Olympiad problems

2023 Olimphíada, 4

We say that a prime $p$ is $n$-$\textit{rephinado}$ if $n | p - 1$ and all $1, 2, \ldots , \lfloor \sqrt[\delta]{p}\rfloor$ are $n$-th residuals modulo $p$, where $\delta = \varphi+1$. Are there infinitely many $n$ for which there are infinitely many $n$-$\textit{rephinado}$ primes? Notes: $\varphi =\frac{1+\sqrt{5}}{2}$. We say that an integer $a$ is a $n$-th residue modulo $p$ if there is an integer $x$ such that $$x^n \equiv a \text{ (mod } p\text{)}.$$

2023 Romania EGMO TST, P2

Tags: number theory , prime number , divisibility , power of 2 , prime divisor

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

2015 Canada National Olympiad, 5

Tags: number theory , prime number

Let $p$ be a prime number for which $\frac{p-1}{2}$ is also prime, and let $a,b,c$ be integers not divisible by $p$. Prove that there are at most $1+\sqrt {2p}$ positive integers $n$ such that $n<p$ and $p$ divides $a^n+b^n+c^n$.

2024 Israel TST, P3

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

Let $n$ be a positive integer and $p$ be a prime number of the form $8k+5$. A polynomial $Q$ of degree at most $2023$ and nonnegative integer coefficients less than or equal to $n$ will be called "cool" if \[p\mid Q(2)\cdot Q(3) \cdot \ldots \cdot Q(p-2)-1.\] Prove that the number of cool polynomials is even.

2016 IFYM, Sozopol, 3

Tags: number theory , prime number

Find the least natural number $n\geq 5$, for which $x^n\equiv 16\, (mod\, p)$ has a solution for any prime number $p$.

2012 IMO Shortlist, N5

Tags: algebra , number theory , prime number , polynomial

For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.

2020 Regional Olympiad of Mexico Southeast, 6

Tags: prime number , sequence , number theory

Prove that for all $a, b$ and $x_0$ positive integers, in the sequence $x_1, x_2, x_3, \cdots$ defined by $$x_{n+1}=ax_n+b, n\geq 0$$ Exist an $x_i$ that is not prime for some $i\geq 1$

2016 Purple Comet Problems, 14

Tags: number theory , prime number

Find the greatest possible value of $pq + r$, where p, q, and r are (not necessarily distinct) prime numbers satisfying $pq + qr + rp = 2016$.

2002 AMC 10, 14

Tags: quadratic , vieta , algebra , number theory , prime number , special factorizations

Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$

2004 Irish Math Olympiad, 1

Tags: number theory , prime number

Determine all pairs of prime numbers $(p, q)$, with $2 \leq p, q < 100$, such that $p+6, p+10, q+4, q+10$ and $p+q+1$ are all prime numbers.

2012 Belarus Team Selection Test, 1

Tags: number theory , prime number

Find all primes numbers $p$ such that $p^2-p-1$ is the cube of some integer.

2024 Singapore MO Open, Q5

Tags: combinatorics , grid , number theory , prime number , prime

Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

2024 Francophone Mathematical Olympiad, 4

Tags: prime number , number theory , divisor

Let $p$ be a fixed prime number. Find all integers $n \ge 1$ with the following property: One can partition the positive divisors of $n$ in pairs $(d,d')$ satisfying $d<d'$ and $p \mid \left\lfloor \frac{d'}{d}\right\rfloor$.

2018 Kürschák Competition, 2

Tags: vector , analytic geometry , number theory , prime number

Given a prime number $p$ and let $\overline{v_1},\overline{v_2},\dotsc ,\overline{v_n}$ be $n$ distinct vectors of length $p$ with integer coordinates in an $\mathbb{R}^3$ Cartesian coordinate system. Suppose that for any $1\leqslant j<k\leqslant n$, there exists an integer $0<\ell <p$ such that all three coordinates of $\overline{v_j} -\ell \cdot \overline{v_k} $ is divisible by $p$. Prove that $n\leqslant 6$.

2019 India PRMO, 21

Tags: number theory , prime number , set , prime

Consider the set $E = \{5, 6, 7, 8, 9\}$. For any partition ${A, B}$ of $E$, with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$. Let $N$ be the largest prime number amonh these numbers. Find the sum of the digits of $N$.

2016 Mathematical Talent Reward Programme, MCQ: P 8

Tags: prime number , number theory

Let $p$ be a prime such that $16p+1$ is a perfect cube. A possible choice for $p$ is [list=1] [*] 283 [*] 307 [*] 593 [*] 691 [/list]

1991 India Regional Mathematical Olympiad, 7

Tags: induction , modular arithmetic , number theory , prime number

Prove that $n^4 + 4^{n}$ is composite for all values of $n$ greater than $1$.

2014 German National Olympiad, 1

Tags: prime number , number theory

For which non-negative integers $n$ is \[K=5^{2n+3} + 3^{n+3} \cdot 2^n\] prime?

1995 AMC 12/AHSME, 29

Tags: counting , distinguishability , number theory , prime number

For how many three-element sets of positive integers $\{a,b,c\}$ is it true that $a \times b \times c = 2310$? $\textbf{(A)}\ 32 \qquad \textbf{(B)}\ 36 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 43 \qquad \textbf{(E)}\ 45$

2022 VIASM Summer Challenge, Problem 1

Tags: number theory , prime number

Find all prime number pairs $(p,q)$ such that $p(p^2-p-1)=q(2q+3).$

2020 MMATHS, I3

Tags: number theory , prime number

Suppose that three prime numbers $p,q,$ and $r$ satisfy the equations $pq + qr + rp = 191$ and $p + q = r - 1$. Find $p + q + r$. [i]Proposed by Andrew Wu[/i]

1974 IMO Longlists, 2

Tags: prime number , number theory

Let ${u_n}$ be the Fibonacci sequence, i.e., $u_0=0,u_1=1,u_n=u_{n-1}+u_{n-2}$ for $n>1$. Prove that there exist infinitely many prime numbers $p$ that divide $u_{p-1}$.

2018 PUMaC Individual Finals A, 3

Tags: number theory , prime number

We say that the prime numbers $p_1,\dots,p_n$ construct the graph $G$ if we can assign to each vertex of $G$ a natural number whose prime divisors are among $p_1,\dots,p_n$ and there is an edge between two vertices in $G$ if and only if the numbers assigned to the two vertices have a common divisor greater than $1$. What is the minimal $n$ such that there exist prime numbers $p_1,\dots,p_n$ which construct any graph $G$ with $N$ vertices?

2020 ITAMO, 5

Tags: function , number theory , prime number

Le $S$ be the set of positive integers greater than or equal to $2$. A function $f: S\rightarrow S$ is italian if $f$ satifies all the following three conditions: 1) $f$ is surjective 2) $f$ is increasing in the prime numbers(that is, if $p_1<p_2$ are prime numbers, then $f(p_1)<f(p_2)$) 3) For every $n\in S$ the number $f(n)$ is the product of $f(p)$, where $p$ varies among all the primes which divide $n$ (For instance, $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$). Determine the maximum and the minimum possible value of $f(2020)$, when $f$ varies among all italian functions.

2018 VJIMC, 2

Tags: number theory , prime number

Find all prime numbers $p$ such that $p^3$ divides the determinant \[\begin{vmatrix} 2^2 & 1 & 1 & \dots & 1\\1 & 3^2 & 1 & \dots & 1\\ 1 & 1 & 4^2 & & 1\\ \vdots & \vdots & & \ddots & \\1 & 1 & 1 & & (p+7)^2 \end{vmatrix}.\]