Found problems: 15460
Bangladesh Mathematical Olympiad 2020 Final, #9
You have 2020 piles of coins in front Of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins and so on. So, the 2020th pile contains 2020 coins. Guess a positive integer[b] k[/b], in which piles contain at least[b] k [/b]coins, take away exact[b] k[/b] coins from these piles. Find the [b]minimum number of turns[/b] you need to take way all of these coins?
2018 Romania Team Selection Tests, 4
Let $D$ be a non-empty subset of positive integers and let $d$ be the greatest common divisor of $D$, and let $d\mathbb{Z}=[dn: n \in \mathbb{Z} ]$. Prove that there exists a bijection $f: \mathbb{Z} \rightarrow d\mathbb{Z} $ such that $| f(n+1)-f(n)|$ is member of $D$ for every integer $n$.
2002 AMC 10, 15
The digits $ 1$, $ 2$, $ 3$, $ 4$, $ 5$, $ 6$, $ 7$, and $ 9$ are used to form four two-digit prime numbers, with each digit used exactly once. What is the sum of these four primes?
$ \text{(A)}\ 150 \qquad
\text{(B)}\ 160 \qquad
\text{(C)}\ 170 \qquad
\text{(D)}\ 180 \qquad
\text{(E)}\ 190$
2004 Regional Olympiad - Republic of Srpska, 1
Prove that the cube of any positive integer greater than 1 can be represented as a difference of the squares
of two positive integers.
2023 Indonesia TST, N
Let $p,q,r$ be primes such that for all positive integer $n$,
$$n^{pqr}\equiv n (\mod{pqr})$$
Prove that this happens if and only if $p,q,r$ are pairwise distinct and $LCM(p-1,q-1,r-1)|pqr-1$
2005 International Zhautykov Olympiad, 1
Prove that the equation $ x^{5} \plus{} 31 \equal{} y^{2}$ has no integer solution.
1993 IberoAmerican, 1
A number is called [i]capicua[/i] if when it is written in decimal notation, it can be read equal from left to right as from right to left; for example: $8, 23432, 6446$. Let $x_1<x_2<\cdots<x_i<x_{i+1},\cdots$ be the sequence of all capicua numbers. For each $i$ define $y_i=x_{i+1}-x_i$. How many distinct primes contains the set $\{y_1,y_2, \ldots\}$?
1985 Spain Mathematical Olympiad, 4
Prove that for each positive integer $k $ there exists a triple $(a,b,c)$ of positive integers such that $abc = k(a+b+c)$. In all such cases prove that $a^3+b^3+c^3$ is not a prime.
2017 BMT Spring, 7
There are $86400$ seconds in a day, which can be deduced from the conversions between seconds, minutes, hours, and days. However, the leading scientists decide that we should decide on $3$ new integers $x, y$, and $z$, such that there are $x$ seconds in a minute, $y$ minutes in an hour, and $z$ hours in a day, such that $xyz = 86400$ as before, but such that the sum $x + y + z$ is minimized. What is the smallest possible value of that sum?
2005 Abels Math Contest (Norwegian MO), 1a
A positive integer $m$ is called triangular if $m = 1 + 2 + ... + n$, for an integer $n$. Show that a positive integer $m$ is triangular if and only if $8m + 1$ is the square of an integer.
2003 Greece Junior Math Olympiad, 1
Find all positive integers $n$ for which number $A = n^3-n^2+n-1$ is prime
2017 Princeton University Math Competition, A7
Compute the number of ordered pairs of integers $(a, b)$, where $0 \le a < 17$ and $0 \le b < 17$, such that $y^2 \equiv x^3 +ax +b \pmod{17}$ has an even number of solutions $(x, y)$, where $0 \le x < 17$ and $0 \le y < 17$ are integers.
2016 IFYM, Sozopol, 7
Let $S$ be a set of integers which has the following properties:
1) There exists $x,y\in S$ such that $(x,y)=(x-2,y-2)=1$;
2) For $\forall$ $x,y\in S, x^2-y\in S$.
Prove that $S\equiv \mathbb{Z}$ .
2012 ELMO Shortlist, 1
Find all positive integers $n$ such that $4^n+6^n+9^n$ is a square.
[i]David Yang, Alex Zhu.[/i]
2016 239 Open Mathematical Olympiad, 4
The sequences of natural numbers $p_n$ and $q_n$ are given such that
$$p_1 = 1,\ q_1 = 1,\ p_{n + 1} = 2q_n^2-p_n^2,\ q_{n + 1} = 2q_n^2+p_n^2 $$
Prove that $p_n$ and $q_m$ are coprime for any m and n.
2000 Tuymaada Olympiad, 1
Let $d(n)$ denote the number of positive divisors of $n$ and let
$e(n)=\left[2000\over n\right]$ for positive integer $n$. Prove that
\[d(1)+d(2)+\dots+d(2000)=e(1)+e(2)+\dots+e(2000).\]
2004 Switzerland - Final Round, 3
Let $p$ be an odd prime number. Find all natural numbers $k$ such that
$$\sqrt{k^2 - pk}$$
is a positive integer.
2015 Iberoamerican Math Olympiad, 1
The number $125$ can be written as a sum of some pairwise coprime integers larger than $1$. Determine the largest number of terms that the sum may have.
2010 China Team Selection Test, 2
Given integer $a_1\geq 2$. For integer $n\geq 2$, define $a_n$ to be the smallest positive integer which is not coprime to $a_{n-1}$ and not equal to $a_1,a_2,\cdots, a_{n-1}$. Prove that every positive integer except 1 appears in this sequence $\{a_n\}$.
2022 HMNT, 3
Alice is bored in class, so she thinks of a positive integer. Every second after that, she subtracts from her current number its smallest prime divisor, possibly itself. After 2022 seconds, she realizes that her number is prime. Find the sum of all possible values of her initial number.
2004 All-Russian Olympiad Regional Round, 9.4
Three natural numbers are such that the product of any two of them is divided by the sum of these two numbers. Prove that these three numbers have a common divisor greater than one.
MathLinks Contest 3rd, 2
Let $a_1, a_2, ..., a_{2004}$ be integer numbers such that for all positive integers $n$ the number $A_n = a^n_1 + a^n_2 + ...+ a^n_{2004}$ is a perfect square. What is the minimal number of zeros within the $2004$ numbers?
2021 Israel TST, 4
Let $r$ be a positive integer and let $a_r$ be the number of solutions to the equation $3^x-2^y=r$ ,such that $0\leq x,y\leq 5781$ are integers. What is the maximal value of $a_r$?
2024 Iran MO (2nd Round), 1
Kimia has a weird clock; the clock's second hand moves 34 or 47 seconds forward instead of each regular second, at random. As an example, if the clock displays the time as $\text{12:23:05}$, the following times could be displayed in this order:
$$\text{12:23:39, 12:24:13, 12:25:00, 12:25:34, 12:26:21,\dots}$$
Prove that the clock's second hand would eventually land on a perfect square.
2014 Rioplatense Mathematical Olympiad, Level 3, 4
A pair (a,b) of positive integers is [i]Rioplatense [/i]if it is true that $b + k$ is a multiple of $a + k$ for all $k \in\{ 0 , 1 , 2 , 3 , 4 \}$. Prove that there is an infinite set $A$ of positive integers such that for any two elements $a$ and $b$ of $A$, with $a < b$, the pair $(a,b)$ is [i]Rioplatense[/i].