1. Theorems

Try to prove these first!

• We denote $v_p(n)$ as the greatest power of $p$ that divides $n$.
• (Legendre) $v_p(n!) = \sum_{i=1}^{\infty} \lfloor \frac{n}{p^i} \rfloor$
• (LTE) If we have integers $a, b$ such that $p | a-b$ then $v_p(a^n-b^n) = v_p(a-b) + v_p(n)$. (Is this true for $p = 2$, is there an extra condition?)
• We say that the order of an element $a$ is $d \pmod n \iff d$ is the smallest number such that $a^d \equiv 1 \pmod n$. We also denote $d$ by $\operatorname{ord}_n(a)$.
• Let $d = \operatorname{ord}_n(a)$ then $a^k \equiv 1 \pmod n \iff d | k$. Therefore $d | \phi(n)$.

2. Problems

• (Lemma) Let $n$ be a natural number. Prove that $\tau(n) < 2\sqrt{n}$. Where $\tau(n)$ is the number of divisors of $n$.
• Let $a, b, c$ be positive reals such that $3 \le a+b+c \le 6$. Prove $\frac{a}{2+bc} + \frac{b}{2+ca} + \frac{c}{2+ab} \ge 1$.
• Let $a, b, c$ be positive integers with $\gcd(a,b,c)=1$ and $a^2+b^2+c^2 = 2(ab+bc+ca)$. Prove that $a, b, c$ are perfect squares.
• Let $0 < x < 1$. The sequence $x_0, x_1, x_2, \dots$ is given by $x_0=1$ and $x_{n+1} = x^{x_n}$ for every $n \ge 0$. Now fix an $n > 1$, find the number of indices $k < n$ satisfying $x_k < x_n$.
• Find all pairs $(n, k)$ of non-negative integers satisfying: $n^{k+1} = (n-2)!$
• Find all positive integers $n$ such that $n = 5 \tau(n)$. Where $\tau(n)$ is the number of divisors of $n$.
• Each point on the plane has been colored one of $2022$ colors, prove that there is a rectangle with 4 points all of the same color.
• 8 *. (Lemma) Suppose that $a > b \ge 3$ are integers. Prove that $b^a > a^b$.
• Do there exist four different natural numbers such that $ad=bc$ and $n^2 \le a, b, c, d < (n+1)^2?$
• Find all prime numbers $p$ and $q$ such that $1 + \frac{p}{q}$ is a prime number.
• Let $a,b,c > 0$. Prove: $\frac{a^3}{b^3} + \frac{b^3}{c^3} + \frac{c^3}{a^3} \ge \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2}$
• 12 *. Find all prime numbers $p$ such that $p^2-p+1$ is a perfect cube.
• It is given that $a+b+c \le 4 \le ab+bc+ca$. Prove that at least two of the following quantities are not more than 2: $|a-b|, |b-c|, |c-a|.$
• Let $p$ be a prime number, $a \ge 2$, $m \ge 1$, $a^m \equiv 1 \pmod p$, $a^p \equiv 1 \pmod{p^2}$. Prove that $a^m \equiv 1 \pmod{p^2}$.
• Let $p$ be a prime number, $a$ a fixed number not divisible by $p$. Prove that the sequence $(a^n-n)$, $n \ge 1$ has infinitely many terms divisible by $p$.
• 16. Let $x,y,z$ be positive integers. Find all solutions $(x, y, z)$ satisfying $x^2+y^2=3z^2$.
• 17 **. (Pell's equation) Prove that there are infinitely many solutions to the equation $a^2-2b^2=1$.
• 18 **. Five positive reals $a, b, c, d, e$ have a product of $1$. Prove that $a^2b^2 + b^2c^2 + c^2d^2 + d^2e^2 + e^2a^2 \ge a+b+c+d+e$.