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.

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Found problems: 2008

2007 JBMO Shortlist, 5
Tags: quadratic , modular arithmetic , number theory
Prove that if $ p$ is a prime number, then $ 7p+3^{p}-4$ is not a perfect square.

2016 China National Olympiad, 3
Tags: algebra , polynomial , modular arithmetic , number theory
Let $p$ be an odd prime and $a_1, a_2,...,a_p$ be integers. Prove that the following two conditions are equivalent: 1) There exists a polynomial $P(x)$ with degree $\leq \frac{p-1}{2}$ such that $P(i) \equiv a_i \pmod p$ for all $1 \leq i \leq p$ 2) For any natural $d \leq \frac{p-1}{2}$, $$ \sum_{i=1}^p (a_{i+d} - a_i )^2 \equiv 0 \pmod p$$ where indices are taken $\pmod p$

1972 IMO Shortlist, 6
Tags: modular arithmetic , number theory , decimal representation , digit
Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.

2023 AMC 12/AHSME, 5
Tags: modular arithmetic
You are playing a game. A $2 \times 1$ rectangle covers two adjacent squares (oriented either horizontally or vertically) of a $3 \times 3$ grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensue that at least one of your guessed squares is covered by the rectangle? $\textbf{(A)}~3\qquad\textbf{(B)}~5\qquad\textbf{(C)}~4\qquad\textbf{(D)}~8\qquad\textbf{(E)}~6$

1985 IMO Shortlist, 13
Tags: algorithm , relatively prime , modular arithmetic , combinatorics , invariant
Let $m$ boxes be given, with some balls in each box. Let $n < m$ be a given integer. The following operation is performed: choose $n$ of the boxes and put $1$ ball in each of them. Prove: [i](a) [/i]If $m$ and $n$ are relatively prime, then it is possible, by performing the operation a finite number of times, to arrive at the situation that all the boxes contain an equal number of balls. [i](b)[/i] If $m$ and $n$ are not relatively prime, there exist initial distributions of balls in the boxes such that an equal distribution is not possible to achieve.

2010 AMC 12/AHSME, 19
Tags: modular arithmetic , ratio , geometric sequence
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $ 100$ points. What was the total number of points scored by the two teams in the first half? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$

2012 JBMO ShortLists, 7
Tags: modular arithmetic
Find all $a , b , c \in \mathbb{N}$ for which \[1997^a+15^b=2012^c\]

1999 Dutch Mathematical Olympiad, 2
Tags: geometry , rectangle , modular arithmetic
A $9 \times 9$ square consists of $81$ unit squares. Some of these unit squares are painted black, and the others are painted white, such that each $2 \times 3$ rectangle and each $3 \times 2$ rectangle contain exactly 2 black unit squares and 4 white unit squares. Determine the number of black unit squares.

2001 IMO, 4
Tags: modular arithmetic , combinatorics , factorial , global
Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$ be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of $\{1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist permutations $a \neq b$ of $\{1,2,\ldots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.

2000 Spain Mathematical Olympiad, 1
Tags: modular arithmetic , floor function , number theory
Find the largest integer $N$ satisfying the following two conditions: [b](i)[/b] $\left[ \frac N3 \right]$ consists of three equal digits; [b](ii)[/b] $\left[ \frac N3 \right] = 1 + 2 + 3 +\cdots + n$ for some positive integer $n.$

2014 JBMO Shortlist, 2
Tags: diophantine equation , modular arithmetic
Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.

2011 Romania Team Selection Test, 3
Tags: ratio , floor function , modular arithmetic , number theory
Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.

2013 Saint Petersburg Mathematical Olympiad, 7
Tags: modular arithmetic , number theory
Given is a natural number $a$ with $54$ digits, each digit equal to $0$ or $1$. Prove the remainder of $a$ when divide by $ 33\cdot 34\cdots 39 $ is larger than $100000$. [hide](It's mean: $a \equiv r \pmod{33\cdot 34\cdots 39 }$ with $ 0<r<33\cdot 34\cdots 39 $ then prove that $r>100000$ )[/hide] M. Antipov

2002 Moldova National Olympiad, 2
Tags: modular arithmetic
Does there exist a positive integer $ n>1$ such that $ n$ is a power of $ 2$ and one of the numbers obtained by permuting its (decimal) digits is a power of $ 3$ ?

2003 AMC 8, 12
Tags: probability , modular arithmetic
When a fair six-sided dice is tossed on a table top, the bottom face cannot be seen. What is the probability that the product of the $5$ faces than can be seen is divisible by $6$? $\textbf{(A)}\ 1/3 \qquad \textbf{(B)}\ 1/2 \qquad \textbf{(C)}\ 2/3 \qquad \textbf{(D)}\ 5/6 \qquad \textbf{(E)}\ 1$

2007 Hong kong National Olympiad, 2
Tags: algebra , polynomial , modular arithmetic , number theory
is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?

1998 Nordic, 4
Tags: combinatorics , modular arithmetic , binomial coefficient
Let $n$ be a positive integer. Count the number of numbers $k \in \{0, 1, 2, . . . , n\}$ such that $\binom{n}{k}$ is odd. Show that this number is a power of two, i.e. of the form $2^p$ for some nonnegative integer $p$.

2002 Austrian-Polish Competition, 5
Tags: polynomial , modular arithmetic , algebra
Let $A$ be the set $\{2,7,11,13\}$. A polynomial $f$ with integer coefficients possesses the following property: for each integer $n$ there exists $p \in A$ such that $p|f(n)$. Prove that there exists $p \in A$ such that $p|f(n)$ for all integers $n$.

2007 International Zhautykov Olympiad, 3
Tags: induction , modular arithmetic , number theory
Show that there are an infinity of positive integers $n$ such that $2^{n}+3^{n}$ is divisible by $n^{2}$.

1990 AIME Problems, 10
Tags: trigonometry , number theory , least common multiple , greatest common divisor , modular arithmetic , complex number , relatively prime
The sets $A = \{z : z^{18} = 1\}$ and $B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A \ \text{and} \ w \in B\}$ is also a set of complex roots of unity. How many distinct elements are in $C$?

2004 Germany Team Selection Test, 2
Tags: modular arithmetic , induction , number theory
Find all pairs of positive integers $\left(n;\;k\right)$ such that $n!=\left( n+1\right)^{k}-1$.

2008 ITest, 100
Tags: algebra , polynomial , modular arithmetic , function , galois theory
Let $\alpha$ be a root of $x^6-x-1$, and call two polynomials $p$ and $q$ with integer coefficients $\textit{equivalent}$ if $p(\alpha)\equiv q(\alpha)\pmod3$. It is known that every such polynomial is equivalent to exactly one of $0,1,x,x^2,\ldots,x^{727}$. Find the largest integer $n<728$ for which there exists a polynomial $p$ such that $p^3-p-x^n$ is equivalent to $0$.

2003 Baltic Way, 20
Tags: modular arithmetic , number theory
Suppose that the sum of all positive divisors of a natural number $n$, $n$ excluded, plus the number of these divisors is equal to $n$. Prove that $n = 2m^2$ for some integer $m$.

2013 India Regional Mathematical Olympiad, 6
Tags: function , modular arithmetic , combinatorics
For a natural number $n$, let $T(n)$ denote the number of ways we can place $n$ objects of weights $1,2,\cdots, n$ on a balance such that the sum of the weights in each pan is the same. Prove that $T(100) > T(99)$.

1996 AIME Problems, 10
Tags: trigonometry , modular arithmetic , algorithm , function
Find the smallest positive integer solution to $\tan 19x^\circ=\frac{\cos 96^\circ+\sin 96^\circ}{\cos 96^\circ-\sin 96^\circ}.$