Found problems: 2008
PEN S Problems, 29
What is the rightmost nonzero digit of $1000000!$?
2008 AIME Problems, 3
Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $ 74$ kilometers after biking for $ 2$ hours, jogging for $ 3$ hours, and swimming for $ 4$ hours, while Sue covers $ 91$ kilometers after jogging for $ 2$ hours, swimming for $ 3$ hours, and biking for $ 4$ hours. Their biking, jogging, and swimming rates are all whole numbers of kilometers per hour. Find the sum of the squares of Ed's biking, jogging, and swimming rates.
2006 Greece National Olympiad, 2
Let $n$ be a positive integer. Prove that the equation
\[x+y+\frac{1}{x}+\frac{1}{y}=3n\]
does not have solutions in positive rational numbers.
2006 Bulgaria National Olympiad, 1
Let $p$ be a prime such that $p^2$ divides $2^{p-1}-1$. Prove that for all positive integers $n$ the number $\left(p-1\right)\left(p!+2^n\right)$ has at least $3$ different prime divisors.
[i]Aleksandar Ivanov[/i]
2005 CentroAmerican, 6
Let $n$ be a positive integer and $p$ a fixed prime. We have a deck of $n$ cards, numbered $1,\ 2,\ldots,\ n$ and $p$ boxes for put the cards on them. Determine all posible integers $n$ for which is possible to distribute the cards in the boxes in such a way the sum of the numbers of the cards in each box is the same.
2008 Germany 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]
2005 AMC 12/AHSME, 10
The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence?
$ \textbf{(A)}\ 29\qquad
\textbf{(B)}\ 55\qquad
\textbf{(C)}\ 85\qquad
\textbf{(D)}\ 133\qquad
\textbf{(E)}\ 250$
2005 Danube Mathematical Olympiad, 4
Let $k$ and $n$ be positive integers. Consider an array of $2\left(2^n-1\right)$ rows by $k$ columns. A $2$-coloring of the elements of the array is said to be [i]acceptable[/i] if any two columns agree on less than $2^n-1$ entries on the same row.
Given $n$, determine the maximum value of $k$ for an acceptable $2$-coloring to exist.
PEN A Problems, 96
Find all positive integers $n$ that have exactly $16$ positive integral divisors $d_{1},d_{2} \cdots, d_{16}$ such that $1=d_{1}<d_{2}<\cdots<d_{16}=n$, $d_6=18$, and $d_{9}-d_{8}=17$.
2005 Romania Team Selection Test, 3
Let $n\geq 0$ be an integer and let $p \equiv 7 \pmod 8$ be a prime number. Prove that
\[ \sum^{p-1}_{k=1} \left \{ \frac {k^{2^n}}p - \frac 12 \right\} = \frac {p-1}2 . \]
[i]Călin Popescu[/i]
2008 ITest, 47
Find $a+b+c$, where $a,b,$ and $c$ are the hundreds, tens, and units digits of the six-digit number $123abc$, which is a multiple of $990$.
1990 India National Olympiad, 4
Consider the collection of all three-element subsets drawn from the set $ \{1,2,3,4,\dots,299,300\}$.
Determine the number of those subsets for which the sum of the elements is a multiple of 3.
2013 Tuymaada Olympiad, 7
Solve the equation $p^2-pq-q^3=1$ in prime numbers.
[i]A. Golovanov[/i]
2013 ELMO Shortlist, 2
For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$?
[i]Proposed by Andre Arslan[/i]
2007 Iran MO (3rd Round), 1
Let $ n$ be a natural number, such that $ (n,2(2^{1386}\minus{}1))\equal{}1$. Let $ \{a_{1},a_{2},\dots,a_{\varphi(n)}\}$ be a reduced residue system for $ n$. Prove that:\[ n|a_{1}^{1386}\plus{}a_{2}^{1386}\plus{}\dots\plus{}a_{\varphi(n)}^{1386}\]
PEN D Problems, 13
Let $\Gamma$ consist of all polynomials in $x$ with integer coefficients. For $f$ and $g$ in $\Gamma$ and $m$ a positive integer, let $f \equiv g \pmod{m}$ mean that every coefficient of $f-g$ is an integral multiple of $m$. Let $n$ and $p$ be positive integers with $p$ prime. Given that $f,g,h,r$ and $s$ are in $\Gamma$ with $rf+sg\equiv 1 \pmod{p}$ and $fg \equiv h \pmod{p}$, prove that there exist $F$ and $G$ in $\Gamma$ with $F \equiv f \pmod{p}$, $G \equiv g \pmod{p}$, and $FG \equiv h \pmod{p^n}$.
2014 Iran MO (3rd Round), 2
We say two sequence of natural numbers A=($a_1,...,a_n$) , B=($b_1,...,b_n$)are the exchange and we write $A\sim B$.
if $503\vert a_i - b_i$ for all $1\leq i\leq n$.
also for natural number $r$ : $A^r$ = ($a_1^r,a_2^r,...,a_n^r$).
Prove that there are natural number $k,m$ such that :
$i$)$250 \leq k $
$ii$)There are different permutations $\pi _1,...,\pi_k$ from {$1,2,3,...,502$} such that for $1\leq i \leq k-1$ we have $\pi _i^m\sim \pi _{i+1}$
(15 points)
1999 National Olympiad First Round, 6
If $ a,b,c\in {\rm Z}$ and
\[ \begin{array}{l} {x\equiv a\, \, \, \pmod{14}} \\
{x\equiv b\, \, \, \pmod {15}} \\
{x\equiv c\, \, \, \pmod {16}} \end{array}
\]
, the number of integral solutions of the congruence system on the interval $ 0\le x < 2000$ cannot be
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$
2011 Singapore MO Open, 5
Find all pairs of positive integers $(m,n)$ such that
\[m+n-\frac{3mn}{m+n}=\frac{2011}{3}.\]
2002 Mexico National Olympiad, 3
Let $n$ be a positive integer. Does $n^2$ has more positive divisors of the form $4k+1$ or of the form $4k-1$?
1970 IMO Longlists, 59
For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$
2013 AIME Problems, 11
Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$, of play blocks which satisfies the conditions:
(a) If $16$, $15$, or $14$ students are present, then in each case all the blocks can be distributed in equal numbers to each student, and
(b) There are three integers $0 < x < y < z < 14$ such that when $x$, $y$, or $z$ students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.
Find the sum of the distinct prime divisors of the least possible value of $N$ satisfying the above conditions.
2014 ELMO Shortlist, 2
Define the Fibanocci sequence recursively by $F_1=1$, $F_2=1$ and $F_{i+2} = F_i + F_{i+1}$ for all $i$. Prove that for all integers $b,c>1$, there exists an integer $n$ such that the sum of the digits of $F_n$ when written in base $b$ is greater than $c$.
[i]Proposed by Ryan Alweiss[/i]
2010 Baltic Way, 17
Find all positive integers $n$ such that the decimal representation of $n^2$ consists of odd digits only.
2008 All-Russian Olympiad, 2
The columns of an $ n\times n$ board are labeled $ 1$ to $ n$. The numbers $ 1,2,...,n$ are arranged in the board so that the numbers in each row and column are pairwise different. We call a cell "good" if the number in it is greater than the label of its column. For which $ n$ is there an arrangement in which each row contains equally many good cells?