Found problems: 2008
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
$\min{\{x,y,z\}} \le ab+bc+ca$.
2013 Kazakhstan National Olympiad, 1
On the board written numbers from 1 to 25 . Bob can pick any three of them say $a,b,c$ and replace by $a^3+b^3+c^3$ . Prove that last number on the board can not be $2013^3$.
2011 Purple Comet Problems, 27
Find the smallest prime number that does not divide \[9+9^2+9^3+\cdots+9^{2010}.\]
2005 Mexico National Olympiad, 3
Already the complete problem:
Determine all pairs $(a,b)$ of integers different from $0$ for which it is possible to find a positive integer $x$ and an integer $y$ such that $x$ is relatively prime to $b$ and in the following list there is an infinity of integers:
$\rightarrow\qquad\frac{a + xy}{b}$, $\frac{a + xy^2}{b^2}$, $\frac{a + xy^3}{b^3}$, $\ldots$, $\frac{a + xy^n}{b^n}$, $\ldots$
One idea?
:arrow: [b][url=http://www.mathlinks.ro/Forum/viewtopic.php?t=61319]View all the problems from XIX Mexican Mathematical Olympiad[/url][/b]
2011 China Western Mathematical Olympiad, 1
Does there exist any odd integer $n \geq 3$ and $n$ distinct prime numbers $p_1 , p_2, \cdots p_n$ such that all $p_i + p_{i+1} (i = 1,2,\cdots , n$ and $p_{n+1} = p_{1})$ are perfect squares?
2013 Online Math Open Problems, 45
Let $N$ denote the number of ordered 2011-tuples of positive integers $(a_1,a_2,\ldots,a_{2011})$ with $1\le a_1,a_2,\ldots,a_{2011} \le 2011^2$ such that there exists a polynomial $f$ of degree $4019$ satisfying the following three properties:
[list] [*] $f(n)$ is an integer for every integer $n$; [*] $2011^2 \mid f(i) - a_i$ for $i=1,2,\ldots,2011$; [*] $2011^2 \mid f(n+2011) - f(n)$ for every integer $n$. [/list]
Find the remainder when $N$ is divided by $1000$.
[i]Victor Wang[/i]
1971 AMC 12/AHSME, 12
For each integer $N>1$, there is a mathematical system in which two or more positive integers are defined to be congruent if they leave the same non-negative remainder when divided by $N$. If $69,90,$ and $125$ are congruent in one such system, then in that same system, $81$ is congruent to
$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad \textbf{(E) }8$
2001 JBMO ShortLists, 3
Find all the three-digit numbers $\overline{abc}$ such that the $6003$-digit number $\overline{abcabc\ldots abc}$ is divisible by $91$.
1958 November Putnam, B2
Hi everybody!
I've an interesting problem!
Can you solve it?
Prove [b]Erdös-Ginzburg-Ziv Theorem[/b]: [i]"Among any $2n-1$ integers, there are some $n$ whose sum is divisible by $n$."[/i]
2013 National Olympiad First Round, 3
If the remainder is $2013$ when a polynomial with coefficients from the set $\{0,1,2,3,4,5\}$ is divided by $x-6$, what is the least possible value of the coefficient of $x$ in this polynomial?
$
\textbf{(A)}\ 5
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ 1
$
1993 China Team Selection Test, 1
Find all integer solutions to $2 x^4 + 1 = y^2.$
1999 CentroAmerican, 2
Find a positive integer $n$ with 1000 digits, all distinct from zero, with the following property: it's possible to group the digits of $n$ into 500 pairs in such a way that if the two digits of each pair are multiplied and then add the 500 products, it results a number $m$ that is a divisor of $n$.
2010 China Second Round Olympiad, 2
Given a fixed integer $k>0,r=k+0.5$,define
$f^1(r)=f(r)=r[r],f^l(r)=f(f^{l-1}(r))(l>1)$
where $[x]$ denotes the smallest integer not less than $x$.
prove that there exists integer $m$ such that $f^m(r)$ is an integer.
2003 Italy TST, 1
Find all triples of positive integers $(a,b,p)$ with $a,b$ positive integers and $p$ a prime number such that $2^a+p^b=19^a$
2005 Junior Balkan Team Selection Tests - Romania, 12
Find all positive integers $n$ and $p$ if $p$ is prime and \[ n^8 - p^5 = n^2+p^2 . \]
[i]Adrian Stoica[/i]
1983 Dutch Mathematical Olympiad, 2
Prove that if $ n$ is an odd positive integer, then the last two digits of $ 2^{2n}(2^{2n\plus{}1}\minus{}1)$ in base $ 10$ are $ 28$.
2005 Baltic Way, 9
A rectangle is divided into $200\times 3$ unit squares. Prove that the number of ways of splitting this rectangle into rectangles of size $1\times 2$ is divisible by $3$.
2011 Romanian Master of Mathematics, 6
The cells of a square $2011 \times 2011$ array are labelled with the integers $1,2,\ldots, 2011^2$, in such a way that every label is used exactly once. We then identify the left-hand and right-hand edges, and then the top and bottom, in the normal way to form a torus (the surface of a doughnut).
Determine the largest positive integer $M$ such that, no matter which labelling we choose, there exist two neighbouring cells with the difference of their labels at least $M$.
(Cells with coordinates $(x,y)$ and $(x',y')$ are considered to be neighbours if $x=x'$ and $y-y'\equiv\pm1\pmod{2011}$, or if $y=y'$ and $x-x'\equiv\pm1\pmod{2011}$.)
[i](Romania) Dan Schwarz[/i]
1999 China National Olympiad, 1
Let $m$ be a positive integer. Prove that there are integers $a, b, k$, such that both $a$ and $b$ are odd, $k\geq0$ and
\[2m=a^{19}+b^{99}+k\cdot2^{1999}\]
2017 South East Mathematical Olympiad, 2
Let $x_i \in \{0,1\}(i=1,2,\cdots ,n)$,if the value of function $f=f(x_1,x_2, \cdots ,x_n)$ can only be $0$ or $1$,then we call $f$ a $n$-var Boole function,and we denote $D_n(f)=\{(x_1,x_2, \cdots ,x_n)|f(x_1,x_2, \cdots ,x_n)=0\}.$
$(1)$ Find the number of $n$-var Boole function;
$(2)$ Let $g$ be a $n$-var Boole function such that $g(x_1,x_2, \cdots ,x_n) \equiv 1+x_1+x_1x_2+x_1x_2x_3 +\cdots +x_1x_2 \cdots x_n \pmod 2$,
Find the number of elements of the set $D_n(g)$,and find the maximum of $n \in \mathbb{N}_+$ such that
$\sum_{(x_1,x_2, \cdots ,x_n) \in D_n(g)}(x_1+x_2+ \cdots +x_n) \le 2017.$
2013 Online Math Open Problems, 6
Find the number of integers $n$ with $n \ge 2$ such that the remainder when $2013$ is divided by $n$ is equal to the remainder when $n$ is divided by $3$.
[i]Proposed by Michael Kural[/i]
2010 Romania Team Selection Test, 3
Given a positive integer $a$, prove that $\sigma(am) < \sigma(am + 1)$ for infinitely many positive integers $m$. (Here $\sigma(n)$ is the sum of all positive divisors of the positive integer number $n$.)
[i]Vlad Matei[/i]
2006 Tuymaada Olympiad, 4
For a positive integer, we define it's [i]set of exponents[/i] the unordered list of all the exponents of the primes, in it`s decomposition. For example, $18=2\cdot 3^{2}$ has it`s set of exponents $1,2$ and $300=2^{2}\cdot 3\cdot 5^{2}$ has it`s set of exponents $1,2,2$. There are given two arithmetical progressions $\big(a_{n}\big)_{n}$ and $\big(b_{n}\big)_{n}$, such that for any positive integer $n$, $a_{n}$ and $b_{n}$ have the same set of exponents. Prove that the progressions are proportional (that is, there is $k$ such that $a_{n}=kb_{n}$ for any $n$).
[i]Proposed by A. Golovanov[/i]
1997 All-Russian Olympiad, 1
Do there exist two quadratic trinomials $ax^2 +bx+c$ and $(a+1)x^2 +(b + 1)x + (c + 1)$ with integer coeficients, both of which have two integer roots?
[i]N. Agakhanov[/i]
1998 Putnam, 6
Prove that, for any integers $a,b,c$, there exists a positive integer $n$ such that $\sqrt{n^3+an^2+bn+c}$ is not an integer.