Olympiad problems

2002 India IMO Training Camp, 14

Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$. Show that there are $p^2+1$ triples of integers $(x,y,z)$ with $0 \le x,y,z < p$ and such that $(x+y+z)^2 \equiv axyz \pmod p$

2003 May Olympiad, 1

Tags: number theory , multiple

Four digits $a, b, c, d$, different from each other and different from zero, are chosen and the list of all the four-digit numbers that are obtained by exchanging the digits $a, b, c, d$ is written. What digits must be chosen so that the list has the greatest possible number of four-digit numbers that are multiples of $36$?

2003 China Team Selection Test, 2

Tags: number theory , modular arithmetic , quadratic

Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.

1998 Czech And Slovak Olympiad IIIA, 4

Tags: max , number theory , power of 3

For each date of year $1998$, we calculate day$^{month}$ −year and determine the greatest power of $3$ that divides it. For example, for April $21$ we get $21^4 - 1998 =192483 = 3^3 \cdot 7129$, which is divisible by $3^3$ and not by $3^4$ . Find all dates for which this power of $3$ is the greatest.

1992 Chile National Olympiad, 2

Tags: number theory , algebra , sum

For a finite set of naturals $(C)$, the product of its elements is going to be noted $P(C)$. We are going to define $P (\phi) = 1$. Calculate the value of the expression $$\sum_{C \subseteq \{1,2,...,n\}} \frac{1}{P(C)}$$

2019 ELMO Problems, 5

Tags: number theory

Let $S$ be a nonempty set of positive integers such that, for any (not necessarily distinct) integers $a$ and $b$ in $S$, the number $ab+1$ is also in $S$. Show that the set of primes that do not divide any element of $S$ is finite. [i]Proposed by Carl Schildkraut[/i]

2018 NZMOC Camp Selection Problems, 1

Tags: multiple , number theory , divide

Suppose that $a, b, c$ and $d$ are four different integers. Explain why $$(a - b)(a - c)(a - d)(b - c)(b -d)(c - d)$$ must be a multiple of $12$.

1998 Romania National Olympiad, 1

Tags: algebra , perfect square , number theory , inequalities

Let $n$ be a positive integer and $x_1,x_2,...,x_n$ be integer numbers such that $$x_1^2+x_2^2+...+x_n^2+ n^3 \le (2n - 1)(x_1+x_2+...+x_n ) + n^2$$ . Show that : a) $x_1,x_2,...,x_n$ are non-negative integers b) the number $x_1+x_2+...+x_n+n+1$ is not a perfect square.

2015 Caucasus Mathematical Olympiad, 4

Tags: number theory , sum , prime number , consecutive

We call a number greater than $25$, [i] semi-prime[/i] if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be [i]semi-prime[/i]?

1996 Yugoslav Team Selection Test, Problem 3

Tags: sequence , number theory

The sequence $\{x_n\}$ is given by $$x_n=\frac14\left(\left(2+\sqrt3\right)^{2n-1}+\left(2-\sqrt3\right)^{2n-1}\right),\qquad n\in\mathbb N.$$Prove that each $x_n$ is equal to the sum of squares of two consecutive integers.

2013 Purple Comet Problems, 7

Tags: number theory

How many seven-digit positive integers do not either start or end with $7$?

PEN A Problems, 9

Tags: relatively prime , divisibility theory , number theory

Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.

1970 Czech and Slovak Olympiad III A, 1

Tags: number theory , prime , fraction , harmonic , number , prime divisor

Let $p>2$ be a prime and $a,b$ positive integers such that \[\frac ab=1+\frac12+\frac13+\cdots+\frac{1}{p-1}.\] Show that $p$ is a divisor of $a.$

1992 Turkey Team Selection Test, 1

Tags: prime number , number theory

Is there $14$ consecutive positive integers such that each of these numbers is divisible by one of the prime numbers $p$ where $2\leq p \leq 11$.

2020 Kosovo National Mathematical Olympiad, 4

Tags: sequence , algebra , number theory

Let $a_0$ be a fixed positive integer. We define an infinite sequence of positive integers $\{a_n\}_{n\ge 1}$ in an inductive way as follows: if we are given the terms $a_0,a_1,...a_{n-1}$ , then $a_n$ is the smallest positive integer such that $\sqrt[n]{a_0\cdot a_1\cdot ...\cdot a_n}$ is a positive integer. Show that the sequence $\{a_n\}_{n\ge 1}$ is eventually constant. [b]Note:[/b] The sequence $\{a_n\}_{n\ge 1}$ is eventually constant if there exists a positive integer $k$ such that $a_n=c$, for every $n\ge k$.

2006 JBMO ShortLists, 4

Tags: number theory , parameterization

Determine the biggest possible value of $ m$ for which the equation $ 2005x \plus{} 2007y \equal{} m$ has unique solution in natural numbers.

1986 Tournament Of Towns, (129) 4

Tags: number theory , divisible , factorial , divisor , divide

We define $N !!$ to be $N(N - 2)(N -4)...5 \cdot 3 \cdot 1$ if $N$ is odd and $N(N -2)(N -4)... 6\cdot 4\cdot 2$ if $N$ is even . For example, $8 !! = 8 \cdot 6\cdot 4\cdot 2$ , and $9 !! = 9v 7 \cdot 5\cdot 3 \cdot 1$ . Prove that $1986 !! + 1985 !!$ i s divisible by $1987$. (V.V . Proizvolov , Moscow)

2019 Saudi Arabia JBMO TST, 5

Tags: algebra , number theory , sum

Let non-integer real numbers $a, b,c,d$ are given, such that the sum of each $3$ of them is integer. May it happen that $ab + cd$ is an integer.

1993 IMO Shortlist, 2

Tags: number theory , divisibility , prime number , composite number , modular arithmetic

A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$ a.) Show that every prime number $n$ has property $P.$ b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$

2014 PUMaC Number Theory A, 7

Tags: greatest common divisor , integration , number theory , floor function , modular arithmetic , calculus

Find the number of positive integers $n \le 2014$ such that there exists integer $x$ that satisfies the condition that $\frac{x+n}{x-n}$ is an odd perfect square.

1999 Turkey MO (2nd round), 1

Tags: number theory

Find the number of ordered quadruples $(x,y,z,w)$ of integers with $0\le x,y,z,w\le 36$ such that ${{x}^{2}}+{{y}^{2}}\equiv {{z}^{3}}+{{w}^{3}}\text{ (mod 37)}$.

1996 All-Russian Olympiad, 5

Tags: number theory

Do there exist three natural numbers greater than 1, such that the square of each, minus one, is divisible by each of the others? [i]A. Golovanov[/i]

2019 Romanian Master of Mathematics, 6

Tags: number theory , graph theory , combinatorics

Find all pairs of integers $(c, d)$, both greater than 1, such that the following holds: For any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p > c(2c+1)$, there exists a set $S$ of at most $\big(\tfrac{2c-1}{2c+1}\big)p$ integers, such that \[\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}\] contains a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$).

2023 China Second Round, 2

Tags: number theory

For some positive integer $n$, $n$ is considered a $\textbf{unique}$ number if for any other positive integer $m\neq n$, $\{\dfrac{2^n}{n^2}\}\neq\{\dfrac{2^m}{m^2}\}$ holds. Prove that there is an infinite list consisting of composite unique numbers whose elements are pairwise coprime.

2015 India Regional MathematicaI Olympiad, 4

Tags: algebra , geometric sequence , number theory

Find all three digit natural numbers of the form $(abc)_{10}$ such that $(abc)_{10}$, $(bca)_{10}$ and $(cab)_{10}$ are in geometric progression. (Here $(abc)_{10}$ is representation in base $10$.)