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: 15460

2007 Pre-Preparation Course Examination, 4
Tags: number theory
$a,b \in \mathbb Z$ and for every $n \in \mathbb{N}_0$, the number $2^na+b$ is a perfect square. Prove that $a=0$.

1995 Cono Sur Olympiad, 3
Tags: floor function , function , modular arithmetic , inequalities , number theory , relatively prime
Let $n$ be a natural number and $f(n) = 2n - 1995 \lfloor \frac{n}{1000} \rfloor$($\lfloor$ $\rfloor$ denotes the floor function). 1. Show that if for some integer $r$: $f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times), then $n$ is multiple of $1995$. 2. Show that if $n$ is multiple of 1995, then there exists r such that:$f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times). Determine $r$ if $n=1995.500=997500$

2012 Danube Mathematical Competition, 1
Tags: perfect square , number theory
a) Exist $a, b, c, \in N$, such that the numbers $ab+1,bc+1$ and $ca+1$ are simultaneously even perfect squares ? b) Show that there is an infinity of natural numbers (distinct two by two) $a, b, c$ and $d$, so that the numbers $ab+1,bc+1, cd+1$ and $da+1$ are simultaneously perfect squares.

2010 Contests, 2
Tags: number theory , perfect power , additive number theory
For a positive integer $k,$ call an integer a $pure$ $k-th$ $power$ if it can be represented as $m^k$ for some integer $m.$ Show that for every positive integer $n,$ there exists $n$ distinct positive integers such that their sum is a pure $2009-$th power and their product is a pure $2010-$th power.

2023 Puerto Rico Team Selection Test, 1
Tags: number theory
A number is [i]capicua [/i] if it is read equally from left to right as it is from right to the left. For example, $23432$ and $111111$ are capicua numbers. (a) How many $2023$-digit capicua numbers can be formed if you want them to have at least $2022$ equal digits? (b) How many $2023$-digit capicua numbers can be formed if you want them to have at least $2021$ equal digits?

2000 AIME Problems, 13
Tags: geometry , calculus , number theory , relatively prime
In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of this region is $m/n$ square miles, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

2023 Singapore Junior Math Olympiad, 2
Tags: number theory
What is the maximum number of integers that can be chosen from $1,2,\dots,99$ so that the chosen integers can be arranged in a circle with the property that the product of every pair of neighbouring integers is 3-digit number?

2015 Azerbaijan JBMO TST, 2
Tags: number theory
All letters in the word $VUQAR$ are different and chosen from the set $\{1,2,3,4,5\}$. Find all solutions to the equation \[\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\]

2007 Iran Team Selection Test, 2
Tags: algebra , polynomial , search , number theory
Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication. [i]By Mohsen Jamali[/i]

2017 Harvard-MIT Mathematics Tournament, 3
Tags: number theory
Find the number of integers $n$ with $1 \le n \le 2017$ so that $(n-2)(n-0)(n-1)(n-7)$ is an integer multiple of $1001$.

2024 Myanmar IMO Training, 6
Tags: induction , number theory
Prove that for all integers $n \geq 3$, there exist odd positive integers $x$, $y$ such that $7x^2 + y^2 = 2^n$.

1992 Czech And Slovak Olympiad IIIA, 3
Tags: sum of digits , sum , digit , number theory
Let $S(n)$ denote the sum of digits of $n \in N$. Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$

2012 Princeton University Math Competition, A7
Tags: number theory
Let $a, b$, and $c$ be positive integers satisfying $a^4 + a^2b^2 + b^4 = 9633$ $2a^2 + a^2b^2 + 2b^2 + c^5 = 3605$. What is the sum of all distinct values of $a + b + c$?

1981 Spain Mathematical Olympiad, 8
Tags: number theory , sum of squares , divisible
If $a$ is an odd number, show that $$a^4 + 4a^3 + 11a^2 + 6a+ 2$$ is a sum of three squares and is divisible by $4$.

2006 Italy TST, 2
Tags: modular arithmetic , euler , number theory
Let $n$ be a positive integer, and let $A_{n}$ be the the set of all positive integers $a\le n$ such that $n|a^{n}+1$. a) Find all $n$ such that $A_{n}\neq \emptyset$ b) Find all $n$ such that $|{A_{n}}|$ is even and non-zero. c) Is there $n$ such that $|{A_{n}}| = 130$?

2012 Princeton University Math Competition, B3
Tags: number theory
Find, with proof, all pairs $(x, y)$ of integers satisfying the equation $3x^2+ 4 = 2y^3$.

2020 SJMO, 6
Tags: number theory , combinatorics
We say a positive integer $n$ is [i]$k$-tasty[/i] for some positive integer $k$ if there exists a permutation $(a_0, a_1, a_2, \ldots , a_n)$ of $(0,1,2, \ldots, n)$ such that $|a_{i+1} - a_i| \in \{k, k+1\}$ for all $0 \le i \le n-1$. Prove that for all positive integers $k$, there exists a constant $N$ such that all integers $n \geq N$ are $k$-tasty. [i]Proposed by Anthony Wang[/i]

2003 Korea Junior Math Olympiad, 5
Tags: integer , number theory
Four odd positive intgers $a, b, c, d (a\leq b \leq c\leq d)$ are given. Choose any three numbers among them and divide their sum by the un-chosen number, and you will always get the remainder as $1$. Find all $(a, b, c, d)$ that satisfies this.

Russian TST 2015, P1
Tags: number theory , prime number
Prove that there exist two natural numbers $a,b$ such that $|a-m|+|b-n|>1000$ for any relatively prime natural numbers $m,n$.

1992 Rioplatense Mathematical Olympiad, Level 3, 2
Tags: diophantine , diophantine equation , number theory
Determine the integers $0 \le a \le b \le c \le d$ such that: $$2^n= a^2 + b^2 + c^2 + d^2.$$

2023 Regional Olympiad of Mexico West, 4
Tags: number theory , perfect square , perfect cube , perfect power
Prove that you can pick $15$ distinct positive integers between $1$ and $2023$, such that each one of them and the sum between some of them is never a perfect square, nor a perfect cube or any other greater perfect power.

2016 Postal Coaching, 5
Tags: set theory , group theory , binary operation , number theory
Is it possible to define an operation $\star$ on $\mathbb Z$ such that[list=a][*] for any $a, b, c$ in $\mathbb Z, (a \star b) \star c = a \star (b \star c)$ holds; [*] for any $x, y$ in $\mathbb Z, x \star x \star y = y \star x \star x=y$?[/list]

2007 Regional Olympiad of Mexico Center Zone, 6
Tags: number theory , digit
Certain tickets are numbered as follows: $1, 2, 3, \dots, N $. Exactly half of the tickets have the digit $ 1$ on them. If $N$ is a three-digit number, determine all possible values ​​of $N $.

2012 International Zhautykov Olympiad, 1
Tags: function , induction , number theory
Do there exist integers $m, n$ and a function $f\colon \mathbb R \to \mathbb R$ satisfying simultaneously the following two conditions? $\bullet$ i) $f(f(x))=2f(x)-x-2$ for any $x \in \mathbb R$; $\bullet$ ii) $m \leq n$ and $f(m)=n$.

1969 IMO Shortlist, 63
Tags: quadratic , number theory , additive number theory
$(SWE 6)$ Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.