Olympiad problems

2007 Harvard-MIT Mathematics Tournament, 27

Tags: modular arithmetic

Find the number of $7$-tuples $(n_1,\ldots,n_7)$ of integers such that \[\sum_{i=1}^7 n_i^6=96957.\]

2009 District Round (Round II), 1

Tags: modular arithmetic , number theory

given a 4-digit number $(abcd)_{10}$ such that both$(abcd)_{10}$and$(dcba)_{10}$ are multiples of $7$,having the same remainder modulo $37$.find $a,b,c,d$.

2013 Online Math Open Problems, 45

Tags: algebra , polynomial , modular arithmetic , calculus , integration

Let $N$ denote the number of ordered 2011-tuples of positive integers $(a_1,a_2,\ldots,a_{2011})$ with $1\le a_1,a_2,\ldots,a_{2011} \le 2011^2$ such that there exists a polynomial $f$ of degree $4019$ satisfying the following three properties: [list] [*] $f(n)$ is an integer for every integer $n$; [*] $2011^2 \mid f(i) - a_i$ for $i=1,2,\ldots,2011$; [*] $2011^2 \mid f(n+2011) - f(n)$ for every integer $n$. [/list] Find the remainder when $N$ is divided by $1000$. [i]Victor Wang[/i]

2011 USAJMO, 1

Tags: modular arithmetic , quadratic , number theory

Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.

2005 Germany Team Selection Test, 3

Tags: inequalities , modular arithmetic , algorithm , induction , number theory

We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.

2007 Pre-Preparation Course Examination, 9

Tags: modular arithmetic , number theory

Solve the equation $4xy-x-y=z^2$ in positive integers.

1998 IberoAmerican, 3

Tags: floor function , modular arithmetic , algebra

Let $\lambda$ the positive root of the equation $t^2-1998t-1=0$. It is defined the sequence $x_0,x_1,x_2,\ldots,x_n,\ldots$ by $x_0=1,\ x_{n+1}=\lfloor\lambda{x_n}\rfloor\mbox{ for }n=1,2\ldots$ Find the remainder of the division of $x_{1998}$ by $1998$. Note: $\lfloor{x}\rfloor$ is the greatest integer less than or equal to $x$.

2008 AIME Problems, 8

Tags: trigonometry , modular arithmetic

Let $ a\equal{}\pi/2008$. Find the smallest positive integer $ n$ such that \[ 2[\cos(a)\sin(a)\plus{}\cos(4a)\sin(2a)\plus{}\cos(9a)\sin(3a)\plus{}\cdots\plus{}\cos(n^2a)\sin(na)]\] is an integer.

2014 NIMO Problems, 5

Tags: modular arithmetic

Let a positive integer $n$ be $\textit{nice}$ if there exists a positive integer $m$ such that \[ n^3 < 5mn < n^3 +100. \] Find the number of [i]nice[/i] positive integers. [i]Proposed by Akshaj[/i]

2015 VJIMC, 2

Tags: number theory , modular arithmetic , real analysis

[b]Problem 2[/b] Determine all pairs $(n, m)$ of positive integers satisfying the equation $$5^n = 6m^2 + 1\ . $$

2018 PUMaC Number Theory A, 8

Tags: number theory , modular arithmetic

Let $p$ be a prime. Let $f(x)$ be the number of ordered pairs $(a, b)$ of positive integers less than $p$, such that $a^b \equiv x \pmod p$. Suppose that there do not exist positive integers $x$ and $y$, both less than $p$, such that $f(x) = 2f(y)$, and that the maximum value of $f$ is greater than $2018$. Find the smallest possible value of $p$.

1986 Canada National Olympiad, 5

Tags: pigeonhole principle , modular arithmetic , induction , algebra

Let $u_1$, $u_2$, $u_3$, $\dots$ be a sequence of integers satisfying the recurrence relation $u_{n + 2} = u_{n + 1}^2 - u_n$. Suppose $u_1 = 39$ and $u_2 = 45$. Prove that 1986 divides infinitely many terms of the sequence.

2001 National Olympiad First Round, 15

Tags: modular arithmetic

How many different solutions does the congruence $x^3+3x^2+x+3 \equiv 0 \pmod{25}$ have? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6 $

1993 Turkey Team Selection Test, 1

Tags: modular arithmetic , arithmetic sequence , number theory

Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference.

2017 Iran Team Selection Test, 4

Tags: number theory , prime number , modular arithmetic

We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$. Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$. Is there always a number $x$ that satisfies all the equations? [i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]

1987 India National Olympiad, 3

Tags: modular arithmetic , algebra

Let $ T$ be the set of all triplets $ (a,b,c)$ of integers such that $ 1 \leq a < b < c \leq 6$ For each triplet $ (a,b,c)$ in $ T$, take number $ a\cdot b \cdot c$. Add all these numbers corresponding to all the triplets in $ T$. Prove that the answer is divisible by 7.

2012 China Western Mathematical Olympiad, 1

Tags: modular arithmetic , prime number , number theory

Find the smallest positive integer $m$ satisfying the following condition: for all prime numbers $p$ such that $p>3$,have $105|9^{ p^2}-29^p+m.$ (September 28, 2012, Hohhot)

2007 ITest, 20

Tags: modular arithmetic

Find the largest integer $n$ such that $2007^{1024}-1$ is divisible by $2^n$. $\textbf{(A) }1\hspace{14em}\textbf{(B) }2\hspace{14em}\textbf{(C) }3$ $\textbf{(D) }4\hspace{14em}\textbf{(E) }5\hspace{14em}\textbf{(F) }6$ $\textbf{(G) }7\hspace{14em}\textbf{(H) }8\hspace{14em}\textbf{(I) }9$ $\textbf{(J) }10\hspace{13.7em}\textbf{(K) }11\hspace{13.5em}\textbf{(L) }12$ $\textbf{(M) }13\hspace{13.3em}\textbf{(N) }14\hspace{13.4em}\textbf{(O) }15$ $\textbf{(P) }16\hspace{13.6em}\textbf{(Q) }55\hspace{13.4em}\textbf{(R) }63$ $\textbf{(S) }64\hspace{13.7em}\textbf{(T) }2007$

2001 USA Team Selection Test, 4

Tags: modular arithmetic , induction , pigeonhole principle , combinatorics

There are 51 senators in a senate. The senate needs to be divided into $n$ committees so that each senator is on one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does [i]not[/i] necessarily hate senator A.) Find the smallest $n$ such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee.

2008 Romania Team Selection Test, 1

Tags: modular arithmetic , number theory , relatively prime , combinatorics

Let $ n$ be an integer, $ n\geq 2$. Find all sets $ A$ with $ n$ integer elements such that the sum of any nonempty subset of $ A$ is not divisible by $ n\plus{}1$.

2008 JBMO Shortlist, 7

Tags: modular arithmetic , quadratic , number theory

Determine the minimum value of prime $p> 3$ for which there is no natural number $n> 0$ such that $2^n+3^n\equiv 0\pmod{p} $.

1979 IMO Longlists, 45

Tags: modular arithmetic , function , number theory

For any positive integer $n$, we denote by $F(n)$ the number of ways in which $n$ can be expressed as the sum of three different positive integers, without regard to order. Thus, since $10 = 7+2+1 = 6+3+1 = 5+4+1 = 5+3+2$, we have $F(10) = 4$. Show that $F(n)$ is even if $n \equiv 2$ or $4 \pmod 6$, but odd if $n$ is divisible by $6$.

2013 Romania Team Selection Test, 1

Tags: geometry , 3d geometry , modular arithmetic , symmetry , algebra , polynomial , number theory

Given an integer $n\geq 2,$ let $a_{n},b_{n},c_{n}$ be integer numbers such that \[ \left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}. \] Prove that $c_{n}\equiv 1\pmod{3} $ if and only if $n\equiv 2\pmod{3}.$

1982 IMO Longlists, 40

Tags: geometry , rectangle , analytic geometry , modular arithmetic , combinatorics

We consider a game on an infinite chessboard similar to that of solitaire: If two adjacent fields are occupied by pawns and the next field is empty (the three fields lie on a vertical or horizontal line), then we may remove these two pawns and put one of them on the third field. Prove that if in the initial position pawns fill a $3k \times n$ rectangle, then it is impossible to reach a position with only one pawn on the board.

1998 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 10

Tags: modular arithmetic

The number of pairs of integers $ (m,n)$ satisfying the equation \[ m^3 \plus{} 6m^2 \plus{} 5m \equal{} 27n^3 \plus{} 9n^2 \plus{} 9n \plus{} 1\] is $ \text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ \text{Infinitely many}$