Found problems: 1048
Three members of the Euclid Middle School girls' softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all $2$-digit primes.
Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month.
Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.
Ashley: And the sum of you two uniform numbers is today's date.
What number does Caitlin wear?
$\textbf{(A) }11\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad \textbf{(E) }23$
Let $n?$ denote the product of all primes smaller than $n$.
Prove that $n? > n$ holds for any natural number $n > 3$.
Prove that there exists a prime number $p$, such that the sum of digits of $p$ is a composite odd integer. Find the smallest such $p$.
Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers
The ratio of prime numbers $p$ and $q$ does not exceed 2 ($p\ne q$).
Prove that there are two consecutive positive integers such that
the largest prime divisor of one of them is $p$ and that of the other is $q$.
Determine all integers $x$ such that $2x^2-x-36$ is a perfect square of a prime.
Let $p \ge 5$ be a prime number. For a positive integer $k$, let $R(k)$ be the remainder when $k$ is divided by $p$, with $0 \le R(k) \le p-1$. Determine all positive integers $a < p$ such that, for every $m = 1, 2, \cdots, p-1$, $$ m + R(ma) > a. $$
Find all prime numbers of the form $\tfrac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \textrm{ ones}}$, where $n$ is a natural number.
Find all positive $k$ such that product of the first $k$ odd prime numbers, reduced by 1 is exactly degree of natural number (which more than one).
Let $ p,q,r$ be 3 prime numbers such that $ 5\leq p <q<r$. Knowing that $ 2p^2\minus{}r^2 \geq 49$ and $ 2q^2\minus{}r^2\leq 193$, find $ p,q,r$.
Let $p$ be a prime number, determine all positive integers $(x, y, z)$ such that:
$x^p + y^p = p^z$
Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.
Let $n$ be a positive integer and $p > n+1$ a prime.
Prove that $p$ divides the following sum $S = 1^n + 2^n +...+ (p - 1)^n$
Let $n$ be a positive integer. Prove that for every odd prime $p$ dividing $n^2 + n + 2$, there exist integers $a, b$ such that $p = a^2 + 7b^2$.
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.$
Given a prime $p \geq 5$ , show that there exist at least two distinct primes $q$ and $r$ in the range $2, 3, \ldots p-2$ such that $q^{p-1} \not\equiv 1 \pmod{p^2}$ and $r^{p-1} \not\equiv 1 \pmod{p^2}$.
Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$
Determine all positive integers $a$ and $b$ for which $a^4 + 4b^4$ be a prime number.
An integer $n > 1$ has the following property: for every (positive) divisor $d$ of $n, d + 1$ is a divisor of $n + 1$.
Prove that $n$ is prime.
Let a pair of positive integers $(n, m)$ that are relatively prime be called [i]intertwined[/i] if among any two divisors of $n$ greater than $1$, there exists a divisor of $m$ and among any two divisors of $m$ greater than $1$, there exists a divisor of $n$. For example, pair $(63, 64)$ is intertwined.
[b]a)[/b] Find the largest integer $k$ for which there exists an intertwined pair $(n, m)$ such that the product $nm$ is equal to the product of the first $k$ prime numbers.
[b]b)[/b] Prove that there does [b]not[/b] exist an intertwined pair $(n, m)$ such that the product $nm$ is the product of $2025$ distinct prime numbers.
[b]c)[/b] Prove that there exists an intertwined pair $(n, m)$ such that the number of divisors of $n$ is greater than $2025$.
[i]Proposed by Stijn Cambie, Belgium[/i]
Let $a, b, c$ be non zero integers,$ a\ne c$ such that $$\frac{a}{c}=\frac{a^2+b^2}{c^2+b^2}$$
Prove that $a^2 +b^2 +c^2$ cannot be a prime number.
Which of the following expressions is never a prime number when $p$ is a prime number?
$\textbf{(A) } p^2+16 \qquad \textbf{(B) } p^2+24 \qquad \textbf{(C) } p^2+26 \qquad \textbf{(D) } p^2+46 \qquad \textbf{(E) } p^2+96$
Determine all sequences of strictly positive integers $a_1, a_2, a_3, \ldots$ satisfying the following two conditions:
[list]
[*]There exists an integer $M > 0$ such that, for all indices $n \geqslant 1$, $0 < a_n \leqslant M$.
[*]For any prime number $p$ and for any index $n \geqslant 1$, the number
\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]
is a multiple of $p$.
[/list]
Let $p$ be a prime number. Prove that there exists $n \in Z$ such that $p | n^2 -n+3$ if and only if there exists $m \in Z$ such that $p | m^2 -m+25$.
Does there exist a natural number $n$ with exactly 3 different prime divisors $p$, $q$, and $r$, so that $p-1\mid n$, $qr-1\mid n$, $q-1\nmid n$, $r-1\nmid n$, and $3\nmid q+r$?