Let $p$ be an odd prime number, and $k$ be an odd number not divisible by $p$. Consider a field $K$ be a field with $kp+1$ elements, and $A = \{x_1,x_2, \dots, x_t\}$ be the set of elements of $K^*$, whose order is not $k$ in the multiplicative group $(K^*,\cdot)$. Prove that the polynomial $P(X)=(X+x_1)(X+x_2)\dots(X+x_t)$ has at least $p$ coefficients equal to $1$.

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

Find $$\sum^{100}_{i=1}i \gcd(i ,100).$$

Let $a_1,a_2, \cdots$ be a sequence of integers that satisfies: $a_1=1$ and $a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor} , \forall n\geq 1 $. Prove that for all positive $k$, there is $m \geq 1$ such that $k \mid a_m$.

Two distinct points $A$ and $B$ are chosen at random from 15 points equally spaced around a circle centered at $O$ such that each pair of points $A$ and $B$ has the same probability of being chosen. The probability that the perpendicular bisectors of $OA$ and $OB$ intersect strictly inside the circle can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers. Find $m+n$. [i]Ray Li.[/i]

You are given $n \ge 2$ distinct positive integers. For every pair $a<b$ of them, Vlada writes on the board the largest power of $2$ that divides $b-a$. At most how many distinct powers of $2$ could Vlada have written? [i]Proposed by Oleksiy Masalitin[/i]

Prove that the composition of the sets of one of the following two forms is finite: (a) $2^{2^n}+1$ (b) $6^{2^n}+1$

Given are distinct natural numbers $a$, $b$, and $c$. Prove that \[ \gcd(ab+1, ac+1, bc+1)\le \frac{a+b+c}{3}\]

(a) Find all prime numbers $p$ such that $4p^2+1$ and $6p^2+1$ are also primes. (b)Find real numbers $x,y,z,u$ such that \[xyz+xy+yz+zx+x+y+z=7\]\[yzu+yz+zu+uy+y+z+u=10\]\[zux+zu+ux+xz+z+u+x=10\]\[uxy+ux+xy+yu+u+x+y=10\]

For each positive integer $k,$ let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n,$ such that all the differences \[t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)\] are divisible by 4. [i]Proposed by Gerhard Wöginger, Austria[/i]

Find the number of positive integers $n < 100$ such that $\gcd(n^2,2023) \neq \gcd(n,2023^2).$

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

Let $A$ be a set of $20$ distinct positive integers which are all no greater than $397$. Prove that for any positive integer $n$ it is possible to pick four (not necessarily distinct) elements $x_1, x_2, x_3, x_4$ of $A$ satisfying $x_1 \neq x_2$ and $$(x_1-x_2)n\equiv x_3-x_4 \pmod{397}.$$

Let $A = \{1,2,..., 100\}$ and $B$ is a subset of $A$ having $48$ elements. Show that $B$ has two distint elements $x$ and $y$ whose sum is divisible by $11$.

Prove that among any $2n+1$ irrational numbers there are $n+1$ numbers such that the sum of any $k$ of them is irrational, for all $k \in \{1,2,3,\ldots, n+1 \}$.

Prove that there exists a positive integer $n$, such that the number $k^2+k+n$ does not have a prime divisor less than $2008$ for any integer $k$.

Prove that for every natural number \( n>2 \) there exists an integer \( k \) that can be written as the sum of \( i \) positive perfect squares, for every \( i \) between \( 2 \) and \( n \).

For any positive integer $n$, consider its binary representation. Denote by $f(n)$ the number we get after removing all the $0$'s in its binary representation, and $g(n)$ the number of $1$'s in the binary representation. For example, $f(19) = 7$ and $g(19) = 3.$ Find all positive integers $n$ that satisfy $$n = f(n)^{g(n)}.$$ [i] Proposed by usjl[/i]

A bunch of lilac consists of flowers with 4 or 5 petals. The number of flowers and the total number of petals are perfect squares. Can the number of flowers with 4 petals be divisible by the number of flowers with 5 petals?

Determine all tuples of integers $(a,b,c)$ such that: $$(a-b)^3(a+b)^2 = c^2 + 2(a-b) + 1$$

We define the set $M = \{1^2,2^2,3^2,..., 99^2, 100^2\}$. a) What is the smallest positive integer that divides exactly two elements of $M$? b) What is the largest positive integer that divides exactly two elements of $M$?

Positive integer numbers a and b satisfy $(a^2- 9b^2)^2 - 33b = 1$. a) Prove $|a -3b|\ge 1$. b) Find all pairs of positive integers $(a, b)$ satisfying the equality.

Let $n, m$ be positive integers such that \[n(4n+1)=m(5m+1)\] (a) Show that the difference $n-m$ is a perfect square of a positive integer. (b) Find a pair of positive integers $(n, m)$ which satisfies the above relation. Additional part (not asked in the TST): Find all such pairs $(n,m)$.

Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$ [list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good. [*][b](b)[/b] Show that all 2010-good pairs are very good.[/list] [i]Proposed by Okan Tekman, Turkey[/i]

Let $k>2$ be a positive integer. Elise and Xavier play a game that has four steps, in this order. [list=1] [*]Elise picks $2$ nonzero digits $(1-9)$, called $e$ and $f$. [*]Xavier then picks $k$ nonzero digits $(1-9)$, called $x_1,\cdots,x_k$. [*]Elise picks any positive integer $d$. [*]Xaiver picks an integer $b>10$.[/list] Each player's choices are known to the other player when the choices are made. The winner is determined as follows. Elise writes down the two-digit base $b$ number $ef_b$. Next, Xavier writes the $k$-digit base $b$ number that is constructed by concatenating his digits, \[(x_1\cdots x_k)_b.\] They then compute the greatest common divisor (gcd) of these two numbers. If this gcd is greater than or equal to the integer $d$ then Xavier wins. Otherwise Elise wins. (As an example game for $k=3$, Elise chooses the digits $(e, f) = (2, 4)$, Xavier chooses $(4, 4, 8)$, and then Elise picks $d = 100$. Xavier picks base $b = 25$. The base-25 numbers $2425$ and $44825$ are, respectively, equal to $54$ and $2608$. The greatest common divisor of these two is $2$, which is much less than $100$, so Elise wins handily.) Find all $k$ for which Xavier can force a win, no matter how Elise plays.