Olympiad problems

PEN D Problems, 16

Determine all positive integers $n \ge 2$ that satisfy the following condition; For all integers $a, b$ relatively prime to $n$, \[a \equiv b \; \pmod{n}\Longleftrightarrow ab \equiv 1 \; \pmod{n}.\]

PEN D Problems, 15

Tags: congruence , modular arithmetic , inequalities

Let $n_{1}, \cdots, n_{k}$ and $a$ be positive integers which satify the following conditions:[list][*] for any $i \neq j$, $(n_{i}, n_{j})=1$, [*] for any $i$, $a^{n_{i}} \equiv 1 \pmod{n_i}$, [*] for any $i$, $n_{i}$ does not divide $a-1$. [/list] Show that there exist at least $2^{k+1}-2$ integers $x>1$ with $a^{x} \equiv 1 \pmod{x}$.

1997 IMO Shortlist, 12

Tags: algebra , polynomial , modular arithmetic , congruence

Let $ p$ be a prime number and $ f$ an integer polynomial of degree $ d$ such that $ f(0) = 0,f(1) = 1$ and $ f(n)$ is congruent to $ 0$ or $ 1$ modulo $ p$ for every integer $ n$. Prove that $ d\geq p - 1$.

PEN D Problems, 22

Tags: congruence , logarithm

Prove that $1980^{1981^{1982}} + 1982^{1981^{1980}}$ is divisible by $1981^{1981}$.

PEN D Problems, 5

Tags: modular arithmetic , congruence

Prove that for $n\geq 2$, \[\underbrace{2^{2^{\cdots^{2}}}}_{n\text{ terms}}\equiv \underbrace{2^{2^{\cdots^{2}}}}_{n-1\text{ terms}}\; \pmod{n}.\]

1992 Spain Mathematical Olympiad, 1

Tags: number theory , divisibility , combinatorics , congruence

Determine the smallest number N, multiple of 83, such that N^2 has 63 positive divisors.

KoMaL A Problems 2018/2019, A. 746

Tags: number theory , congruence

Let $p$ be a prime number. How many solutions does the congruence $x^2+y^2+z^2+1\equiv 0\pmod{p}$ have among the modulo $p$ remainder classes? [i]Proposed by: Zoltán Gyenes, Budapest[/i]

PEN D Problems, 12

Tags: congruence , modular arithmetic , floor function , abstract algebra , number theory , relatively prime , group theory

Suppose that $m>2$, and let $P$ be the product of the positive integers less than $m$ that are relatively prime to $m$. Show that $P \equiv -1 \pmod{m}$ if $m=4$, $p^n$, or $2p^{n}$, where $p$ is an odd prime, and $P \equiv 1 \pmod{m}$ otherwise.

PEN D Problems, 1

Tags: floor function , modular arithmetic , congruence

If $p$ is an odd prime, prove that \[{k \choose p}\equiv \left\lfloor \frac{k}{p}\right\rfloor \pmod{p}.\]

PEN D Problems, 4

Tags: congruence , modular arithmetic

Let $n$ be a positive integer. Prove that $n$ is prime if and only if \[{{n-1}\choose k}\equiv (-1)^{k}\pmod{n}\] for all $k \in \{ 0, 1, \cdots, n-1 \}$.

2019 Oral Moscow Geometry Olympiad, 2

Tags: congruent , quadrilateral , congruence , diagonal , geometry

The angles of one quadrilateral are equal to the angles another quadrilateral. In addition, the corresponding angles between their diagonals are equal. Are these quadrilaterals necessarily similar?

Russian TST 2022, P3

Tags: number theory , congruence

Let $n = 2k + 1$ be an odd positive integer, and $m$ be an integer realtively prime to $n{}$. For each $j =1,2,\ldots,k$ we define $p_j$ as the unique integer from the interval $[-k, k]$ congruent to $m\cdot j$ modulo $n{}$. Prove that there are equally many pairs $(i,j)$ for which $1\leqslant i<j\leqslant k$ which satisfy $|p_i|>|p_j|$ as those which satisfy $p_ip_j<0$.