Found problems: 1766
2003 Federal Competition For Advanced Students, Part 2, 1
Consider the polynomial $P(n) = n^3 -n^2 -5n+ 2$. Determine all integers $n$ for which $P(n)^2$ is a square of a prime.
[hide="Remark."]I'm not sure if the statement of this problem is correct, because if $P(n)^2$ be a square of a prime, then $P(n)$ should be that prime, and I don't think the problem means that.[/hide]
2012 Gulf Math Olympiad, 4
Fawzi cuts a spherical cheese completely into (at least three) slices of equal thickness. He starts at one end, making successive parallel cuts, working through the cheese until the slicing is complete. The discs exposed by the first two cuts have integral areas.
[list](i) Prove that all the discs that he cuts have integral areas.
(ii) Prove that the original sphere had integral surface area if, and only if, the area of the second disc that he exposes is even.[/list]
2021 Romania Team Selection Test, 1
Find all pairs $(m,n)$ of positive odd integers, such that $n \mid 3m+1$ and $m \mid n^2+3$.
2011 Rioplatense Mathematical Olympiad, Level 3, 1
Given a positive integer $n$, an operation consists of replacing $n$ with either $2n-1$, $3n-2$ or $5n-4$. A number $b$ is said to be a [i]follower[/i] of number $a$ if $b$ can be obtained from $a$ using this operation multiple times. Find all positive integers $a < 2011$ that have a common follower with $2011$.
2002 Tournament Of Towns, 6
In an infinite increasing sequence of positive integers, every term from the $2002^{\text{th}}$ term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.
1988 IberoAmerican, 2
Let $a,b,c,d,p$ and $q$ be positive integers satisfying $ad-bc=1$ and $\frac{a}{b}>\frac{p}{q}>\frac{c}{d}$.
Prove that:
$(a)$ $q\ge b+d$
$(b)$ If $q=b+d$, then $p=a+c$.
2013 China National Olympiad, 3
Let $m,n$ be positive integers. Find the minimum positive integer $N$ which satisfies the following condition. If there exists a set $S$ of integers that contains a complete residue system module $m$ such that $| S | = N$, then there exists a nonempty set $A \subseteq S$ so that $n\mid {\sum\limits_{x \in A} x }$.
2013 Turkey Team Selection Test, 1
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.
2008 Junior Balkan Team Selection Tests - Romania, 2
Prove that for every $ n \in \mathbb{N}^*$ exists a multiple of $ n$, having sum of digits equal to $ n$.
1992 Turkey Team Selection Test, 1
Is there $14$ consecutive positive integers such that each of these numbers is divisible by one of the prime numbers $p$ where $2\leq p \leq 11$.
1992 Hungary-Israel Binational, 5
We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$
\[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n}, \]
where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof.
Show that $L_{2n+1}+(-1)^{n+1}(n \geq 1)$ can be written as a product of three (not necessarily distinct) Fibonacci numbers.
2009 Olympic Revenge, 6
Let $a, n \in \mathbb{Z}^{*}_{+}$. $a$ is defined inductively in the base $n$-[i]recursive[/i]. We first write $a$ in the base $n$, e.g., as a sum of terms of the form $k_tn^t$, with $0 \le k_t < n$. For each exponent $t$, we write $t$ in the base $n$-[i]recursive[/i], until all the numbers in the representation are less than $n$. For instance,
$1309 = 3^6 + 2.3^5 + 1.3^4 + 1.3^2 + 1.3 + 1$
$ = 3^{2.3} + 2.3^{3+2} + 1.3^{3+1} + 1.3^2 + 1$
Let $x_1 \in \mathbb{Z}$ arbitrary. We define $x_n$ recursively, as following: if $x_{n-1} > 0$, we write $x_{n-1}$ in the base $n$-[i]recursive[/i] and we replace all the numbers $n$ for $n+1$ (even the exponents!), so we obtain the successor of $x_n$. If $x_{n-1} = 0$, then $x_n = 0$.
Example:
$x_1 = 2^{2^{2} + 2 + 1} + 2^{2+1} + 2 + 1$
$\Rightarrow x_2 = 3^{3^{3} + 3 + 1} + 3^{3+1} + 3$
$\Rightarrow x_3 = 4^{4^{4} + 4 + 1} + 4^{4+1} + 3$
$\Rightarrow x_4 = 5^{5^{5} + 5 + 1} + 5^{5+1} + 2$
$\Rightarrow x_5 = 6^{6^{6} + 6 + 1} + 6^{6+1} + 1$
$\Rightarrow x_6 = 7^{7^{7} + 7 + 1} + 7^{7+1}$
$\Rightarrow x_7 = 8^{8^{8} + 8 + 1} + 7.8^8 + 7.8^7 + 7.8^6 + ... + 7$
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Prove that $\exists N : x_N = 0$.
1996 All-Russian Olympiad, 1
Can the number obtained by writing the numbers from 1 to $n$ in order ($n > 1$) be the same when read left-to-right and right-to-left?
[i]N. Agakhanov[/i]
2001 Federal Competition For Advanced Students, Part 2, 2
Determine all integers $m$ for which all solutions of the equation $3x^3-3x^2+m = 0$ are rational.
2008 Kyiv Mathematical Festival, 3
Prove that among any 7 integers there exist three numbers $ a,b,c$ such that $ a^2\plus{}b^2\plus{}c^2\minus{}ab\minus{}bc\minus{}ac$ is divisible by 7.
2003 Vietnam National Olympiad, 1
Find the largest positive integer $n$ such that the following equations have integer solutions in $x, y_{1}, y_{2}, ... , y_{n}$ :
$(x+1)^{2}+y_{1}^{2}= (x+2)^{2}+y_{2}^{2}= ... = (x+n)^{2}+y_{n}^{2}.$
2005 Iran MO (3rd Round), 4
$k$ is an integer. We define the sequence $\{a_n\}_{n=0}^{\infty}$ like this:
\[a_0=0,\ \ \ a_1=1,\ \ \ a_n=2ka_{n-1}-(k^2+1)a_{n-2}\ \ (n \geq 2)\]
$p$ is a prime number that $p\equiv 3(\mbox{mod}\ 4)$
a) Prove that $a_{n+p^2-1}\equiv a_n(\mbox{mod}\ p)$
b) Prove that $a_{n+p^3-p}\equiv a_n(\mbox{mod}\ p^2)$
2015 ITAMO, 4
Determine all pairs of integers $(a, b)$ that solve the equation $a^3 + b^3 + 3ab = 1$.
2010 India IMO Training Camp, 12
Prove that there are infinitely many positive integers $m$ for which there exists consecutive odd positive integers $p_m<q_m$ such that $p_m^2+p_mq_m+q_m^2$ and $p_m^2+m\cdot p_mq_m+q_m^2$ are both perfect squares. If $m_1, m_2$ are two positive integers satisfying this condition, then we have $p_{m_1}\neq p_{m_2}$
2002 Romania National Olympiad, 1
For any number $n\in\mathbb{N},n\ge 2$, denote by $P(n)$ the number of pairs $(a,b)$ whose elements are of positive integers such that
\[\frac{n}{a}\in (0,1),\quad \frac{a}{b}\in (1,2)\quad \text{and}\quad \frac{b}{n}\in (2,3). \]
$a)$ Calculate $P(3)$.
$b)$ Find $n$ such that $P(n)=2002$.
2014 Contests, 2
Determine the minimum possible amount of distinct prime divisors of $19^{4n}+4$, for a positive integer $n$.
2002 China Western Mathematical Olympiad, 3
Assume that $ \alpha$ and $ \beta$ are two roots of the equation: $ x^2\minus{}x\minus{}1\equal{}0$. Let $ a_n\equal{}\frac{\alpha^n\minus{}\beta^n}{\alpha\minus{}\beta}$, $ n\equal{}1, 2, \cdots$.
(1) Prove that for any positive integer $ n$, we have $ a_{n\plus{}2}\equal{}a_{n\plus{}1}\plus{}a_n$.
(2) Find all positive integers $ a$ and $ b$, $ a<b$, satisfying $ b \mid a_n\minus{}2na^n$ for any positive integer $ n$.
2004 ITAMO, 3
(a) Is $2005^{2004}$ the sum of two perfect squares?
(b) Is $2004^{2005}$ the sum of two perfect squares?
1998 Irish Math Olympiad, 3
Show that no integer of the form $ xyxy$ in base $ 10$ can be a perfect cube. Find the smallest base $ b>1$ for which there is a perfect cube of the form $ xyxy$ in base $ b$.
2012 China Western Mathematical Olympiad, 1
Find the smallest positive integer $m$ satisfying the following condition: for all prime numbers $p$ such that $p>3$,have $105|9^{ p^2}-29^p+m.$
(September 28, 2012, Hohhot)