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: 15460

2019 Taiwan TST Round 2, 2
Tags: number theory
Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.

2019 Kosovo National Mathematical Olympiad, 1
Tags: number theory
Find last three digits of the number $\frac{2019!}{2^{1009}}$ .

2015 Harvard-MIT Mathematics Tournament, 9
Tags: divisibility , algebra , number theory
Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A,B,C,D)\in\{1,2,\ldots,N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3+Bn^2+2Cn+D$ is divisible by $N$.

2003 Federal Competition For Advanced Students, Part 2, 3
Tags: geometry , number theory
For every lattice point $(x, y)$ with $x, y$ non-negative integers, a square of side $\frac{0.9}{2^x5^y}$ with center at the point $(x, y)$ is constructed. Compute the area of the union of all these squares.

2009 Italy TST, 3
Tags: geometry , number theory
Find all pairs of integers $(x,y)$ such that \[ y^3=8x^6+2x^3y-y^2.\]

2020-IMOC, N5
Tags: functional equation , number theory
$\textbf{N5.}$ Find all $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all $a,b,c \in \mathbb{N}$ $f(a)+f(b)+f(c)-ab-bc-ca \mid af(a)+bf(b)+cf(c)-3abc$

2023 Argentina National Olympiad Level 2, 2
Tags: number theory , combinatorics
Given the number $720$, Juan must choose $4$ numbers that are divisors of $720$. He wins if none of the four chosen numbers is a divisor of the product of the other three. Decide whether Juan can win.

2007 Pre-Preparation Course Examination, 14
Tags: inequalities , relatively prime , symmetry , number theory
Find all $a,b,c \in \mathbb{N}$ such that \[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad c^2a|a^3+b^3+c^3.\] [PS: The original problem was this: Find all $a,b,c \in \mathbb{N}$ such that \[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad \color{red}{c^2b}|a^3+b^3+c^3.\] But I think the author meant $c^2a|a^3+b^3+c^3$, just because of symmetry]

2013 CHMMC (Fall), 10
Tags: number theory
Compute the lowest positive integer $k$ such that none of the numbers in the sequence $$\{1, 1 +k, 1 + k + k^2 , 1 + k + k^2 + k^3, ... \}$$ are prime.

2021 Philippine MO, 4
Tags: polynomial , number theory , algebra
Determine the set of all polynomials $P(x)$ with real coefficients such that the set $\{P(n) | n \in \mathbb{Z}\}$ contains all integers, except possibly finitely many of them.

1998 Taiwan National Olympiad, 5
Tags: number theory
For a positive integer $n$, let $\omega(n)$ denote the number of positive prime divisors of $n$. Find the smallest positive tinteger $k$ such that $2^{\omega(n)}\leq k\sqrt[4]{n}\forall n\in\mathbb{N}$.

1952 Moscow Mathematical Olympiad, 224
Tags: number theory , perfect square
a) Prove that if the square of a number begins with $0.\underbrace{\hbox{9...9}}_{\hbox{100}}$, then the number itself begins with $0.\underbrace{\hbox{9...9}}_{\hbox{100}}$,. b) Calculate $\sqrt{0.9...9}$ ($60$ nines) to $60$ decimal places

2010 Middle European Mathematical Olympiad, 4
Tags: inequalities , number theory
Find all positive integers $n$ which satisfy the following tow conditions: (a) $n$ has at least four different positive divisors; (b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)[/i]

2001 Slovenia National Olympiad, Problem 2
Tags: number theory
Let $p(n)$ denote the product of decimal digits of a positive integer $n$. Computer the sum $p(1)+p(2)+\ldots+p(2001)$.

2009 USA Team Selection Test, 5
Tags: algebra , polynomial , quadratic , vieta , inequalities , number theory
Find all pairs of positive integers $ (m,n)$ such that $ mn \minus{} 1$ divides $ (n^2 \minus{} n \plus{} 1)^2$. [i]Aaron Pixton.[/i]

2005 Junior Balkan Team Selection Tests - Romania, 2
Tags: inequalities , number theory
Find the largest positive integer $n>10$ such that the residue of $n$ when divided by each perfect square between $2$ and $\dfrac n2$ is an odd number.

2011 HMNT, 4
Tags: number theory
Determine the remainder when $$2^{\frac {1 \cdot 2}{2}} + 2^{\frac {2 \cdot 3}{2}}+ ...+ 2^{\frac {2011 \cdot 2012}{2}}$$ is divided by $7$.

2016 International Zhautykov Olympiad, 2
Tags: number theory
$a_1,a_2,...,a_{100}$ are permutation of $1,2,...,100$. $S_1=a_1, S_2=a_1+a_2,...,S_{100}=a_1+a_2+...+a_{100}$Find the maximum number of perfect squares from $S_i$

2025 Belarusian National Olympiad, 8.3
Tags: number theory
A positive integer with three digits is written on the board. Each second the number $n$ on the board gets replaced by $n+\frac{n}{p}$, where $p$ is the largest prime divisor of $n$. Prove that either after 999 seconds or 1000 second the number on the board will be a power of two. [i]A. Voidelevich[/i]

2021 Bangladesh Mathematical Olympiad, Problem 7
Tags: combinatorics , number theory
A binary string is a word containing only $0$s and $1$s. In a binary string, a $1-$run is a non extendable substring containing only $1$s. Given a positive integer $n$, let $B(n)$ be the number of $1-$runs in the binary representation of $n$. For example, $B(107)=3$ since $107$ in binary is $1101011$ which has exactly three $1-$runs. What is the following expression equal to? $$B(1)+B(2)+B(3)+ \dots + B(255)$$

1986 IMO Longlists, 67
Tags: number theory , decimal representation , rational number , algebra
Let $f(x) = x^n$ where $n$ is a fixed positive integer and $x =1, 2, \cdots .$ Is the decimal expansion $a = 0.f (1)f(2)f(3) . . .$ rational for any value of $n$ ? The decimal expansion of a is defined as follows: If $f(x) = d_1(x)d_2(x) \cdots d_{r(x)}(x)$ is the decimal expansion of $f(x)$, then $a = 0.1d_1(2)d_2(2) \cdots d_{r(2)}(2)d_1(3) . . . d_{r(3)}(3)d_1(4) \cdots .$

2004 AIME Problems, 13
Tags: relatively prime , number theory , ratio , geometry
Let $ABCDE$ be a convex pentagon with $AB\parallel CE$, $BC\parallel AD$, $AC\parallel DE$, $\angle ABC=120^\circ$, $AB=3$, $BC=5$, and $DE=15$. Given that the ratio between the area of triangle $ABC$ and the area of triangle $EBD$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

2022 May Olympiad, 3
Tags: number theory
Choose nine of the digits from $0$ to $9$ and place them in the boxes in the figure so that there are no repeated digits and the indicated sum is correct. [img]https://cdn.artofproblemsolving.com/attachments/6/2/7f06575ec70eb9ddd58c6cf9dd3cb60d306e7c.png[/img] Which digit was not used? You can fill in the boxes so that the unused digit is other?

1989 Tournament Of Towns, (236) 4
Tags: number theory , digit
The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether? (G. Galperin)

2025 Malaysian IMO Training Camp, 1
Tags: number theory
Given two primes $p$ and $q$, is $v_p(q^n+n^q)$ unbounded as $n$ varies? [i](Proposed by Ivan Chan Kai Chin)[/i]