Olympiad problems

2012 BMT Spring, 7

Let $ a $ , $ b $ , $ c $ , $ d $ , $ (a + b + c + 18 + d) $ , $ (a + b + c + 18 - d) $ , $ (b + c) $ , and $ (c + d) $ be distinct prime numbers such that $ a + b + c = 2010 $, $ a $, $ b $, $ c $, $ d \neq 3 $ , and $ d \le 50 $. Find the maximum value of the diﬀerence between two of these prime numbers.

2021 Kyiv City MO Round 1, 11.5

Tags: number theory , prime number

For positive integers $m, n$ define the function $f_n(m) = 1^{2n} + 2^{2n} + 3^{2n} + \ldots +m^{2n}$. Prove that there are only finitely many pairs of positive integers $(a, b)$ such that $f_n(a) + f_n(b)$ is a prime number. [i]Proposed by Nazar Serdyuk[/i]

2019 Dutch IMO TST, 4

Tags: number theory , function , find all functions , prime number , functional equation

Find all functions $f : Z \to Z$ satisfying $\bullet$ $ f(p) > 0$ for all prime numbers $p$, $\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.

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]?

2002 AMC 10, 15

Tags: number theory , prime number

The digits $ 1$, $ 2$, $ 3$, $ 4$, $ 5$, $ 6$, $ 7$, and $ 9$ are used to form four two-digit prime numbers, with each digit used exactly once. What is the sum of these four primes? $ \text{(A)}\ 150 \qquad \text{(B)}\ 160 \qquad \text{(C)}\ 170 \qquad \text{(D)}\ 180 \qquad \text{(E)}\ 190$

2014 CentroAmerican, 1

Tags: prime number , number theory

A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?

1994 Bulgaria National Olympiad, 3

Tags: number theory , prime number

Let $p$ be a prime number, determine all positive integers $(x, y, z)$ such that: $x^p + y^p = p^z$

2016 Korea Summer Program Practice Test, 3

Tags: number theory , prime number , prime , primitive root , decimal representation

Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.

2009 Princeton University Math Competition, 5

Tags: number theory , prime number

Find the sum of all prime numbers $p$ which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct) $a$, $b$ and $c$.

2012 European Mathematical Cup, 1

Tags: prime number , number theory

Find all positive integers $a$, $b$, $n$ and prime numbers $p$ that satisfy \[ a^{2013} + b^{2013} = p^n\text{.}\] [i]Proposed by Matija Bucić.[/i]

2009 Germany Team Selection Test, 1

Tags: prime number , number theory

For which $ n \geq 2, n \in \mathbb{N}$ are there positive integers $ A_1, A_2, \ldots, A_n$ which are not the same pairwise and have the property that the product $ \prod^n_{i \equal{} 1} (A_i \plus{} k)$ is a power for each natural number $ k.$

2011 Akdeniz University MO, 1

Tags: number theory , prime number

Let $m,n$ positive integers and $p$ prime number with $p=3k+2$. If $p \mid {(m+n)^2-mn}$ , prove that $$p \mid m,n$$

2019 Moroccan TST, 4

Tags: number theory , prime number

Let $p$ be a prime number. Find all the positive integers $n$ such that $p+n$ divides $pn$

2002 AMC 12/AHSME, 17

Tags: number theory , prime number

Several sets of prime numbers, such as $ \{ 7, 83, 421, 659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have? $ \textbf{(A)}\ 193\qquad\textbf{(B)}\ 207\qquad\textbf{(C)}\ 225\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 447$

2012-2013 SDML (High School), 5

Tags: number theory , prime number

Palmer correctly computes the product of the first $1,001$ prime numbers. Which of the following is NOT a factor of Palmer's product? $\text{(A) }2,002\qquad\text{(B) }3,003\qquad\text{(C) }5,005\qquad\text{(D) }6,006\qquad\text{(E) }7,007$

2014 IMAC Arhimede, 5

Tags: number theory , numerical systems , system , binomial , product , modulo , prime number

Let $p$ be a prime number. The natural numbers $m$ and $n$ are written in the system with the base $p$ as $n = a_0 + a_1p +...+ a_kp^k$ and $m = b_0 + b_1p +..+ b_kp^k$. Prove that $${n \choose m} \equiv \prod_{i=0}^{k}{a_i \choose b_i} (mod p)$$

2019 Moldova Team Selection Test, 12

Tags: number theory , prime number

Let $p\ge 5$ be a prime number. Prove that there exist positive integers $m$ and $n$ with $m+n\le \frac{p+1}{2}$ for which $p$ divides $2^n\cdot 3^m-1.$

2014 Iran Team Selection Test, 2

Tags: function , prime number , number theory

is there a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that $i) \exists n\in \mathbb{N}:f(n)\neq n$ $ii)$ the number of divisors of $m$ is $f(n)$ if and only if the number of divisors of $f(m)$ is $n$

2015 Thailand TSTST, 1

Tags: number theory , prime number

Find all primes $1 < p < 100$ such that the equation $x^2-6y^2=p$ has an integer solution $(x, y)$.

2011 Putnam, B2

Tags: number theory , prime number , algebra , polynomial , rational root theorem

Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2+qx+r=0.$ Which primes appear in seven or more elements of $S?$

2009 Bosnia and Herzegovina Junior BMO TST, 3

Tags: number theory , prime , division , prime number

Let $p$ be a prime number, $p\neq 3$ and let $a$ and $b$ be positive integers such that $p \mid a+b$ and $p^2\mid a^3+b^3$. Show that $p^2 \mid a+b$ or $p^3 \mid a^3+b^3$

2020 IMC, 6

Tags: number theory , prime number

Find all prime numbers $p$ such that there exists a unique $a \in \mathbb{Z}_p$ for which $a^3 - 3a + 1 = 0.$

1973 Miklós Schweitzer, 3

Tags: floor function , prime number , number theory , logarithm

Find a constant $ c > 1$ with the property that, for arbitrary positive integers $ n$ and $ k$ such that $ n>c^k$, the number of distinct prime factors of $ \binom{n}{k}$ is at least $ k$. [i]P. Erdos[/i]

2006 National Olympiad First Round, 2

Tags: number theory , prime number

If $p$ and $p^2+2$ are prime numbers, at most how many prime divisors can $p^3+3$ have? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

2023 IMC, 4

Tags: algebra , polynomial , number theory , modular arithmetic , prime number

Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_i=i^k+i$ for $i=0,1, \ldots,p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_2$ upon division by $p$?