This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 2008

2005 Germany Team Selection Test, 3
Tags: modular arithmetic , combinatorics , circles , intersection , double counting , geometry
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]

2007 Italy TST, 2
Tags: modular arithmetic , combinatorics
In a competition, there were $2n+1$ teams. Every team plays exatly once against every other team. Every match finishes with the victory of one of the teams. We call cyclical a 3-subset of team ${ A,B,C }$ if $A$ won against $B$, $B$ won against $C$ , $C$ won against $A$. (a) Find the minimum of cyclical 3-subset (depending on $n$); (b) Find the maximum of cyclical 3-subset (depending on $n$).

2003 Italy TST, 1
Tags: modular arithmetic , number theory
Find all triples of positive integers $(a,b,p)$ with $a,b$ positive integers and $p$ a prime number such that $2^a+p^b=19^a$

2009 IMO Shortlist, 1
Tags: induction , modular arithmetic , number theory
Let $ n$ be a positive integer and let $ a_1,a_2,a_3,\ldots,a_k$ $ ( k\ge 2)$ be distinct integers in the set $ { 1,2,\ldots,n}$ such that $ n$ divides $ a_i(a_{i + 1} - 1)$ for $ i = 1,2,\ldots,k - 1$. Prove that $ n$ does not divide $ a_k(a_1 - 1).$ [i]Proposed by Ross Atkins, Australia [/i]

1978 AMC 12/AHSME, 30
Tags: ratio , modular arithmetic
In a tennis tournament, $n$ women and $2n$ men play, and each player plays exactly one match with every other player. If there are no ties and the ratio of the number of matches won by women to the number of matches won by men is $7/5$, then $n$ equals $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad \textbf{(E) }\text{none of these}$

1983 AIME Problems, 5
Tags: geometry , 3d geometry , modular arithmetic , algebra , polynomial , complex number
Suppose that the sum of the squares of two complex numbers $x$ and $y$ is 7 and the sum of the cubes is 10. What is the largest real value that $x + y$ can have?

PEN S Problems, 35
Tags: modular arithmetic
Counting from the right end, what is the $2500$th digit of $10000!$?

1980 IMO Shortlist, 6
Tags: modular arithmetic , decimal representation , digit , combinatorics
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]

2004 Iran Team Selection Test, 1
Tags: modular arithmetic , quadratic , special factorizations , number theory
Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that: \[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]

2007 Turkey MO (2nd round), 1
Tags: modular arithmetic , number theory
Let $k>1$ be an integer, $p=6k+1$ be a prime number and $m=2^{p}-1$ . Prove that $\frac{2^{m-1}-1}{127m}$ is an integer.

2008 Germany Team Selection Test, 1
Tags: floor function , modular arithmetic , inequalities
Show that there is a digit unequal to 2 in the decimal represesentation of $ \sqrt [3]{3}$ between the $ 1000000$-th und $ 3141592$-th position after decimal point.

2002 AIME Problems, 14
Tags: modular arithmetic , set
A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is the greatet number of elements that $\mathcal{S}$ can have?

2010 Nordic, 1
Tags: inequalities , function , modular arithmetic , number theory , relatively prime , algebra
A function $f : \mathbb{Z}_+ \to \mathbb{Z}_+$, where $\mathbb{Z}_+$ is the set of positive integers, is non-decreasing and satisfies $f(mn) = f(m)f(n)$ for all relatively prime positive integers $m$ and $n$. Prove that $f(8)f(13) \ge (f(10))^2$.

2003 China Second Round Olympiad, 2
Tags: geometry , perimeter , floor function , calculus , integration , modular arithmetic , algebra
Let the three sides of a triangle be $\ell, m, n$, respectively, satisfying $\ell>m>n$ and $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$, where $\{x\}=x-\lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$. Find the minimum perimeter of such a triangle.

2014 Bundeswettbewerb Mathematik, 4
Tags: modular arithmetic , number theory
Find all postive integers $n$ for which the number $\frac{4n+1}{n(2n-1)}$ has a terminating decimal expansion.

2005 Czech-Polish-Slovak Match, 3
Tags: polynomial , modular arithmetic , quadratic , algebra
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.

2015 AMC 10, 23
Tags: factorial , modular arithmetic , floor function
Let $n$ be a positive integer greater than 4 such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$? $ \textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11 $

2002 Czech and Slovak Olympiad III A, 3
Tags: modular arithmetic , number theory
Show that a given natural number $A$ is the square of a natural number if and only if for any natural number $n$, at least one of the differences \[(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A, \cdots , (A + n)^2 - A\] is divisible by $n$.

1988 IMO Shortlist, 31
Tags: modular arithmetic , combinatorics , invariant
Around a circular table an even number of persons have a discussion. After a break they sit again around the circular table in a different order. Prove that there are at least two people such that the number of participants sitting between them before and after a break is the same.

2010 Contests, 4
Tags: modular arithmetic , algebra , polynomial , linear algebra , matrix , combinatorics
Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$

2012 Junior Balkan MO, 4
Tags: modular arithmetic , number theory , greatest common divisor , algebra
Find all positive integers $x,y,z$ and $t$ such that $2^x3^y+5^z=7^t$.

2000 Korea - Final Round, 1
Tags: floor function , modular arithmetic , quadratic , number theory
Let $p$ be a prime such that $p \equiv 1 (\text {mod}4)$. Evaluate \[\sum_{k=1}^{p-1} \left( \left \lfloor \frac{2k^2}{p}\right \rfloor - 2 \left \lfloor {\frac{k^2}{p}}\right \rfloor \right)\]

2013 Brazil Team Selection Test, 3
Tags: modular arithmetic , number theory , divisibility
Let $x$ and $y$ be positive integers. If ${x^{2^n}}-1$ is divisible by $2^ny+1$ for every positive integer $n$, prove that $x=1$.

2010 Postal Coaching, 6
Tags: polynomial , modular arithmetic , induction , number theory , relatively prime , algebra
Find all polynomials $P$ with integer coefficients which satisfy the property that, for any relatively prime integers $a$ and $b$, the sequence $\{P (an + b) \}_{n \ge 1}$ contains an infinite number of terms, any two of which are relatively prime.

2009 Hong kong National Olympiad, 4
Tags: modular arithmetic , number theory
find all pairs of non-negative integer pairs $(m,n)$,satisfies $107^{56}(m^{2}-1)+2m+3=\binom{113^{114}}{n}$