Found problems: 37
Russian TST 2022, P3
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 $p_ip_j<0$.
Novosibirsk Oral Geo Oly VII, 2023.4
Inside the convex pentagon $ABCDE$, a point $O$ was chosen, and it turned out that all five triangles $AOB$, $BOC$, $COD$, $DOE$ and $EOA$ are congrunet to each other. Prove that these triangles are isosceles or right-angled.
PEN D Problems, 1
If $p$ is an odd prime, prove that \[{k \choose p}\equiv \left\lfloor \frac{k}{p}\right\rfloor \pmod{p}.\]
PEN D Problems, 22
Prove that $1980^{1981^{1982}} + 1982^{1981^{1980}}$ is divisible by $1981^{1981}$.
KoMaL A Problems 2018/2019, A. 746
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]
1992 Spain Mathematical Olympiad, 1
Determine the smallest number N, multiple of 83, such that N^2 has 63 positive divisors.
PEN D Problems, 21
Determine the last three digits of \[2003^{2002^{2001}}.\]
2021 Balkan MO Shortlist, N3
Let $n$ be a positive integer. Determine, in terms of $n$, the greatest integer which divides
every number of the form $p + 1$, where $p \equiv 2$ mod $3$ is a prime number which does not divide $n$.
PEN D Problems, 7
Somebody incorrectly remembered Fermat's little theorem as saying that the congruence $a^{n+1} \equiv a \; \pmod{n}$ holds for all $a$ if $n$ is prime. Describe the set of integers $n$ for which this property is in fact true.
2015 Romania Team Selection Tests, 3
Define a sequence of integers by $a_0=1$ , and $a_n=\sum_{k=0}^{n-1} \binom{n}{k}a_k$ , $n \geq 1$ . Let $m$ be a positive integer , let $p$ be a prime , and let $q$ and $r$ be non-negative integers . Prove that :
$$a_{p^mq+r} \equiv a_{p^{m-1}q+r} \pmod{p^m}$$
PEN D Problems, 19
Let $a_{1}$, $\cdots$, $a_{k}$ and $m_{1}$, $\cdots$, $m_{k}$ be integers with $2 \le m_{1}$ and $2m_{i}\le m_{i+1}$ for $1 \le i \le k-1$. Show that there are infinitely many integers $x$ which do not satisfy any of congruences \[x \equiv a_{1}\; \pmod{m_{1}}, x \equiv a_{2}\; \pmod{m_{2}}, \cdots, x \equiv a_{k}\; \pmod{m_{k}}.\]
PEN D Problems, 23
Let $p$ be an odd prime of the form $p=4n+1$. [list=a][*] Show that $n$ is a quadratic residue $\pmod{p}$. [*] Calculate the value $n^{n}$ $\pmod{p}$. [/list]
PEN D Problems, 6
Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[2, \; 2^{2}, \; 2^{2^{2}}, \; 2^{2^{2^{2}}}, \cdots \pmod{n}\] is eventually constant.
PEN D Problems, 17
Determine all positive integers $n$ such that $ xy+1 \equiv 0 \; \pmod{n} $ implies that $ x+y \equiv 0 \; \pmod{n}$.
PEN D Problems, 12
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.
2023 Novosibirsk Oral Olympiad in Geometry, 4
Inside the convex pentagon $ABCDE$, a point $O$ was chosen, and it turned out that all five triangles $AOB$, $BOC$, $COD$, $DOE$ and $EOA$ are congrunet to each other. Prove that these triangles are isosceles or right-angled.
2015 Serbia National Math Olympiad, 6
In nonnegative set of integers solve the equation:
$$(2^{2015}+1)^x + 2^{2015}=2^y+1$$
PEN D Problems, 4
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 $k \in \{ 0, 1, \cdots, n-1 \}$.
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, 5
Prove that for $n\geq 2$, \[\underbrace{2^{2^{\cdots^{2}}}}_{n\text{ terms}}\equiv \underbrace{2^{2^{\cdots^{2}}}}_{n-1\text{ terms}}\; \pmod{n}.\]
PEN D Problems, 20
Show that $1994$ divides $10^{900}-2^{1000}$.
PEN D Problems, 15
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 $a^{x} \equiv 1 \pmod{x}$.
Kyiv City MO 1984-93 - geometry, 1992.9.2
Two lines divide a square into $4$ figures of the same area. Prove that all these figures are congruent.
PEN D Problems, 2
Suppose that $p$ is an odd prime. Prove that \[\sum_{j=0}^{p}\binom{p}{j}\binom{p+j}{j}\equiv 2^{p}+1\pmod{p^{2}}.\]
2022 Romania Team Selection Test, 3
Consider a prime number $p\geqslant 11$. We call a triple $a,b,c$ of natural numbers [i]suitable[/i] if they give non-zero, pairwise distinct residues modulo $p{}$. Further, for any natural numbers $a,b,c,k$ we define \[f_k(a,b,c)=a(b-c)^{p-k}+b(c-a)^{p-k}+c(a-b)^{p-k}.\]Prove that there exist suitable $a,b,c$ for which $p\mid f_2(a,b,c)$. Furthermore, for each such triple, prove that there exists $k\geqslant 3$ for which $p\nmid f_k(a,b,c)$ and determine the minimal $k{}$ with this property.
[i]Călin Popescu and Marian Andronache[/i]