Found problems: 15460
2016 China Girls Math Olympiad, 8
Let $\mathbb{Q}$ be the set of rational numbers, $\mathbb{Z}$ be the set of integers. On the coordinate plane, given positive integer $m$, define $$A_m = \left\{ (x,y)\mid x,y\in\mathbb{Q}, xy\neq 0, \frac{xy}{m}\in \mathbb{Z}\right\}.$$
For segment $MN$, define $f_m(MN)$ as the number of points on segment $MN$ belonging to set $A_m$.
Find the smallest real number $\lambda$, such that for any line $l$ on the coordinate plane, there exists a constant $\beta (l)$ related to $l$, satisfying: for any two points $M,N$ on $l$, $$f_{2016}(MN)\le \lambda f_{2015}(MN)+\beta (l)$$
2023 CMIMC Algebra/NT, 6
Compute the sum of all positive integers $N$ for which there exists a unique ordered triple of non-negative integers $(a,b,c)$ such that $2a+3b+5c=200$ and $a+b+c=N$.
[i]Proposed by Kyle Lee[/i]
1977 Yugoslav Team Selection Test, Problem 1
Determine the set of all real numbers $\alpha$ with the following property: For each positive $c$ there exists a rational number $\frac mn~(m\in\mathbb Z,n\in\mathbb N)$ different than $\alpha$ such that
$$\left|\alpha-\frac mn\right|<\frac cn.$$
1998 Junior Balkan MO, 3
Find all pairs of positive integers $ (x,y)$ such that
\[ x^y \equal{} y^{x \minus{} y}.
\]
[i]Albania[/i]
2023 LMT Fall, 1
George has $150$ cups of flour and $200$ eggs. He can make a cupcake with $3$ cups of flour and $2$ eggs, or he can make an omelet with $4$ eggs. What is the maximum number of treats (both omelets and cupcakes) he canmake?
2014 Tuymaada Olympiad, 1
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?
[i](A. Golovanov)[/i]
Oliforum Contest V 2017, 5
Find the smallest integer $n > 3$ such that, for each partition of $\{3, 4,..., n\}$ in two sets, at least one of these sets contains three (not necessarily distinct) numbers $ a, b, c$ for which $ab = c$.
(Alberto Alfarano)
1990 IMO Longlists, 79
Determine all integers $ n > 1$ such that
\[ \frac {2^n \plus{} 1}{n^2}
\]
is an integer.
2023 Brazil National Olympiad, 1
Show an infinite sequence $a_1, a_2, \ldots$ of integers with both of the following properties:
• $a_i \neq 0$ for every positive integer $i$, that is, no term in the sequence is equal to zero;
• for all positive integer $n$, $a_n + a_{2n} + \ldots + a_{2023n} = 0$.
1995 Canada National Olympiad, 4
Let $n$ be a constant positive integer. Show that for only non-negative integers $k$, the Diophantine equation $\sum_{i=1 }^{n}{ x_i ^3}=y^{3k+2}$ has infinitely many solutions in the positive integers $x_i, y$.
2005 IMO Shortlist, 6
Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$.
[i]Proposed by Mohsen Jamali, Iran[/i]
1999 Portugal MO, 5
Each of the numbers $a_1,...,a_n$ is equal to $1$ or $-1$. If $a_1a_2 + a_2a_3 + ··· + a_{n-1}a_n + a_na_1 = 0$, proves that $n$ is divisible by $4$.
2010 Serbia National Math Olympiad, 3
Let $a_0$ and $a_n$ be different divisors of a natural number $m$, and $a_0, a_1, \ldots, a_n$ be a sequence of natural numbers such that it satisfies
\[a_{i+1} = |a_i\pm a_{i-1}|\text{ for }0 < i < n\]
If $gcd(a_0,a_1,\ldots, a_n) = 1$, show that there exists a term of the sequence that is smaller than $\sqrt{m}$ .
[i]Proposed by Dusan Djukic[/i]
2015 IFYM, Sozopol, 5
A positive interger number $k$ is called “$t-m$”-property if forall positive interger number $a$, there exists a positive integer number $n$ such that
${{1}^{k}}+{{2}^{k}}+{{3}^{k}}+...+{{n}^{k}} \equiv a (\bmod m).$
a) Find all positive integer numbers $k$ which has $t-20$-property.
b) Find smallest positive integer number $k$ which has $t-{{20}^{15}}$-property.
2001 Croatia Team Selection Test, 3
Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.
2017 Argentina National Olympiad, 4
For a positive integer $n$ we denote $D_2(n)$ to the number of divisors of $n$ which are perfect squares and $D_3(n)$ to the number of divisors of $n$ which are perfect cubes. Prove that there exists such that $D_2(n)=999D_3(n).$
Note. The perfect squares are $1^2,2^2,3^2,4^2,…$ , the perfect cubes are $1^3,2^3,3^3,4^3,…$ .
2010 AIME Problems, 4
Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $ 100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks $ 400$ feet or less to the new gate be a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
1959 Poland - Second Round, 4
Given a sequence of numbers $ 13, 25, 43, \ldots $ whose $ n $-th term is defined by the formula
$$a_n =3(n^2 + n) + 7$$
Prove that this sequence has the following properties:
1) Of every five consecutive terms of the sequence, exactly one is divisible by $ 5 $,
2( No term of the sequence is the cube of an integer.
2021 Harvard-MIT Mathematics Tournament., 8
For positive integers $a$ and $b$, let $M(a,b) = \tfrac{\text{lcm}(a,b)}{\gcd(a,b)},$ and for each positive integer $n \ge 2,$ define
\[x_n = M(1, M(2, M(3, \dots , M(n - 2, M(n - 1, n))\cdots))).\]
Compute the number of positive integers $n$ such that $2 \le n \le 2021$ and $5x_n^2 + 5x_{n+1}^2 = 26x_nx_{n+1}.$
2007 All-Russian Olympiad Regional Round, 11.5
Find all positive integers $ n$ for which there exist integers $ a,b,c$ such that $ a\plus{}b\plus{}c\equal{}0$ and the number $ a^{n}\plus{}b^{n}\plus{}c^{n}$ is prime.
2023 Dutch IMO TST, 4
Find all positive integers $n$, such that $\sigma(n) =\tau(n) \lceil {\sqrt{n}} \rceil$.
2018 Pan African, 3
For any positive integer $x$, we set
$$
g(x) = \text{ largest odd divisor of } x,
$$
$$
f(x) = \begin{cases}
\frac{x}{2} + \frac{x}{g(x)} & \text{ if } x \text{ is even;} \\
2^{\frac{x+1}{2}} & \text{ if } x \text{ is odd.}
\end{cases}
$$
Consider the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_1 = 1$, $x_{n + 1} = f(x_n)$. Show that the integer $2018$ appears in this sequence, determine the least integer $n$ such that $x_n = 2018$, and determine whether $n$ is unique or not.
2005 MOP Homework, 3
Prove that the equation $a^3-b^3=2004$ does not have any solutions in positive integers.
2019 IFYM, Sozopol, 2
Does there exist a strictly increasing function $f:\mathbb{N}\rightarrow \mathbb{N}$, such that for $\forall$ $n\in \mathbb{N}$:
$f(f(f(n)))=n+2f(n)$?
1983 IMO, 3
Let $a,b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x,y,z$ are non-negative integers.