Found problems: 1362
2013 CentroAmerican, 1
Juan writes the list of pairs $(n, 3^n)$, with $n=1, 2, 3,...$ on a chalkboard. As he writes the list, he underlines the pairs $(n, 3^n)$ when $n$ and $3^n$ have the same units digit. What is the $2013^{th}$ underlined pair?
2006 Pan African, 4
For every positive integer $k$ let $a(k)$ be the largest integer such that $2^{a(k)}$ divides $k$. For every positive integer $n$ determine $a(1)+a(2)+\cdots+a(2^n)$.
2018 Latvia Baltic Way TST, P13
Determine whether there exists a prime $q$ so that for any prime $p$ the number
$$\sqrt[3]{p^2+q}$$
is never an integer.
2007 China Team Selection Test, 3
Let $ n$ be a positive integer, let $ A$ be a subset of $ \{1, 2, \cdots, n\}$, satisfying for any two numbers $ x, y\in A$, the least common multiple of $ x$, $ y$ not more than $ n$. Show that $ |A|\leq 1.9\sqrt {n} \plus{} 5$.
2009 ELMO Problems, 1
Let $a,b,c$ be positive integers such that $a^2 - bc$ is a square. Prove that $2a + b + c$ is not prime.
[i]Evan o'Dorney[/i]
1987 IMO Longlists, 54
Let $n$ be a natural number. Solve in integers the equation
\[x^n + y^n = (x - y)^{n+1}.\]
2006 USAMO, 5
A mathematical frog jumps along the number line. The frog starts at $1$, and jumps according to the following rule: if the frog is at integer $n$, then it can jump either to $n+1$ or to $n + 2^{m_n+1}$ where $2^{m_n}$ is the largest power of $2$ that is a factor of $n.$ Show that if $k \geq 2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^ik$ is greater than the minimum number of jumps needed to reach $2^i.$
2006 Estonia National Olympiad, 3
The sequence $ (F_n)$ of Fibonacci numbers satisfies $ F_1 \equal{} 1, F_2 \equal{} 1$ and $ F_n \equal{} F_{n\minus{}1} \plus{}F_{n\minus{}2}$ for all $ n \ge 3$. Find all pairs of positive integers $ (m, n)$, such that $ F_m . F_n \equal{} mn$.
2003 Tournament Of Towns, 3
Find all positive integers $k$ such that there exist two positive integers $m$ and $n$ satisfying
\[m(m + k) = n(n + 1).\]
2009 Iran MO (3rd Round), 6
Let $z$ be a complex non-zero number such that $Re(z),Im(z)\in \mathbb{Z}$.
Prove that $z$ is uniquely representable as $a_0+a_1(1+i)+a_2(1+i)^2+\dots+a_n(1+i)^n$ where $n\geq 0$ and $a_j \in \{0,1\}$ and $a_n=1$.
Time allowed for this problem was 1 hour.
1984 IMO Longlists, 49
Let $n > 1$ and $x_i \in \mathbb{R}$ for $i = 1,\cdots, n$. Set
\[S_k = x_1^k+ x^k_2+\cdots+ x^k_n\]
for $k \ge 1$. If $S_1 = S_2 =\cdots= S_{n+1}$, show that $x_i \in \{0, 1\}$ for every $i = 1, 2,\cdots, n.$
1972 IMO Longlists, 16
Consider the set $S$ of all the different odd positive integers that are not multiples of $5$ and that are less than $30m, m$ being a positive integer. What is the smallest integer $k$ such that in any subset of $k$ integers from $S$ there must be two integers one of which divides the other? Prove your result.
Oliforum Contest I 2008, 2
Find all non-negative integers $ x,y,z$ such that $ 5^x \plus{} 7^y \equal{} 2^z$.
:lol:
([i]Daniel Kohen, University of Buenos Aires - Buenos Aires,Argentina[/i])
2003 France Team Selection Test, 2
A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.
1993 Taiwan National Olympiad, 3
Find all $ x,y,z\in\mathbb{N}_{0}$ such that $ 7^{x} \plus{} 1 \equal{} 3^{y} \plus{} 5^{z}$.
[i]Alternative formulation:[/i] Solve the equation $ 1\plus{}7^{x}\equal{}3^{y}\plus{}5^{z}$ in nonnegative integers $ x$, $ y$, $ z$.
2001 Tournament Of Towns, 1
The natural number $n$ can be replaced by $ab$ if $a + b = n$, where $a$ and $b$ are natural numbers. Can the number $2001$ be obtained from $22$ after a sequence of such replacements?
2012 Indonesia TST, 4
Given a non-zero integer $y$ and a positive integer $n$. If $x_1, x_2, \ldots, x_n \in \mathbb{Z} - \{0, 1\}$ and $z \in \mathbb{Z}^+$ satisfy $(x_1x_2 \ldots x_n)^2y \le 2^{2(n+1)}$ and $x_1x_2 \ldots x_ny = z + 1$, prove that there is a prime among $x_1, x_2, \ldots, x_n, z$.
[color=blue]It appears that the problem statement is incorrect; suppose $y = 5, n = 2$, then $x_1 = x_2 = -1$ and $z = 4$. They all satisfy the problem's conditions, but none of $x_1, x_2, z$ is a prime. What should the problem be, or did I misinterpret the problem badly?[/color]
2010 Costa Rica - Final Round, 2
Consider the sequence $x_n>0$ defined with the following recurrence relation:
\[x_1 = 0\]
and for $n>1$ \[(n+1)^2x_{n+1}^2 + (2^n+4)(n+1)x_{n+1}+ 2^{n+1}+2^{2n-2} = 9n^2x_n^2+36nx_n+32.\]
Show that if $n$ is a prime number larger or equal to $5$, then $x_n$ is an integer.
1999 Finnish National High School Mathematics Competition, 1
Show that the equation $x^3 + 2y^2 + 4z = n$ has an integral solution $(x, y, z)$ for all integers $n.$
1972 IMO Longlists, 41
The ternary expansion $x = 0.10101010\cdots$ is given. Give the binary expansion of $x$.
Alternatively, transform the binary expansion $y = 0.110110110 \cdots$ into a ternary expansion.
2007 Estonia Math Open Junior Contests, 8
Call a k-digit positive integer a [i]hyperprime[/i] if all its segments consisting of $ 1, 2, ..., k$ consecutive digits are prime. Find all hyperprimes.
2006 Taiwan National Olympiad, 2
$x,y,z,a,b,c$ are positive integers that satisfy $xy \equiv a \pmod z$, $yz \equiv b \pmod x$, $zx \equiv c \pmod y$. Prove that
$\min{\{x,y,z\}} \le ab+bc+ca$.
2010 Dutch BxMO TST, 3
Let $N$ be the number of ordered 5-tuples $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ of positive integers satisfying
$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\frac{1}{a_{4}}+\frac{1}{a_{5}}=1$
Is $N$ even or odd?
Oh and [b]HINTS ONLY[/b], please do not give full solutions. Thanks.
1985 IMO Longlists, 14
Let $k$ be a positive integer. Define $u_0 = 0, u_1 = 1$, and $u_n=ku_{n-1}-u_{n-2} , n \geq 2.$ Show that for each integer $n$, the number $u_1^3 + u_2^3 +\cdots+ u_n^3 $ is a multiple of $u_1 + u_2 +\cdots+ u_n.$
2007 Croatia Team Selection Test, 2
Prove that the sequence $a_{n}=\lfloor n\sqrt 2 \rfloor+\lfloor n\sqrt 3 \rfloor$ contains infintely many even and infinitely many odd numbers.