a_2 \equal{} 12^{12}, \, a_3 \equal{} 13^{13}$, and $ a_n \equal{} |a_{n \minus{} 1} \minus{} a_{n \minus{} 2}| \plus{} |a_{n \minus{} 2} \minus{} a_{n \minus{} 3}|, n \geq 4.$ Determine $ a_{14^{14}}$.
2011 Brazil Team Selection Test, 2
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]
2016 Hong Kong TST, 3
Let $p$ be a prime number greater than 5. Suppose there is an integer $k$ satisfying that $k^2+5$ is divisible by $p$. Prove that there are positive integers $m$ and $n$ such that $p^2=m^2+5n^2$
2022 BMT, 22
Set $n = 425425$. Let $S$ be the set of proper divisors of $n$. Compute the remainder when $$ \sum_{k\in S} \phi (k) {2n/k \choose n/k}$$ is divided by $2n$, where $\phi (x)$ is the number of positive integers at most $x$ that are relatively prime to it.
2012 India Regional Mathematical Olympiad, 3
Solve for real $x$ : $2^{2x} \cdot 2^{3\{x\}} = 11 \cdot 2^{5\{x\}} + 5 \cdot 2^{2[x]}$
(For a real number $x, [x]$ denotes the greatest integer less than or equal to x. For instance, $[2.5] = 2$, $[-3.1] = -4$, $[\pi ] = 3$. For a real number $x, \{x\}$ is defined as $x - [x]$.)
2019 Saint Petersburg Mathematical Olympiad, 5
Call the [i]improvement [/i] of a positive number its replacement by a power of two. (i.e. one of the numbers $1, 2, 4, 8, ...$), for which it increases, but not more than than $3$ times. Given $2^{100}$ positive numbers with a sum of $2^{100}$. Prove that you can erase some of them, and [i]improve [/i] each of the other numbers so that the sum the resulting numbers were again $2^{100}$.
2013 India Regional Mathematical Olympiad, 5
Let $a_1,b_1,c_1$ be natural numbers. We define \[a_2=\gcd(b_1,c_1),\,\,\,\,\,\,\,\,b_2=\gcd(c_1,a_1),\,\,\,\,\,\,\,\,c_2=\gcd(a_1,b_1),\] and \[a_3=\operatorname{lcm}(b_2,c_2),\,\,\,\,\,\,\,\,b_3=\operatorname{lcm}(c_2,a_2),\,\,\,\,\,\,\,\,c_3=\operatorname{lcm}(a_2,b_2).\] Show that $\gcd(b_3,c_3)=a_2$.
2022 VN Math Olympiad For High School Students, Problem 8
Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$.
Prove that: $k(m)$ is even for all $m>2.$
2009 IMAR Test, 4
Given any $n$ positive integers, and a sequence of $2^n$ integers (with terms among them), prove there exists a subsequence made of consecutive terms, such that the product of its terms is a perfect square. Also show that we cannot replace $2^n$ with any lower value (therefore $2^n$ is the threshold value for this property).
2022 MMATHS, 8
Let $S = \{1, 2, 3, 5, 6, 10, 15, 30\}$. For each of the $64$ ordered pairs $(a, b)$ of elements of $S$, AJ computes $gcd(a, b)$. They then sum all of the numbers they computed. What is AJ’s sum?
2021 Baltic Way, 18
Find all integer triples $(a, b, c)$ satisfying the equation
$$
5 a^2 + 9 b^2 = 13 c^2.
$$
1907 Eotvos Mathematical Competition, 3
Let $$\frac{r}{s}= 0.k_1k_2k_3 ...$$
be the decimal expansion of a rational number (If this is a terminating decimal, all $k_i$ from a certain one on are $0$). Prove that at least two of the numbers
$$\sigma_1 = 10\frac{r}{s} - k_i, \sigma_2 = 10^2- (10k_1 + k_2),$$
$$\sigma_3 = 10^2 - (10^2k_1 + 10k_2 + k_3), ...$$
are equal.
2021 IMO Shortlist, N7
Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$
2007 iTest Tournament of Champions, 1
Find the remainder when $3^{2007}$ is divided by $2007$.
2024 VJIMC, 4
Let $(b_n)_{n \ge 0}$ be a sequence of positive integers satisfying $b_n=d\left(\sum_{i=0}^{n-1} b_k\right)$ for all $n \ge 1$. (By $d(m)$ we denote the number of positive divisors of $m$.)
a) Prove that $(b_n)_{n \ge 0}$ is unbounded.
b) Prove that there are infinitely many $n$ such that $b_n>b_{n+1}$.
1997 Poland - Second Round, 4
There is a set with three elements: (2,3,5). It has got an interesting property: (2*3) mod 5=(2*5) mod 3=(3*5) mod 2. Prove that it is the only one set with such property.
2023 May Olympiad, 3
The $49$ numbers $2,3,4,...,49,50$ are written on the blackboard . An allowed operation consists of choosing two different numbers $a$ and $b$ of the blackboard such that $a$ is a multiple of $b$ and delete exactly one of the two. María performs a sequence of permitted operations until she observes that it is no longer possible to perform any more. Determine the minimum number of numbers that can remain on the board at that moment.