The diagonals of a cyclic quadrilateral meet at point $M$. A circle $\omega$ touches segments $MA$ and $MD$ at points $P,Q$ respectively and touches the circumcircle of $ABCD$ at point $X$. Prove that $X$ lies on the radical axis of circles $ACQ$ and $BDP$. [i](Proposed by Ivan Frolov)[/i]

Let $T$ be the set of ordered triples $(x,y,z)$, where $x,y,z$ are integers with $0\leq x,y,z\leq9$. Players $A$ and $B$ play the following guessing game. Player $A$ chooses a triple $(x,y,z)$ in $T$, and Player $B$ has to discover $A$[i]'s[/i] triple in as few moves as possible. A [i]move[/i] consists of the following: $B$ gives $A$ a triple $(a,b,c)$ in $T$, and $A$ replies by giving $B$ the number $\left|x+y-a-b\right |+\left|y+z-b-c\right|+\left|z+x-c-a\right|$. Find the minimum number of moves that $B$ needs to be sure of determining $A$[i]'s[/i] triple.

[b]p1.[/b] Arrange the whole numbers $1$ through $15$ in a row so that the sum of any two adjacent numbers is a perfect square. In how many ways this can be done? [b]p2.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 - q^2$ is divisible by $24$. [b]p3.[/b] If a polyleg has even number of legs he always tells truth. If he has an odd number of legs he always lies. Once a green polyleg told a dark-blue polyleg ”- I have $8$ legs. And you have only $6$ legs!” The offended dark-blue polyleg replied ”-It is me who has $8$ legs, and you have only $7$ legs!” A violet polyleg added ”-The dark-blue polyleg indeed has $8$ legs. But I have $9$ legs!” Then a stripped polyleg started ”None of you has $8$ legs. Only I have $8$ legs!” Which polyleg has exactly $8$ legs? [b][b]p4.[/b][/b] There is a small puncture (a point) in the wall (as shown in the figure below to the right). The housekeeper has a small flag of the following form (see the figure left). Show on the figure all the points of the wall where you can hammer in a nail such that if you hang the flag it will close up the puncture. [img]https://cdn.artofproblemsolving.com/attachments/a/f/8bb55a3fdfb0aff8e62bc4cf20a2d3436f5d90.png[/img] [b]p5.[/b] Assume $ a, b, c$ are odd integers. Show that the quadratic equation $ax^2 + bx + c = 0$ has no rational solutions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many solutions does the equation: $$[\frac{x}{20}]=[\frac{x}{17}]$$ have over the set of positve integers? $[a]$ denotes the largest integer that is less than or equal to $a$. [i]Proposed by Karl Czakler[/i]

Let $A_{1}A_{2}\ldots A_{2n}$ be a convex polygon and let $P$ be a point in its interior such that it doesn't lie on any of the diagonals of the polygon. Prove that there is a side of the polygon such that none of the lines $PA_{1}$, $\ldots$, $PA_{2n}$ intersects it in its interior.

Find a polynomial with integer coefficients which has $\sqrt2 + \sqrt3$ and $\sqrt2 + \sqrt[3]{3}$ as roots.

Point $B$ is due east of point $A$. Point $C$ is due north of point $B$. The distance between points $A$ and $C$ is $10\sqrt{2}$ meters, and $\angle BAC=45^{\circ}$. Point $D$ is $20$ meters due north of point $C$. The distance $AD$ is between which two integers? $ \textbf{(A)}\ 30\text{ and }31\qquad\textbf{(B)}\ 31\text{ and }32\qquad\textbf{(C)}\ 32\text{ and }33\qquad\textbf{(D)}\ 33\text{ and }34\qquad\textbf{(E)}\ 34\text{ and }35$

How many positive perfect squares less than $2023$ are divisible by $5$? $\textbf{(A) } 8 \qquad\textbf{(B) }9 \qquad\textbf{(C) }10 \qquad\textbf{(D) }11 \qquad\textbf{(E) } 12$

Natural numbers from $1$ to $200$ were divided into $50$ sets. Prove that one of them contains three numbers that are the lengths of the sides some triangle.

In an acute triangle $ABC,$ let $O,G,H$ be its circumcentre, centroid and orthocenter. Let $D\in BC, E\in CA$ and $OD\perp BC, HE\perp CA.$ Let $F$ be the midpoint of $AB.$ If the triangles $ODC, HEA, GFB$ have the same area, find all the possible values of $\angle C.$

Determine all quadruples $ (a,b,c,d)$ of real numbers satisfying the equation: $ 256a^3 b^3 c^3 d^3\equal{}(a^6\plus{}b^2\plus{}c^2\plus{}d^2)(a^2\plus{}b^6\plus{}c^2\plus{}d^2)(a^2\plus{}b^2\plus{}c^6\plus{}d^2)(a^2\plus{}b^2\plus{}c^2\plus{}d^6).$

Let $n$ be a positive integer, let $p$ be prime and let $q$ be a divisor of $(n + 1)^p - n^p$. Show that $p$ divides $q - 1$.

A triangle $ABC$ and a point $P$ inside it are given. $A', B', C'$ are the projections of $P$ onto the straight lines ot the sides $BC,CA,AB$. Prove that the center of the circle circumscribed around the triangle $A'B'C'$ lies inside the triangle $ABC$.

Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$

Find all integer solutions of the equation $$y^2+2y=x^4+20x^3+104x^2+40x+2003.$$ Note: has appeared many times before, see [url=https://artofproblemsolving.com/community/q1_%22x%5E4%2B20x%5E3%2B104x%5E2%22]here[/url]

Let $\left \lfloor x \right \rfloor$ denote the greatest integer less than or equal to $x$. Find the sum of all positive integer solutions to $$\left \lfloor \frac{n^3}{27} \right \rfloor - \left \lfloor \frac{n}{3} \right \rfloor ^3=10.$$ [i]Proposed by Jason Hsu[/i]

Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.

Over all pairs of complex numbers $(x,y)$ satisfying the equations $$x+2y^2=x^4 \quad \text{and} \quad y+2x^2=y^2,$$ compute the minimum possible real part of $x.$

Let \( G \) be a graph colored using \( k \) colors. We say that a vertex is [b]forced[/b] if it has neighbors in all the other \( k - 1 \) colors. Prove that for any \( 2024 \)-regular graph \( G \) that contains no triangles or quadrilaterals, there exists a coloring using \( 2025 \) colors such that at least \( 1013 \) of the colors have a forced vertex of that color. Note: The graph coloring must be valid, this means no \( 2 \) vertices of the same color may be adjacent.

$x_{n+1}= \left ( 1+\frac2n \right )x_n+\frac4n$, for every positive integer $n$. If $x_1=-1$, what is $x_{2000}$? $ \textbf{(A)}\ 1999998 \qquad\textbf{(B)}\ 2000998 \qquad\textbf{(C)}\ 2009998 \qquad\textbf{(D)}\ 2000008 \qquad\textbf{(E)}\ 1999999 $

Let $ABC$ be a triangle inscribed in $(O)$. Two tangents of $(O)$ at $B,C$ meets at $P$. The bisector of angle $BAC $ intersects $(P,PB)$ at point $E$ lying inside triangle $ABC$. Let $M,N$ be the midpoints of arcs $BC$ and $BAC$. Circle with diameter $BC$ intersects line segment $EN$ at $F$. Prove that the orthocenter of triangle $EFM$ lies on $BC$.

Three $3$-legged (distinguishable) Stanfurdians take off their socks and trade them with each other. How many ways is this possible if everyone ends up with exactly $3$ socks and nobody gets any of their own socks? All socks originating from the Stanfurdians are distinguishable from each other. All Stanfurdian feet are indistinguishable from other feet of the same Stanfurdian.

Tanya cut out a convex polygon from the paper, fold it several times and obtained a two-layers quadrilateral. Can the cutted polygon be a heptagon?

[b]a)[/b] Let $ a,b $ two non-negative integers such that $ a^2>b. $ Show that the equation $$ \left\lfloor\sqrt{x^2+2ax+b}\right\rfloor =x+a-1 $$ has an infinite number of solutions in the non-negative integers. Here, $ \lfloor\alpha\rfloor $ denotes the floor of $ \alpha. $ [b]b)[/b] Find the floor of $ m=\sqrt{2+\sqrt{2+\underbrace{\cdots}_{\text{n times}}+\sqrt{2}}} , $ where $ n $ is a natural number. Justify.

Let $X$ be a point inside triangle $ABC$ such that $XA.BC=XB.AC=XC.AC$. Let $I_1, I_2, I_3$ be the incenters of $XBC, XCA, XAB$. Prove that $AI_1, BI_2, CI_3$ are concurrent. [hide]Of course, the most natural way to solve this is the Ceva sin theorem, but there is an another approach that may surprise you;), try not to use the Ceva theorem :))[/hide]