Let $k$ be a positive integer and $a_1,a_2,\dots$ be an infinite sequence of positive integers such that \[a_ia_{i+1} \mid k-a_i^2\] for all integers $i \ge 1$. Prove that there exists a positive integer $M$ such that $a_n=a_{n+1}$ for all integers $n \ge M$.

Suppose 32 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?

A square with horizontal and vertical sides is drawn on the plane. It held several segments parallel to the sides, and there are no two segments which lie on one line or intersect at an interior point for both segments. It turned out that the segments cuts square into rectangles, and any vertical line intersecting the square and not containing segments of the partition intersects exactly $ k $ rectangles of the partition, and any horizontal line intersecting the square and not containing segments of the partition intersects exactly $\ell$ rectangles. How much the number of rectangles can be? [i]I. Bogdanov, D. Fon-Der-Flaass[/i]

Daniel chooses some distinct subsets of $\{1, \dots, 2019\}$ such that any two distinct subsets chosen are disjoint. Compute the maximum possible number of subsets he can choose. [i]Proposed by Ankan Bhattacharya[/i]

On a blackboard a finite number of integers greater than one are written. Every minute, Nordi additionally writes on the blackboard the smallest positive integer greater than every other integer on the blackboard and not divisible by any of the numbers on the blackboard. Show that from some point onwards Nordi only writes primes on the blackboard.

Find all real numbers $x,y,z$ satisfying the following system of equations: \begin{align*} xy+z&=-30\\ yz+x &= 30\\ zx+y &=-18 \end{align*}

Compute the number of positive integers $n$ less than $100$ for which $1+2+\dots+n$ is not divisible by $n.$

Suppose that $n$ triangles are given in the plane such that any three of them have a common vertex, but no four of them do. Find the greatest possible $n$.

Each of $2023$ balls is placed in on of $3$ bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls? $\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } \frac{3}{10} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{1}{4}$

The functions $ f(x) ,\ g(x)$ satify that $ f(x) \equal{} \frac {x^3}{2} \plus{} 1 \minus{} x\int_0^x g(t)\ dt,\ g(x) \equal{} x \minus{} \int_0^1 f(t)\ dt$. Let $ l_1,\ l_2$ be the tangent lines of the curve $ y \equal{} f(x)$, which pass through the point $ (a,\ g(a))$ on the curve $ y \equal{} g(x)$. Find the minimum area of the figure bounded by the tangent tlines $ l_1,\ l_2$ and the curve $ y \equal{} f(x)$ .

Prove that: for all positive integers $n\ge 2,$ the polynomial$$(x^2-1)^2(x^2-1)^2...(x^2-2023)^2+1$$ is irreducible in $\mathbb{Q}[x].$

Let $G$ be a planar graph, which is also bipartite. Is it always possible to assign a vertex to each face of the graph such that no two faces have the same vertex assigned to them? [i]Submitted by Dávid Matolcsi, Budapest[/i]

Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly $ 5\over 4$ times greater than Vasya's result. What minimum number of portraits can be on the wall? [i]Author: V. Frank[/i]

Let $ABC$ be a triangle with $BA=BC$ and $\angle ABC=90^{\circ}$. Let $D$ and $E$ be the midpoints of $CA$ and $BA$ respectively. The point $F$ is inside of $\triangle ABC$ such that $\triangle DEF$ is equilateral. Let $X=BF\cap AC$ and $Y=AF\cap DB$. Prove that $DX=YD$.

Let $ABC$ be a triangle. Let $\omega_1$ be circle tangent to $BC$ at $B$ and passes through $A$. Let $\omega_2$ be circle tangent to $BC$ at $C$ and passes through $A$. Let $\omega_1$ and $\omega_2$ intersect again at $P \neq A$. Let $\omega_1$ intersect $AC$ again at $E\neq A$, and let $\omega_2$ intersect $AB$ again at $F\neq A$. Let $R$ be the reflection of $A$ about $BC$, Prove that lines $BE, CF, PR$ are concurrent.

Let $ n $ $(n\geq2)$ be an integer. Find the greatest possible value of the expression $$E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2}$$ if the positive real numbers $a_1,a_2,\ldots,a_n$ satisfy $a_1+a_2+\ldots+a_n=\frac{n}{2}.$ What are the values of $a_1,a_2,\ldots,a_n$ when the greatest value is achieved?

Find the value of the integral $\int_{-1}^{1} ln \left(x +\sqrt{1 + x^2}\right) dx$.

Show that in a right-angled triangle the bisector of the right angle divides into equal parts the angle between the altitude and the median, drawn from the same vertex.

Let $p=2^{16}+1$ be a prime. Let $N$ be the number of ordered tuples $(A,B,C,D,E,F)$ of integers between $0$ and $p-1$, inclusive, such that there exist integers $x,y,z$ not all divisible by $p$ with $p$ dividing all three of $Ax+Ez+Fy, By+Dz+Fx, Cz+Dy+Ex$. Compute the remainder when $N$ is divided by $10^6$. [i]Proposed by Vincent Huang[/i]

A regular star-shaped hexagon is split into $4$ parts. Construct from them a convex polygon. Note: A regular six-pointed star is a figure that is obtained by combining a regular triangle and a triangle symmetrical to it relative to its center

Each of $K$ friends simultaneously learns one different item of news. They begin to phone one another to tell them their news. Each conversation lasts exactly one hour, during which time it is possible for two friends to tell each other all of their news. What is the minimum number of hours needed in order for all of the friends to know all of the news? Consider in this problem: (a) $K = 64$. (b) $K = 55$. (c) $K = 100$. (A Andjans, Riga) PS. (a) was the junior problem, (a),(b),(c) the senior one

Let $k$ be a positive real number. In the $X-Y$ coordinate plane, let $S$ be the set of all points of the form $(x,x^2+k)$ where $x\in\mathbb{R}$. Let $C$ be the set of all circles whose center lies in $S$, and which are tangent to $X$-axis. Find the minimum value of $k$ such that any two circles in $C$ have at least one point of intersection.

If $-1 < x < 1$ and $-1 < y < 1$, define the "relativistic sum'' $x \oplus y$ to be \[ x \oplus y = \frac{x + y}{1 + xy} \, . \] The operation $\oplus$ is commutative and associative. Let $v$ be the number \[ v = \frac{\sqrt[7]{17} - 1}{\sqrt[7]{17} + 1} \, . \] What is the value of \[ v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \, ? \] (In this expression, $\oplus$ appears 13 times.)

The negation of the statement "all men are honest," is: $ \textbf{(A)}\ \text{no men are honest} \qquad\textbf{(B)}\ \text{all men are dishonest} \\ \textbf{(C)}\ \text{some men are dishonest} \qquad\textbf{(D)}\ \text{no men are dishonest} \\ \textbf{(E)}\ \text{some men are honest}$

[b]p1.[/b] Each point in the interior and on the boundary of a square of side $2$ inches is colored either red or blue. Prove that there exists at least one pair of points of the same color whose distance apart is not less than $-\sqrt5$ inches. [b]p2.[/b] $ABC$ is an equilateral triangle of altitude $h$. A circle with center $0$ and radius $h$ is tangent to side $AB$ at $Z$ and intersects side $AC$ in point $X$ and side $BC$ in point $Y$. Prove that the circular arc $XZY$ has measure $60^o$. [img]https://cdn.artofproblemsolving.com/attachments/b/e/ac70942f7a14cd0759ac682c3af3551687dd69.png[/img] [b]p3.[/b] Find all of the real and complex solutions (if any exist) of the equation $x^7 + 7^7 = (x + 7)^7$ [b]p4.[/b] The four points $A, B, C$, and $D$ are not in the same plane. Given that the three angles, angle $ABC$, angle $BCD$, and angle $CDA$, are all right angles, prove that the fourth angle, angle $DAB$, of this skew quadrilateral is acute. [b]p5.[/b] $A, B, C$ and $D$ are four positive whole numbers with the following properties: (i) each is less than the sum of the other three, and (ii) each is a factor of the sum of the other three. Prove that at least two of the numbers must be equal. (An example of four such numbers: $A = 4$, $B = 4$, $C = 2$, $D = 2$.) [b]p6.[/b] $S$ is a set of six points and $L$ is a set of straight line segments connecting certain pairs of points in $S$ so that each point of $S$ is connected with at least four of the other points. Let $A$ and $B$ denote two arbitrary points of $S$. Show that among the triangles having sides in $L$ and vertices in $S$ there are two with the properties: (i) The two triangles have no common vertex. (ii) $A$ is a vertex of one of the triangles, and $B$ is a vertex of the other. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].