Olympiad problems

2008 Germany Team Selection Test, 1

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.

2021 Alibaba Global Math Competition, 18

Tags: modular arithmetic , linear algebra , vector

Let $p$ be an odd prime number, and let $m \ge 0$ and $N \ge 1$ be integers. Let $\Lambda$ be a free $\mathbb{Z}/p^N\mathbb{Z}$-module of rank $2m+1$, equipped with a perfect symmetric $\mathbb{Z}/p^N\mathbb{Z}$-bilinear form \[(\, ,\,): \Lambda \times \Lambda \to \mathbb{Z}/p^N\mathbb{Z}.\] Here ``perfect'' means that the induced map \[\Lambda \to \text{Hom}_{\mathbb{Z}/p^N\mathbb{Z}}(\Lambda, \mathbb{Z}/p^N\mathbb{Z}), \quad x \mapsto (x,\cdot)\] is an isomorphism. Find the cardinality of the set \[\{x \in \Lambda: (x,x)=0\},\] expressed in terms of $p,m,N$.

2005 Korea - Final Round, 5

Tags: modular arithmetic , number theory

Find all positive integers $m$ and $n$ such that both $3^{m}+1$ and $3^{n}+1$ are divisible by $mn$.

2001 India IMO Training Camp, 2

Tags: modular arithmetic , number theory

Let $p > 3$ be a prime. For each $k\in \{1,2, \ldots , p-1\}$, define $x_k$ to be the unique integer in $\{1, \ldots, p-1\}$ such that $kx_k\equiv 1 \pmod{p}$ and set $kx_k = 1+ pn_k$. Prove that : \[\sum_{k=1}^{p-1}kn_k \equiv \frac{p-1}{2} \pmod{p}\]

1998 Spain Mathematical Olympiad, 2

Tags: number theory , 3d geometry , modular arithmetic , geometry

Find all four-digit numbers which are equal to the cube of the sum of their digits.

2010 Brazil Team Selection Test, 2

Tags: algebra , function , modular arithmetic , divisibility , number theory

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

2014 USAMTS Problems, 2:

Tags: modular arithmetic , search , prime number , number theory

Find all triples $(x, y, z)$ such that $x, y, z, x - y, y - z, x - z$ are all prime positive integers.

2008 Romania Team Selection Test, 2

Tags: number theory , modular arithmetic

Let $ m, n \geq 1$ be two coprime integers and let also $ s$ an arbitrary integer. Determine the number of subsets $ A$ of $ \{1, 2, ..., m \plus{} n \minus{} 1\}$ such that $ |A| \equal{} m$ and $ \sum_{x \in A} x \equiv s \pmod{n}$.

PEN H Problems, 24

Tags: geometry , modular arithmetic , diophantine equation , 3d geometry

Prove that if $n$ is a positive integer such that the equation \[x^{3}-3xy^{2}+y^{3}=n.\] has a solution in integers $(x,y),$ then it has at least three such solutions. Show that the equation has no solutions in integers when $n=2891$.

1995 China Team Selection Test, 1

Tags: number theory , modular arithmetic

Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers.

2008 Regional Competition For Advanced Students, 4

Tags: number theory , logarithm , modular arithmetic , floor function

For every positive integer $ n$ let \[ a_n\equal{}\sum_{k\equal{}n}^{2n}\frac{(2k\plus{}1)^n}{k}\] Show that there exists no $ n$, for which $ a_n$ is a non-negative integer.

2006 National Olympiad First Round, 26

Tags: modular arithmetic

For how many primes $p$, there exists an integr $m$ such that $m^3+3m-2 \equiv 0 \pmod p$ and $m^2+4m+5\equiv 0 \pmod p$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{Infinitely many} $

2003 AIME Problems, 13

Tags: function , probability , counting , distinguishability , limit , vector , modular arithmetic

A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

2002 India IMO Training Camp, 14

Tags: number theory , modular arithmetic

Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$. Show that there are $p^2+1$ triples of integers $(x,y,z)$ with $0 \le x,y,z < p$ and such that $(x+y+z)^2 \equiv axyz \pmod p$

1982 USAMO, 4

Tags: modular arithmetic , number theory

Prove that there exists a positive integer $k$ such that $k\cdot2^n+1$ is composite for every integer $n$.

2010 Postal Coaching, 6

Tags: number theory , algebra , polynomial , relatively prime , modular arithmetic , induction

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.

2007 AMC 12/AHSME, 24

Tags: function , algebra , trigonometry , floor function , domain , arithmetic sequence , modular arithmetic

For each integer $ n > 1,$ let $ F(n)$ be the number of solutions of the equation $ \sin x \equal{} \sin nx$ on the interval $ [0,\pi].$ What is $ \sum_{n \equal{} 2}^{2007}F(n)?$ $ \textbf{(A)}\ 2,014,524 \qquad \textbf{(B)}\ 2,015,028 \qquad \textbf{(C)}\ 2,015,033 \qquad \textbf{(D)}\ 2,016,532 \qquad \textbf{(E)}\ 2,017,033$

2002 AIME Problems, 7

Tags: algebra , binomial theorem , modular arithmetic

The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $ x, y,$ and $ r$ with $ |x| > |y|,$ \[ (x \plus{} y)^r \equal{} x^r \plus{} rx^{r \minus{} 1}y \plus{} \frac {r(r \minus{} 1)}2x^{r \minus{} 2}y^2 \plus{} \frac {r(r \minus{} 1)(r \minus{} 2)}{3!}x^{r \minus{} 3}y^3 \plus{} \cdots \] What are the first three digits to the right of the decimal point in the decimal representation of $ \left(10^{2002} \plus{} 1\right)^{10/7}?$

1968 AMC 12/AHSME, 21

Tags: factorial , modular arithmetic

If $S=1!+2!+3!+ \cdots +99!$, then the units' digit in the value of $S$ is: $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 0$

2012 JBMO ShortLists, 7

Tags: modular arithmetic

Find all $a , b , c \in \mathbb{N}$ for which \[1997^a+15^b=2012^c\]

2009 AIME Problems, 13

Tags: algebra , trigonometry , modular arithmetic , search , polynomial , vieta

Let $ A$ and $ B$ be the endpoints of a semicircular arc of radius $ 2$. The arc is divided into seven congruent arcs by six equally spaced points $ C_1,C_2,\ldots,C_6$. All chords of the form $ \overline{AC_i}$ or $ \overline{BC_i}$ are drawn. Let $ n$ be the product of the lengths of these twelve chords. Find the remainder when $ n$ is divided by $ 1000$.

2010 AMC 12/AHSME, 20

Tags: modular arithmetic , number theory , greatest common divisor

Arithmetic sequences $ (a_n)$ and $ (b_n)$ have integer terms with $ a_1 \equal{} b_1 \equal{} 1 < a_2 \le b_2$ and $ a_nb_n \equal{} 2010$ for some $ n$. What is the largest possible value of $ n$? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 288 \qquad \textbf{(E)}\ 2009$

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 $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad \textbf{(E) }\text{none of these}$

2012 France Team Selection Test, 3

Tags: number theory , modular arithmetic , diophantine equation

Let $p$ be a prime number. Find all positive integers $a,b,c\ge 1$ such that: \[a^p+b^p=p^c.\]

2016 Iran MO (3rd Round), 3

Tags: number theory , modular arithmetic

Let $m$ be a positive integer. The positive integer $a$ is called a [i]golden residue[/i] modulo $m$ if $\gcd(a,m)=1$ and $x^x \equiv a \pmod m$ has a solution for $x$. Given a positive integer $n$, suppose that $a$ is a golden residue modulo $n^n$. Show that $a$ is also a golden residue modulo $n^{n^n}$. [i]Proposed by Mahyar Sefidgaran[/i]