Found problems: 2008
PEN D Problems, 18
Let $p$ be a prime number. Determine the maximal degree of a polynomial $T(x)$ whose coefficients belong to $\{ 0,1,\cdots,p-1 \}$, whose degree is less than $p$, and which satisfies \[T(n)=T(m) \; \pmod{p}\Longrightarrow n=m \; \pmod{p}\] for all integers $n, m$.
2001 Austrian-Polish Competition, 1
Determine the number of positive integers $a$, so that there exist nonnegative integers $x_0,x_1,\ldots,x_{2001}$ which satisfy the equation
\[ \displaystyle a^{x_0} = \sum_{i=1}^{2001} a^{x_i} \]
2012 Romanian Master of Mathematics, 4
Prove that there are infinitely many positive integers $n$ such that $2^{2^n+1}+1$ is divisible by $n$ but $2^n+1$ is not.
[i](Russia) Valery Senderov[/i]
2012 IMO Shortlist, A2
Let $\mathbb{Z}$ and $\mathbb{Q}$ be the sets of integers and rationals respectively.
a) Does there exist a partition of $\mathbb{Z}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint?
b) Does there exist a partition of $\mathbb{Q}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint?
Here $X+Y$ denotes the set $\{ x+y : x \in X, y \in Y \}$, for $X,Y \subseteq \mathbb{Z}$ and for $X,Y \subseteq \mathbb{Q}$.
1966 IMO Longlists, 54
We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$
2014 National Olympiad First Round, 10
How many non-negative integer triples $(m,n,k)$ are there such that $m^3-n^3=9^k+123$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2014 ELMO Shortlist, 11
Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define
\[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \]
Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$.
[i]Proposed by Victor Wang[/i]
2014 Purple Comet Problems, 24
Let $S=2^3+3^4+5^4+7^4+\cdots+17497^4$ be the sum of the fourth powers of the first $2014$ prime numbers. Find the remainder when $S$ is divided by $240$.
2005 Czech-Polish-Slovak Match, 3
Find all integers $n \ge 3$ for which the polynomial
\[W(x) = x^n - 3x^{n-1} + 2x^{n-2} + 6\]
can be written as a product of two non-constant polynomials with integer coefficients.
2005 Germany Team Selection Test, 3
Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible.
[i]Proposed by Horst Sewerin, Germany[/i]
1982 AMC 12/AHSME, 30
Find the units digit of the decimal expansion of \[(15 \plus{} \sqrt{220})^{19} \plus{} (15 \plus{} \sqrt{220})^{82}.\]
$ \textbf{(A)}\ 0\qquad
\textbf{(B)}\ 2\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 9\qquad
\textbf{(E)}\ \text{none of these}$
1995 India Regional Mathematical Olympiad, 3
Prove that among any $18$ consecutive three digit numbers there is at least one number which is divisible by the sum of its digits.
2021 Korea Junior Math Olympiad, 2
Let $\{a_n\}$ be a sequence of integers satisfying the following conditions.
[list]
[*] $a_1=2021^{2021}$
[*] $0 \le a_k < k$ for all integers $k \ge 2$
[*] $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$.
[/list]
Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.
2008 AMC 10, 24
Let $ k\equal{}2008^2\plus{}2^{2008}$. What is the units digit of $ k^2\plus{}2^k$?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 8$
1991 Federal Competition For Advanced Students, P2, 2
Find all functions $ f: \mathbb{Z} \minus{} \{ 0 \} \rightarrow \mathbb{Q}$ satisfying:
$ f \left( \frac{x\plus{}y}{3} \right)\equal{}\frac {f(x)\plus{}f(y)}{2},$ whenever $ x,y,\frac{x\plus{}y}{3} \in \mathbb{Z} \minus{} \{ 0 \}.$
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
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$
1999 Junior Balkan MO, 2
For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$. Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$.
[i]Romania[/i]
2007 All-Russian Olympiad Regional Round, 10.7
Given an integer $ n>6$. Consider those integers $ k\in (n(n\minus{}1),n^{2})$ which are coprime with $ n$. Prove that the greatest common divisor of the considered numbers is $ 1$.
2003 Vietnam Team Selection Test, 3
Let $n$ be a positive integer. Prove that the number $2^n + 1$ has no prime divisor of the form $8 \cdot k - 1$, where $k$ is a positive integer.
2005 Irish Math Olympiad, 4
Find the first digit to the left and the first digit to the right of the decimal point in the expansion of $ (\sqrt{2}\plus{}\sqrt{5})^{2000}.$
2017 Azerbaijan EGMO TST, 4
Find all natural numbers a, b such that $ a^{n}\plus{} b^{n} \equal{} c^{n\plus{}1}$ where c and n are naturals.
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.
2005 South africa National Olympiad, 6
Consider the increasing sequence $1,2,4,5,7,9,10,12,14,16,17,19,\dots$ of positive integers, obtained by concatenating alternating blocks $\{1\},\{2,4\},\{5,7,9\},\{10,12,14,16\},\dots$ of odd and even numbers. Each block contains one more element than the previous one and the first element in each block is one more than the last element of the previous one. Prove that the $n$-th element of the sequence is given by \[2n-\Big\lfloor\frac{1+\sqrt{8n-7}}{2}\Big\rfloor.\]
(Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.)
2002 National Olympiad First Round, 6
The thousands digit of a five-digit number which is divisible by $37$ and $173$ is $3$. What is the hundreds digit of this number?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 2
\qquad\textbf{c)}\ 4
\qquad\textbf{d)}\ 6
\qquad\textbf{e)}\ 8
$
2014 Turkey MO (2nd round), 2
Find all all positive integers x,y,and z satisfying the equation $x^3=3^y7^z+8$