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: 1766

2008 Czech and Slovak Olympiad III A, 1
Tags: modular arithmetic , calculus , number theory proposed , number theory
In decimal representation, we call an integer [i]$k$-carboxylic[/i] if and only if it can be represented as a sum of $k$ distinct integers, all of them greater than $9$, whose digits are the same. For instance, $2008$ is [i]$5$-carboxylic[/i] because $2008=1111+666+99+88+44$. Find, with an example, the smallest integer $k$ such that $8002$ is [i]$k$-carboxylic[/i].

1974 IMO Longlists, 7
Tags: number theory proposed , number theory
Let $p$ be a prime number and $n$ a positive integer. Prove that the product \[{N=\frac{1}{p^{n^2}}} \prod_{i=1;2 \nmid i}^{2n-1} \biggl[ \left( (p-1)! \right) \binom{p^2 i}{pi}\biggr]\] Is a positive integer that is not divisible by $p.$

2014 BMO TST, 5
Tags: modular arithmetic , number theory proposed , number theory
Find all non-negative integers $k,n$ which satisfy $2^{2k+1} + 9\cdot 2^k+5=n^2$.

2011 Canadian Students Math Olympiad, 2
Tags: quadratics , number theory proposed , number theory
For a fixed positive integer $k$, prove that there exist infinitely many primes $p$ such that there is an integer $w$, where $w^2-1$ is not divisible by $p$, and the order of $w$ in modulus $p$ is the same as the order of $w$ in modulus $p^k$. [i]Author: James Rickards[/i]

2008 CentroAmerican, 1
Tags: number theory proposed , number theory
Find the least positive integer $ N$ such that the sum of its digits is 100 and the sum of the digits of $ 2N$ is 110.

1999 Federal Competition For Advanced Students, Part 2, 1
Tags: floor function , logarithms , number theory proposed , number theory
Prove that for each positive integer $n$, the sum of the numbers of digits of $4^n$ and of $25^n$ (in the decimal system) is odd.

2008 Indonesia MO, 1
Tags: number theory proposed , number theory
Let $ m,n > 1$ are integers which satisfy $ n|4^m \minus{} 1$ and $ 2^m|n \minus{} 1$. Is it a must that $ n \equal{} 2^{m} \plus{} 1$?

2007 Indonesia TST, 3
Tags: inequalities , floor function , Diophantine equation , number theory proposed , number theory
For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.

1997 Taiwan National Olympiad, 7
Tags: function , number theory proposed , number theory
Find all positive integers $k$ for which there exists a function $f: \mathbb{N}\to\mathbb{Z}$ satisfying $f(1997)=1998$ and $f(ab)=f(a)+f(b)+kf(\gcd{(a,b)})\forall a,b$.

2024 Germany Team Selection Test, 2
Tags: number theory , number theory proposed , Digits , product of digits , Divisibility
Show that there exists a real constant $C>1$ with the following property: For any positive integer $n$, there are at least $C^n$ positive integers with exactly $n$ decimal digits, which are divisible by the product of their digits. (In particular, these $n$ digits are all non-zero.) [i]Proposed by Jean-Marie De Koninck and Florian Luca[/i]

1993 All-Russian Olympiad, 1
Tags: number theory proposed , number theory
For a positive integer $n$, numbers $2n+1$ and $3n+1$ are both perfect squares. Is it possible for $5n+3$ to be prime?

2001 India IMO Training Camp, 1
Tags: algebra , polynomial , complex numbers , number theory proposed , number theory
Complex numbers $\alpha$ , $\beta$ , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$. Prove that the polynomial \[(x-\alpha)(x-\beta )(x-\gamma )\] has integer coefficients.

2021 Baltic Way, 16
Tags: number theory , number theory proposed
Show that no non-zero integers $a$, $b$, $x$, $y$ satisfy $$ \begin{cases} a x - b y = 16,\\ a y + b x = 1. \end{cases} $$

2010 China Western Mathematical Olympiad, 1
Tags: modular arithmetic , number theory proposed , number theory
Suppose that $m$ and $k$ are non-negative integers, and $p = 2^{2^m}+1$ is a prime number. Prove that [b](a)[/b] $2^{2^{m+1}p^k} \equiv 1$ $(\text{mod } p^{k+1})$; [b](b)[/b] $2^{m+1}p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1$ $(\text{mod } p^{k+1})$.

2004 Czech-Polish-Slovak Match, 6
Tags: greatest common divisor , number theory proposed , number theory
On the table there are $k \ge 3$ heaps of $1, 2, \dots , k$ stones. In the first step, we choose any three of the heaps, merge them into a single new heap, and remove $1$ stone from this new heap. Thereafter, in the $i$-th step ($i \ge 2$) we merge some three heaps containing more than $i$ stones in total and remove $i$ stones from the new heap. Assume that after a number of steps a single heap of $p$ stones remains on the table. Show that the number $p$ is a perfect square if and only if so are both $2k + 2$ and $3k + 1$. Find the least $k$ with this property.

2013 ITAMO, 1
Tags: number theory proposed , number theory
A model car is tested on some closed circuit $600$ meters long, consisting of flat stretches, uphill and downhill. All uphill and downhill have the same slope. The test highlights the following facts: [list] (a) The velocity of the car depends only on the fact that the car is driving along a stretch of uphill, plane or downhill; calling these three velocities $v_s, v_p$ and $v_d$ respectively, we have $v_s <v_p <v_d$; (b) $v_s,v_p$ and $v_d$, expressed in meter per second, are integers. (c) Whatever may be the structure of the circuit, the time taken to complete the circuit is always $50$ seconds. [/list] Find all possible values of $v_s, v_p$ and $v_d$.

2013 Hitotsubashi University Entrance Examination, 1
Tags: number theory proposed , number theory
Find all pairs $(p,\ q)$ of positive integers such that $3p^3-p^2q-pq^2+3q^3=2013.$

2013 IberoAmerican, 5
Tags: number theory proposed , number theory
Let $A$ and $B$ be two sets such that $A \cup B$ is the set of the positive integers, and $A \cap B$ is the empty set. It is known that if two positive integers have a prime larger than $2013$ as their difference, then one of them is in $A$ and the other is in $B$. Find all the possibilities for the sets $A$ and $B$.

1990 IMO Longlists, 84
Tags: number theory proposed , number theory
Let $n \geq 4$ be an integer. $a_1, a_2, \ldots, a_n \in (0, 2n)$ are $n$ distinct integers. Prove that there exists a subset of the set $\{a_1, a_2, \ldots, a_n \}$ such that the sum of its elements is divisible by $2n.$

2021 Baltic Way, 17
Tags: number theory , number theory proposed , primes
Distinct positive integers $a, b, c, d$ satisfy $$\begin{cases} a \mid b^2 + c^2 + d^2,\\ b\mid a^2 + c^2 + d^2,\\ c \mid a^2 + b^2 + d^2,\\ d \mid a^2 + b^2 + c^2,\end{cases}$$ and none of them is larger than the product of the three others. What is the largest possible number of primes among them?

2012 ITAMO, 4
Tags: modular arithmetic , quadratics , number theory proposed , number theory
Let $x_1,x_2,x_3, \cdots$ be a sequence defined by the following recurrence relation: \[ \begin{cases}x_{1}&= 4\\ x_{n+1}&= x_{1}x_{2}x_{3}\cdots x_{n}+5\text{ for }n\ge 1\end{cases} \] The first few terms of the sequence are $x_1=4,x_2=9,x_3=41 \cdots$ Find all pairs of positive integers $\{a,b\}$ such that $x_a x_b$ is a perfect square.

2010 Slovenia National Olympiad, 2
Tags: number theory proposed , number theory
Find all prime numbers $p, q, r$ such that \[15p+7pq+qr=pqr.\]

2024 ITAMO, 3
Tags: number theory , number theory proposed , egyptian fractions , reciprocal sum
A positive integer $n$ is called [i]egyptian[/i] if there exists a strictly increasing sequence $0<a_1<a_2<\dots<a_k=n$ of integers with last term $n$ such that \[\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_k}=1.\] (a) Determine if $n=72$ is egyptian. (b) Determine if $n=71$ is egyptian. (c) Determine if $n=72^{71}$ is egyptian.

2011 ISI B.Math Entrance Exam, 5
Tags: inequalities , number theory proposed , number theory
Consider a sequence denoted by $F_n$ of non-square numbers . $F_1=2$,$F_2=3$,$F_3=5$ and so on . Now , if $m^2\leq F_n<(m+1)^2$ . Then prove that $m$ is the integer closest to $\sqrt{n}$.

2007 Polish MO Finals, 2
Tags: number theory proposed , number theory
2. Positive integer will be called white, if it is equal to $1$ or is a product of even number of primes (not necessarily distinct). Rest of the positive integers will be called black. Determine whether there exists a positive integer which sum of white divisors is equal to sum of black divisors