Found problems: 2008
1991 IMTS, 3
Prove that a positive integer can be expressed in the form $3x^2+y^2$ iff it can also be expressed in form $u^2+uv+v^2$, where $x,y,u,v$ are all positive integers.
2014 ISI Entrance Examination, 8
$n(>1)$ lotus leaves are arranged in a circle. A frog jumps from a particular leaf from another under the following rule:
[list]
[*]It always moves clockwise.
[*]From starting it skips one leaf and then jumps to the next. After that it skips two leaves and jumps to the following. And the process continues. (Remember the frog might come back on a leaf twice or more.)[/list]
Given that it reaches all leaves at least once. Show $n$ cannot be odd.
2011 China Team Selection Test, 2
Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$, such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$. It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$, and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$. Find all possible values of $m,n$.
2009 Singapore Senior Math Olympiad, 2
Find all positive integers $ m,n $ that satisfy the equation \[ 3.2^m +1 = n^2 \]
2012 AIME Problems, 1
Find the number of positive integers with three not necessarily distinct digits, $abc$, with $a \neq 0$, $c \neq 0$ such that both $abc$ and $cba$ are divisible by 4.
2006 District Olympiad, 1
Prove that for all positive integers $n$, $n>1$ the number $\sqrt{ \overline{ 11\ldots 44 \ldots 4 }}$, where 1 appears $n$ times, and 4 appears $2n$ times, is irrational.
2006 Czech-Polish-Slovak Match, 4
Show that for every integer $k \ge 1$ there is a positive integer $n$ such that the decimal representation of $2^n$ contains a block of exactly $k$ zeros, i.e. $2^n = \dots a00 \dots 0b \cdots$ with $k$ zeros and $a, b \ne 0$.
2002 AMC 12/AHSME, 24
Find the number of ordered pairs of real numbers $ (a,b)$ such that $ (a \plus{} bi)^{2002} \equal{} a \minus{} bi$.
$ \textbf{(A)}\ 1001\qquad \textbf{(B)}\ 1002\qquad \textbf{(C)}\ 2001\qquad \textbf{(D)}\ 2002\qquad \textbf{(E)}\ 2004$
2008 Brazil Team Selection Test, 2
Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that:
[b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color,
and
[b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$.
[i]Author: Gerhard Wöginger, Netherlands[/i]
2010 China Girls Math Olympiad, 8
Determine the least odd number $a > 5$ satisfying the following conditions: There are positive integers $m_1,m_2, n_1, n_2$ such that $a=m_1^2+n_1^2$, $a^2=m_2^2+n_2^2$, and $m_1-n_1=m_2-n_2.$
2016 Iran Team Selection Test, 3
Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$.
2002 Tuymaada Olympiad, 1
Each of the points $G$ and $H$ lying from different sides of the plane of hexagon $ABCDEF$ is connected with all vertices of the hexagon.
Is it possible to mark 18 segments thus formed by the numbers $1, 2, 3, \ldots, 18$ and arrange some real numbers at points $A, B, C, D, E, F, G, H$ so that each segment is marked with the difference of the numbers at its ends?
[i]Proposed by A. Golovanov[/i]
2010 Slovenia National Olympiad, 1
Let $a,b,c$ be positive integers. Prove that $a^2+b^2+c^2$ is divisible by $4$, if and only if $a,b,c$ are even.
2024 Turkey Team Selection Test, 4
Find all positive integer pairs $(a,b)$ such that, $$\frac{10^{a!} - 3^b +1}{2^a}$$ is a perfect square.
2004 Regional Olympiad - Republic of Srpska, 3
Given a sequence $(a_n)$ of real numbers such that the set $\{a_n\}$ is finite.
If for every $k>1$ subsequence $(a_{kn})$ is periodic, is it true that the sequence $(a_n)$ must be periodic?
2002 National Olympiad First Round, 26
Which of the following is the set of all perfect squares that can be written as sum of three odd composite numbers?
$\textbf{a)}\ \{(2k + 1)^2 : k \geq 0\}$
$\textbf{b)}\ \{(4k + 3)^2 : k \geq 1\}$
$\textbf{c)}\ \{(2k + 1)^2 : k \geq 3\}$
$\textbf{d)}\ \{(4k + 1)^2 : k \geq 2\}$
$\textbf{e)}\ \text{None of above}$
2009 Finnish National High School Mathematics Competition, 2
A polynomial $P$ has integer coefficients and $P(3)=4$ and $P(4)=3$. For how many $x$ we might have $P(x)=x$?
2006 China Team Selection Test, 2
Find all positive integer pairs $(a,n)$ such that $\frac{(a+1)^n-a^n}{n}$ is an integer.
2012 Online Math Open Problems, 30
Let $P(x)$ denote the polynomial
\[3\sum_{k=0}^{9}x^k + 2\sum_{k=10}^{1209}x^k + \sum_{k=1210}^{146409}x^k.\]Find the smallest positive integer $n$ for which there exist polynomials $f,g$ with integer coefficients satisfying $x^n - 1 = (x^{16} + 1)P(x) f(x) + 11\cdot g(x)$.
[i]Victor Wang.[/i]
2011 Balkan MO, 3
Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.
2002 AMC 12/AHSME, 13
What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1,k_2,\ldots,k_n$ for which \[k_1^2+k_2^2+\ldots+k_n^2=2002?\]
$\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
2001 JBMO ShortLists, 2
Let $P_n \ (n=3,4,5,6,7)$ be the set of positive integers $n^k+n^l+n^m$, where $k,l,m$ are positive integers. Find $n$ such that:
i) In the set $P_n$ there are infinitely many squares.
ii) In the set $P_n$ there are no squares.
PEN H Problems, 13
Find all pairs $(x,y)$ of positive integers that satisfy the equation \[y^{2}=x^{3}+16.\]
2012 Indonesia TST, 4
Determine all natural numbers $n$ such that for each natural number $a$ relatively prime with $n$ and $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$ there exists some integer $x$ with $a \equiv x^2 \mod n$.
Remark: "Natural numbers" is the set of positive integers.
2014 ITAMO, 3
For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers.
(a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$.
(b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.