A triangle $ABC$ with $AB>AC$ is given, $AD$ is the A-angle bisector with point $D$ on $BC$ and point $I$ is the incenter of triangle $ABC$. Point M is the midpoint of segment $AD$ and point $F$ is the second intersection of $MB$ with the circumcirle of triangle $BIC$. Prove that $AF\bot FC$.

Andrew has a four-digit number whose last digit is $2$. Given that this number is divisible by $9$, determine the number of possible values for this number that Andrew could have. Note that leading zeros are not allowed.

In a particular European city, there are only $7$ day tickets and $30$ day tickets to the public transport. The former costs $7.03$ euro and the latter costs $30$ euro. Aina the Algebraist decides to buy at once those tickets that she can travel by the public transport the whole three year (2014-2016, 1096 days) visiting in the city. What is the cheapest solution?

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that: - $f$ is strictly increasing, - there exists $M \in \mathbb{N}$ such that $f(x+1) - f(x) < M$ for all $x \in \mathbb{N}$, - for every $x \in \mathbb{Z}$, there exists $y \in \mathbb{Z}$ such that \[ f(y) = \frac{f(x) + f(x + 2024)}{2}. \] [i]Proposed by Pavle Martinović[/i]

[b]p1.[/b] In triangle $ABC$, the median $BM$ is drawn. The length $|BM| = |AB|/2$. The angle $\angle ABM = 50^o$. Find the angle $\angle ABC$. [b]p2.[/b] Is there a positive integer $n$ which is divisible by each of $1, 2,3,..., 2018$ except for two numbers whose difference is$ 7$? [b]p3.[/b] Twenty numbers are placed around the circle in such a way that any number is the average of its two neighbors. Prove that all of the numbers are equal. [b]p4.[/b] A finite number of frogs occupy distinct integer points on the real line. At each turn, a single frog jumps by $1$ to the right so that all frogs again occupy distinct points. For some initial configuration, the frogs can make $n$ moves in $m$ ways. Prove that if they jump by $1$ to the left (instead of right) then the number of ways to make $n$ moves is also $m$. [b]p5.[/b] A square box of chocolates is divided into $49$ equal square cells, each containing either dark or white chocolate. At each move Alex eats two chocolates of the same kind if they are in adjacent cells (sharing a side or a vertex). What is the maximal number of chocolates Alex can eat regardless of distribution of chocolates in the box? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Three convex polygons are given on a plane. Prove that there is no line cutting all the polygons if and only if each of the polygons can be separated from the other two by a line.

Given a semicirle with center $O$ an arbitrary inner point of the diameter divides it into two segments. Let there be semicircles above the two segments as visible in the below figure. The line $\ell$ passing through the point $A$ intersects the semicircles in $4$ points: $B, C, D$ and $E$. Show that the segments $BC$ and $DE$ have the same length. [img]https://cdn.artofproblemsolving.com/attachments/1/4/86a369d54fef7e25a51fea6481c0b5e7dd45ff.png[/img]

Let $ABCD$ be a quadrilateral such that line $AB$ intersects $CD$ at $X$. Denote circles with inradius $r_1$ and centers $A, B$ as $w_a$ and $w_b$ with inradius $r_2$ and centers $C, D$ as $w_c$ and $w_d$. $w_a$ intersects $w_d$ at $P, Q$. $w_b$ intersects $w_c$ at $R, S$. Prove that if $XA.XB+r_2^2=XC.XD+r_1^2$, then $P,Q,R,S$ are cyclic.

Michael writes down all the integers between $1$ and $N$ inclusive on a piece of paper and discovers that exactly $40\%$ of them have leftmost digit $1$. Given that $N > 2017$, find the smallest possible value of $N$.

In the diagram below, $AB$ is a diameter of circle $O$. Point C is drawn such that $\overline{BC}$ is tangent to circle $O$, and $AB = BC$. A point $F$ is selected on line $AB$ and a point $D$ is selected on circle $O$ such that $\angle CDF = 90^\circ$. Line $\overline{BD}$ is then extended to point $E$ such that $AE$ is tangent to circle $O$. Given that $AE = 5$, calculate the length of $\overline{AF}$. (Diagram not to scale) [asy] size(120); pair A,O,F,B,D,EE,C; A=(-5,0); O=(0,0); B=(5,0); EE=(-5,6); F=(3.8,0); D=(-2.5,4.33); C=(5,10); dot(A^^O^^B^^EE^^F^^D^^C); draw(circle(O,5)); draw(A--EE--F--cycle); draw(D--B--C--cycle); draw(A--B); label("$A$",A,W); label("$O$",O,S); label("$B$",B,E); label("$F$",F,S); label("$E$",EE,N); label("$D$",D,N); label("$C$",C,N); [/asy] $\textbf{(A) }\dfrac92\qquad\textbf{(B) }5\qquad\textbf{(C) }3\sqrt3\qquad\textbf{(D) }7\qquad\textbf{(E) }\text{impossible to determine}$

Let $u_k$ be a sequence of integers, and let $V_n$ be the number of those which are less than or equal to $n$. Show that if $$\sum_{k=1}^{\infty} \frac{1}{u_k } < \infty,$$ then $$\lim_{n \to \infty} \frac{ V_{n}}{n}=0.$$

For real number $r$ let $f(r)$ denote the integer that is the closest to $r$ (if the fractional part of $r$ is $1/2$, let $f(r)$ be $r-1/2$). Let $a>b>c$ rational numbers such that for all integers $n$ the following is true: $f(na)+f(nb)+f(nc)=n$. What can be the values of $a$, $b$ and $c$? [i]Submitted by Gábor Damásdi, Budapest[/i]

There are 2022 marked points on a straight line so that every two adjacent points are the same distance apart. Half of all the points are coloured red and the other half are coloured blue. Can the sum of the lengths of all the segments with a red left endpoint and a blue right endpoint be equal to the sum of the lengths of all the segments with a blue left endpoint and a red right endpoint?

Let $n$ be a positive integer. Let $S$ be a set of ordered pairs $(x, y)$ such that $1\leq x \leq n$ and $0 \leq y \leq n$ in each pair, and there are no pairs $(a, b)$ and $(c, d)$ of different elements in $S$ such that $a^2+b^2$ divides both $ac+bd$ and $ad - bc$. In terms of $n$, determine the size of the largest possible set $S$.

There were finitely many persons at a party among whom some were friends. Among any $4$ of them there were either $3$ who were all friends among each other or $3$ who weren't friend with each other. Prove that you can separate all the people at the party in two groups in such a way that in the first group everyone is friends with each other and that all the people in the second group are not friends to anyone else in second group. (Friendship is a mutual relation).

Let $C(O_1),C(O_2)$ be two externally tangent circles at point $P$. A line $t$ is tangent to $C(O_1)$ in point $R$ and intersects $C(O_2)$ in points $A,B$ such that $A$ is closer to $R$ than $B$ is. The line $AO_1$ intersects the perpendicular to $t$ in $B$ at point $C$, the line $PC$ intersects $AB$ in $Q$. Prove that $QO_1$ passes through the midpoint of $BC$.

Given cubic polynomial with integer coefficients and three irrational roots. Show that none of these roots can be root of quadratic equation with integer coefficients.

The positive sequence $\{a_n\}$ satisfies:$a_1=1+\sqrt 2$ and $(a_n-a_{n-1})(a_n+a_{n-1}-2\sqrt n)=2(n\geq 2).$ (1)Find the general formula of $\{a_n\}$; (2)Find the set of all the positive integers $n$ so that $\lfloor a_n\rfloor=2022$.

Find all solutions of the equation $4^x+3^y=z^2$ in positive integers.

Let $PQRS$ be a cyclic quadrilateral with $\angle PSR = 90^o$, and let $H,K$ be the projections of $Q$ on the lines $PR$ and $PS$, respectively. Prove that the line $HK$ passes through the midpoint of the segment $SQ$.

Assume that the differential equation $$y'''+p(x)y''+q(x)y'+r(x)y=0$$has solutions $y_1(x)$, $y_2(x)$, $y_3(x)$ on the real line such that $$y_1(x)^2+y_2(x)^2+y_3(x)^2=1$$for all real $x$. Let $$f(x)=y_1'(x)^2+y_2'(x)^2+y_3'(x)^2.$$Find constants $A$ and $B$ such that $f(x)$ is a solution to the differential equation $$y'+Ap(x)y=Br(x).$$

In triangle $ABC$, $M\in(BC)$, $\frac{BM}{BC}=\alpha$, $N\in(CA)$, $\frac{CN}{CA}=\beta$, $P\in(AB)$, $\frac{AP}{AB}=\gamma$. Let $AM\cap BN=\{D\}$, $BN\cap CP=\{E\}$, $CP\cap AM=\{F\}$. Prove that $S_{DEF}=S_{BMD}+S_{CNE}+S_{APF}$ iff $\alpha+\beta+\gamma=1$.

Given triangle $ ABC$. The incircle of triangle $ ABC$ is tangent to $ BC,CA,AB$ at $ D,E,F$ respectively. Construct point $ G$ on $ EF$ such that $ DG$ is perpendicular to $ EF$. Prove that $ \frac{FG}{EG} \equal{} \frac{BF}{CE}$.

In an acute triangle $ABC$, let $M$ and $N$ be the midpoints of $AB$ and $AC$ and let $BH$ be its altitude from $B$. Its incircle touches $AC$ at $K$ and the line through $K$ parallel to $MH$ meets $MN$ at $P$. Prove that $AMPK$ has an incircle.

Consider the sequence $b_1$, $b_2$, $b_3$, $ . . .$ of real numbers defined by $b_1 = \frac{3+\sqrt3}{6}$ , $b_2 = 1$, and for $n \ge 3$, $$b_n =\frac{1- b_{n-1} - b_{n-2}}{2b_{n-1}b_{n-2} - b_{n-1} - b_{n-2}}.$$ Compute $b_{2023}$.