Determine whether the equation $x^2 +xy+y^2 = 2$ has a solution $(x,y)$ in rational numbers.

$P(x)$ is a nonzero polynomial with integer coefficients. Prove that there exists infinitely many prime numbers $q$ such that for some natural number $n$, $q|2^n+P(n)$. [i]Proposed by Mohammad Gharakhani[/i]

Let $a + bi$ and $c + di$ be two roots of the equation $x^n = 1990$, where $n \geq 3$ is an integer and $a,b,c,d \in \mathbb R$. Under the linear transformation $f =\left(\begin{array}{cc}a&c\\b &d\end{array}\right)$, we have $(2, 1) \to (1, 2)$. Denote $r$ to be the distance from the image of $(2, 2)$ to the origin. Find the range of $r.$

Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$.

Let's define [i]almost mean[/i] of numbers $a_1, a_2, \ldots, a_n$ as $\frac{a_1 + a_2 + \ldots + a_n}{n+1}$. Oleksiy has positive real numbers $b_1, b_2, \ldots, b_{2023}$, not necessarily distinct. For each pair $(i, j)$ with $1 \leq i, j \leq 2023$, Oleksiy wrote on a board [i]almost mean[/i] of numbers $b_i, b_{i+1}, \ldots, b_j$. Prove that there are at least $45$ distinct numbers on the board. [i]Proposed by Anton Trygub[/i]

What is the value of $1 + 7 + 21 + 35 + 35 + 21 + 7 + 1$?

Let $ C$ be the circle with radius 1 centered on the origin. Fix the endpoint of the string with length $ 2\pi$ on the point $ A(1,\ 0)$ and put the other end point $ P$ on the point $ P_0(1,\ 2\pi)$. From this situation, when we twist the string around $ C$ by moving the point $ P$ in anti clockwise with the string streched tightly, find the length of the curve that the point $ P$ draws from sarting point $ P_0$ to reaching point $ A$.

Suppose that $ n \geq 2$ and $ x_1, x_2, \ldots, x_n$ are real numbers between 0 and 1 (inclusive). Prove that for some index $ i$ between $ 1$ and $ n \minus{} 1$ the inequality \[ x_i (1 \minus{} x_{i\plus{}1}) \geq \frac{1}{4} x_1 (1 \minus{} x_{n})\]

Is there a positive integer $ n$ such that among 200th digits after decimal point in the decimal representations of $ \sqrt{n}$, $ \sqrt{n\plus{}1}$, $ \sqrt{n\plus{}2}$, $ \ldots,$ $ \sqrt{n\plus{}999}$ every digit occurs 100 times? [i]Proposed by A. Golovanov[/i]

Determine a) the smallest number b) the biggest number $n \ge 3$ of non-negative integers $x_1, x_2, ... , x_n$, having the sum $2011$ and satisfying: $x_1 \le | x_2 - x_3 | , x_2 \le | x_3 - x_4 | , ... , x_{n-2} \le | x_{n-1} -x_n | , x_{n-1} \le | x_n - x_1 |$ and $x_n \le | x_1 - x_2 | $.

$f(x)$ is a polynomial of degree $2n$. Show that all polynomials $p(x)$, $q(x)$ of degree at most $n$ such that $f(x)q(x)-p(x)$ has the form \[ \sum\limits_{2n<k\leq 3n} (a^k + x^k) \] have the same $p(x)/q(x)$.

Prove that for all positive integers $n$, all complex roots $r$ of the polynomial \[P(x) = (2n)x^{2n} + (2n-1)x^{2n-1} + \dots + (n+1)x^{n+1} + nx^n + (n+1)x^{n-1} + \dots + (2n-1)x + 2n\] lie on the unit circle (i.e. $|r| = 1$).

For three real numbers $ a,b,c>1, $ prove the inequality: $ \log_{a^2b} a +\log_{b^2c} b +\log_{c^2a} c\le 1. $

Given $a, \theta \in \mathbb R, m \in \mathbb N$, and $P(x) = x^{2m}- 2|a|^mx^m \cos \theta +a^{2m}$, factorize $P(x)$ as a product of $m$ real quadratic polynomials.

Can one find 4004 positive integers such that the sum of any 2003 of them is not divisible by 2003?

If it is known that the equation $$x^4+ax^3+2x^2+bx+1=0$$ has a (real) root, prove the inequality $$a^2+b^2 \ge 8.$$ (A Egorov)

[u]Round 5[/u] [b]p13.[/b] A unit square is rotated $30^o$ counterclockwise about one of its vertices. Determine the area of the intersection of the original square with the rotated one. [b]p14.[/b] Suppose points $A$ and $B$ lie on a circle of radius $4$ with center $O$, such that $\angle AOB = 90^o$. The perpendicular bisectors of segments $OA$ and $OB$ divide the interior of the circle into four regions. Find the area of the smallest region. [b]p15.[/b] Let $ABCD$ be a quadrilateral such that $AB = 4$, $BC = 6$, $CD = 5$, $DA = 3$, and $\angle DAB = 90^o$. There is a point $I$ inside the quadrilateral that is equidistant from all the sides. Find $AI$. [u]Round 6[/u] [i]The answer to each of the three questions in this round depends on the answer to one of the other questions. There is only one set of correct answers to these problems; however, each question will be scored independently, regardless of whether the answers to the other questions are correct. [/i] [b]p16.[/b] Let $C$ be the answer to problem $18$. Compute $$\left( 1 - \frac{1}{2^2} \right) \left( 1 - \frac{1}{3^2} \right) ... \left( 1 - \frac{1}{C^2} \right).$$ [b]p17.[/b] Let $A$ be the answer to problem $16$. Let $PQRS$ be a square, and let point $M$ lie on segment $PQ$ such that $MQ = 7PM$ and point $N$ lie on segment $PS$ such that $NS = 7PN$. Segments $MS$ and $NQ$ meet at point $X$. Given that the area of quadrilateral $PMXN$ is $A - \frac12$, find the side length of the square. [b]p18.[/b] Let $B$ be the answer to problem $17$ and let $N = 6B$. Find the number of ordered triples $(a, b, c)$ of integers between $0$ and $N - 1$, inclusive, such that $a + b + c$ is divisible by $N$. [u]Round 7[/u] [b]p19.[/b] Let $k$ be the units digit of $\underbrace{7^{7^{7^{7^{7^{7^{7}}}}}}}_{Seven \,\,7s}$ . What is the largest prime factor of the number consisting of $k$ $7$’s written in a row? [b]p20.[/b] Suppose that $E = 7^7$ , $M = 7$, and $C = 7·7·7$. The characters $E, M, C, C$ are arranged randomly in the following blanks. $$... \times ... \times ... \times ... $$ Then one of the multiplication signs is chosen at random and changed to an equals sign. What is the probability that the resulting equation is true? [b]p21[/b]. During a recent math contest, Sophy Moore made the mistake of thinking that $133$ is a prime number. Fresh Mann replied, “To test whether a number is divisible by $3$, we just need to check whether the sum of the digits is divisible by $3$. By the same reasoning, to test whether a number is divisible by $7$, we just need to check that the sum of the digits is a multiple of $7$, so $133$ is clearly divisible by $7$.” Although his general principle is false, $133$ is indeed divisible by $7$. How many three-digit numbers are divisible by $7$ and have the sum of their digits divisible by $7$? [u]Round 8[/u] [b]p22.[/b] A [i]look-and-say[/i] sequence is defined as follows: starting from an initial term $a_1$, each subsequent term $a_k$ is found by reading the digits of $a_{k-1}$ from left to right and specifying the number of times each digit appears consecutively. For example, $4$ would be succeeded by $14$ (“One four.”), and $31337$ would be followed by $13112317$ (“One three, one one, two three, one seven.”) If $a_1$ is a random two-digit positive integer, find the probability that $a_4$ is at least six digits long. [b]p23.[/b] In triangle $ABC$, $\angle C = 90^o$. Point $P$ lies on segment $BC$ and is not $B$ or $C$. Point $I$ lies on segment $AP$, and $\angle BIP = \angle PBI = \angle CAB$. If $\frac{AP}{BC} = k$, express $\frac{IP}{CP}$ in terms of $k$. [b]p24.[/b] A subset of $\{1, 2, 3, ... , 30\}$ is called [i]delicious [/i] if it does not contain an element that is $3$ times another element. A subset is called super delicious if it is delicious and no delicious set has more elements than it has. Determine the number of super delicious subsets. PS. You sholud use hide for answers. First rounds have been posted [url=https://artofproblemsolving.com/community/c4h2784267p24464980]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$ c$ is a positive integer. Consider the following recursive sequence: $ a_1\equal{}c, a_{n\plus{}1}\equal{}ca_{n}\plus{}\sqrt{(c^2\minus{}1)(a_n^2\minus{}1)}$, for all $ n \in N$. Prove that all the terms of the sequence are positive integers.

[u]Round 1[/u] [b]p1.[/b] An iscoceles triangle has one angle equal to $100$ degrees, what is the degree measure of one of the two remaining angles. [b]p2.[/b] Tanmay picks four cards from a standard deck of $52$ cards at random. What is the probability he gets exactly one Ace, exactly exactly one King, exactly one Queen, exactly one Jack and exactly one Ten? [b]p3.[/b] What is the sum of all the factors of $2014$? [u]Round 2[/u] [b]p4.[/b] Which number under $1000$ has the greatest number of factors? [b]p5.[/b] How many $10$ digit primes have all distinct digits? [b]p6.[/b] In a far o universe called Manhattan, the distance between two points on the plane $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ is defined as $d(P,Q) = |x_1-x_2|+|y_1-y_2|$. Let $S$ be the region of points that are a distance of $\le 7$ away from the origin $(0, 0)$. What is the area of $S$? [u]Round 3[/u] [b]p7.[/b] How many factors does $13! + 14! + 15!$ have? [b]p8.[/b] How many zeroes does $45!$ have consecutively at the very end in its representation in base $45$? [b]p9.[/b] A sequence of circles $\omega_0$, $\omega_1$, $\omega_2$, ... is drawn such that: $\bullet$ $\omega_0$ has a radius of $1$. $\bullet$ $\omega_{i+1}$ has twice the radius of $\omega_i$. $\bullet$ $\omega_i$ is internally tangent to $\omega_{i+1}$. Let $A$ be a point on $\omega_0$ and $B$ be a point on $\omega_{10}$. What is the maximum possible value of $AB$? [u]Round 4[/u] [b]p10.[/b] A $3-4-5$ triangle is constructed. Then a similar triangle is constructed with the shortest side of the first triangle being the new hypotenuse for the second triangle. This happens an infinite amount of times. What is the maximum area of the resulting figure? [b]p11.[/b] If an unfair coin is flipped $4$ times and has a $3/64$ chance of coming heads exactly thrice, what is the probability the coin comes tails on a single flip. [b]p12.[/b] Find all triples of positive integers $(a, b, c)$ that satisfy $2a = 1+bc$, $2b = 1+ac$, and $2c = 1 + ab$. [u]Round 5[/u] [b]p13.[/b] $6$ numbered points on a plane are placed so that they can create a regular hexagon $P_1P_2P_3P_4P_5P_6$ if connected. If a triangle is drawn to include a certain amount of points in it, how many triangles are there that hold a different set of points? (note: the triangle with $P_1$ and $P_2$ is not the same as the one with $P_3$ and $P_4$). [b]p14.[/b] Let $S$ be the set of all numbers of the form $n(2n + 1)(3n + 2)(4n + 3)(5n + 4)$ for $n \ge 1$. What is the largest number that divides every member of $S$? [b]p15. [/b]Jordan tosses a fair coin until he gets heads at least twice. What is the expected number of flips of the coin that he will make? PS. You should use hide for answers. Rounds 6-10 have been posted [url=https://artofproblemsolving.com/community/c3h3156859p28695035]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let's consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total of $d$ even numbers and $m$ odd numbers. Determine the maximum possible value of $2d + 4m$. [i]Proposed by Gogi Khimshiashvili, Georgia[/i]

Given that, a function $f(n)$, defined on the natural numbers, satisfies the following conditions: (i)$f(n)=n-12$ if $n>2000$; (ii)$f(n)=f(f(n+16))$ if $n \leq 2000$. (a) Find $f(n)$. (b) Find all solutions to $f(n)=n$.

Find all values of $a$ for which the equation $x^3 - x + a = 0$ has three different integer solutions.

Classify up to homeomorphism the topological spaces of the support of functions that are real quadratic polynoms of three variables and and irreducible over the set of real numbers.

Show that \[1 \leq \frac{(x+y)(x^3+y^3)}{(x^2+y^2)^2} \leq \frac98\] holds for all positive real numbers $x,y$.

Evaluate $$\sum^{17}_{n=2} \frac{n^2+n+1}{n^4+2n^3-n^2-2n}.$$