Let $a$ be a rational number and let $n$ be a positive integer. Prove that the polynomial $X^{2^n}(X+a)^{2^n}+1$ is irreducible in the ring $\mathbb{Q}[X]$ of polynomials with rational coefficients. [i]Proposed by Vincent Jugé, École Polytechnique, Paris.[/i]

Find all positive integers $A$ which can be represented in the form: \[ A = \left ( m - \dfrac 1n \right) \left( n - \dfrac 1p \right) \left( p - \dfrac 1m \right) \] where $m\geq n\geq p \geq 1$ are integer numbers. [i]Ioan Bogdan[/i]

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 $1$.

Solve into $N$: $$a^2 = 2^b +c^4$$

Something related to this [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=845756#845756]problem[/url]: Prove that for a set $ S\subset\mathbb N$, there exists a sequence $ \{a_{i}\}_{i \equal{} 0}^{\infty}$ in $ S$ such that for each $ n$, $ \sum_{i \equal{} 0}^{n}a_{i}x^{i}$ is irreducible in $ \mathbb Z[x]$ if and only if $ |S|\geq2$. [i]By Omid Hatami[/i]

Find all integer solutions of the equation \[\frac {x^{7} \minus{} 1}{x \minus{} 1} \equal{} y^{5} \minus{} 1.\]

(a) Prove that (p^2)-1 is divisible by 24 if p is a prime number greater than 3. (b) Prove that (p^2)-(q^2) is divisible by 24 if p and q are prime numbers greater than 3.

Let $ m_1,m_2,m_3,\dots,m_n$ be a rearrangement of the numbers $ 1,2,\dots,n$. Suppose that $ n$ is odd. Prove that the product \[ \left(m_1\minus{}1\right)\left(m_2\minus{}2\right)\cdots \left(m_n\minus{}n\right)\] is an even integer.

Prove that the number $\lfloor (2+\sqrt5)^{2019} \rfloor$ is not prime.

Find all prime numbers $p,q$, for which $p^{q+1}+q^{p+1}$ is a perfect square. [i]Proposed by P. Boyvalenkov[/i]

Let $a$ and $b$ be distinct positive integers such that $3^a + 2$ is divisible by $3^b + 2$. Prove that $a > b^2$. [i]Proposed by Tynyshbek Anuarbekov, Kazakhstan[/i]

Let $p_1, p_2, \ldots$ be a sequence of primes such that $p_1 =2$ and for $n\geq 1, p_{n+1}$ is the largest prime factor of $p_1 p_2 \ldots p_n +1$ . Prove that $p_n \not= 5$ for any $n$.

Find, with proof, all triples of positive integers $(x,y,z)$ satisfying the equation $3^x - 5^y = 4z^2$.

Show that there are infinitely many tuples $(a,b,c,d)$ of natural numbers such that $a^3 + b^4 + c^5 = d^7$.

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] What is $20 \div 2 - 0 \times 1 + 2 \times 5$? [b]p2.[/b] Today is Saturday, January $25$, $2020$. Exactly four hundred years from today, January $25$, $2420$, is again a Saturday. How many weekend days (Saturdays and Sundays) are in February, $2420$? (January has $31$ days and in year $2040$, February has $29$ days.) [b]p3.[/b] Given that there are four people sitting around a circular table, and two of them stand up, what is the probability that the two of them were originally sitting next to each other? [b]p4.[/b] What is the area of a triangle with side lengths $5$, $5$, and $6$? [b]p5.[/b] Six people go to OBA Noodles on Main Street. Each person has $1/2$ probability to order Duck Noodle Soup, $1/3$ probability to order OBA Ramen, and $1/6$ probability to order Kimchi Udon Soup. What is the probability that three people get Duck Noodle Soup, two people get OBA Ramen, and one person gets Kimchi Udon Soup? [b]p6.[/b] Among all positive integers $a$ and $b$ that satisfy $a^b = 64$, what is the minimum possible value of $a+b$? [b]p7.[/b] A positive integer $n$ is called trivial if its tens digit divides $n$. How many two-digit trivial numbers are there? [b]p8.[/b] Triangle $ABC$ has $AB = 5$, $BC = 13$, and $AC = 12$. Square $BCDE$ is constructed outside of the triangle. The perpendicular line from $A$ to side $DE$ cuts the square into two parts. What is the positive difference in their areas? [b]p9.[/b] In an increasing arithmetic sequence, the first, third, and ninth terms form an increasing geometric sequence (in that order). Given that the first term is $5$, find the sum of the first nine terms of the arithmetic sequence. [b]p10.[/b] Square $ABCD$ has side length $1$. Let points $C'$ and $D'$ be the reflections of points $C$ and $D$ over lines $AB$ and $BC$, respectively. Let P be the center of square $ABCD$. What is the area of the concave quadrilateral $PD'BC'$? [b]p11.[/b] How many four-digit palindromes are multiples of $7$? (A palindrome is a number which reads the same forwards and backwards.) [b]p12.[/b] Let $A$ and $B$ be positive integers such that the absolute value of the difference between the sum of the digits of $A$ and the sum of the digits of $(A + B)$ is $14$. What is the minimum possible value for $B$? [b]p13.[/b] Clark writes the following set of congruences: $x \equiv a$ (mod $6$), $x \equiv b$ (mod $10$), $x \equiv c$ (mod $15$), and he picks $a$, $b$, and $c$ to be three randomly chosen integers. What is the probability that a solution for $x$ exists? [b]p14.[/b] Vincent the bug is crawling on the real number line starting from $2020$. Each second, he may crawl from $x$ to $x - 1$, or teleport from $x$ to $\frac{x}{3}$ . What is the least number of seconds needed for Vincent to get to $0$? [b]p15.[/b] How many positive divisors of $2020$ do not also divide $1010$? [b]p16.[/b] A bishop is a piece in the game of chess that can move in any direction along a diagonal on which it stands. Two bishops attack each other if the two bishops lie on the same diagonal of a chessboard. Find the maximum number of bishops that can be placed on an $8\times 8$ chessboard such that no two bishops attack each other. [b]p17.[/b] Let $ABC$ be a right triangle with hypotenuse $20$ and perimeter $41$. What is the area of $ABC$? [b]p18.[/b] What is the remainder when $x^{19} + 2x^{18} + 3x^{17} +...+ 20$ is divided by $x^2 + 1$? [b]p19.[/b] Ben splits the integers from $1$ to $1000$ into $50$ groups of $20$ consecutive integers each, starting with $\{1, 2,...,20\}$. How many of these groups contain at least one perfect square? [b]p20.[/b] Trapezoid $ABCD$ with $AB$ parallel to $CD$ has $AB = 10$, $BC = 20$, $CD = 35$, and $AD = 15$. Let $AD$ and $BC$ intersect at $P$ and let $AC$ and $BD$ intersect at $Q$. Line $PQ$ intersects $AB$ at $R$. What is the length of $AR$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be any positive integer, and let $S(n)$ denote the number of permutations $\tau$ of $\{1,\dots,n\}$ such that $k^4+(\tau(k))^4$ is prime for all $k=1,\dots,n$. Show that $S(n)$ is always a square.

For every integer $n\geqslant2$ prove the inequality $$ \frac{1}{2!}+\frac{2}{3!}+\ldots+\frac{2^{n-2}}{n!}\leqslant\frac{3}{2}, $$ where $k!=1\cdot2\cdot\ldots\cdot k$.

Find all prime numbers $p$ and positive integers $m$ such that $2p^2 + p + 9 = m^2.$

Fix an integer $k>2$. Two players, called Ana and Banana, play the following game of numbers. Initially, some integer $n \ge k$ gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number $m$ just written on the blackboard and replaces it by some number $m'$ with $k \le m' < m$ that is coprime to $m$. The first player who cannot move anymore loses. An integer $n \ge k $ is called good if Banana has a winning strategy when the initial number is $n$, and bad otherwise. Consider two integers $n,n' \ge k$ with the property that each prime number $p \le k$ divides $n$ if and only if it divides $n'$. Prove that either both $n$ and $n'$ are good or both are bad.

Let $\left \{ x_n \right \} _{n\ge 1}$ and $\left \{ y_n \right \} _{n\ge 1}$ be two infinite sequences of integers. Prove that there exists an infinite sequence of integers $\left \{ z_n \right \} _{n\ge 1}$ such that for any positive integer \( n \), the following holds: \[ \sum_{k|n} k \cdot z_k^{\frac{n}{k}} = \left( \sum_{k|n} k \cdot x_k^{\frac{n}{k}} \right) \cdot \left( \sum_{k|n} k \cdot y_k^{\frac{n}{k}} \right). \]

Determine all positive integers $a$, $b$, $c$, $p$, where $p$ and $p+2$ are odd primes and \[2^ap^b=(p+2)^c-1.\]

(a) If $m, n$ are positive integers such that $2^n-1$ divides $m^2 + 9$, prove that $n$ is a power of $2$; (b) If $n$ is a power of $2$, prove that there exists a positive integer $m$ such that $2^n-1$ divides $m^2 + 9$.

A prime number $p$ and integers $x, y, z$ with $0 < x < y < z < p$ are given. Show that if the numbers $x^3, y^3, z^3$ give the same remainder when divided by $p$, then $x^2 + y^2 + z^2$ is divisible by $x + y + z.$

The cells of the $11 \times 111 \times11$ cube contain the numbers $ 1, 2, , . .. . . 1331$, once each number. Two worms are sent from one corner cube to the opposite corner. Each of them can crawl into a cube adjacent to the edge, while the first can crawl if the number in the adjacent cube differs by $8$, the second - if they differ by $ 9$. Is there such an arrangement of numbers that both worms can get to the opposite corner cube?

Five numbers are chosen from the diagram below, such that no two numbers are chosen from the same row or from the same column. Prove that their sum is always the same. \[\begin{array}{|c|c|c|c|c|}\hline 1&4&7&10&13\\ \hline 16&19&22&25&28\\ \hline 31&34&37&40&43\\ \hline 46&49&52&55&58\\ \hline 61&64&67&70&73\\ \hline \end{array}\]