Let $ABC$ be an isosceles triangle with base $AB$ and $D$ be a point on side $AB$ such that the incircle of triangle $ACD$ is congruent to the excircle of triangle $DCB$ across $C$. Prove that the diameter of each of these circles equals half the altitude of $\vartriangle ABC$ from $A$

The [i]liar's guessing game[/i] is a game played between two players $A$ and $B$. The rules of the game depend on two positive integers $k$ and $n$ which are known to both players. At the start of the game $A$ chooses integers $x$ and $N$ with $1 \le x \le N.$ Player $A$ keeps $x$ secret, and truthfully tells $N$ to player $B$. Player $B$ now tries to obtain information about $x$ by asking player $A$ questions as follows: each question consists of $B$ specifying an arbitrary set $S$ of positive integers (possibly one specified in some previous question), and asking $A$ whether $x$ belongs to $S$. Player $B$ may ask as many questions as he wishes. After each question, player $A$ must immediately answer it with [i]yes[/i] or [i]no[/i], but is allowed to lie as many times as she wants; the only restriction is that, among any $k+1$ consecutive answers, at least one answer must be truthful. After $B$ has asked as many questions as he wants, he must specify a set $X$ of at most $n$ positive integers. If $x$ belongs to $X$, then $B$ wins; otherwise, he loses. Prove that: 1. If $n \ge 2^k,$ then $B$ can guarantee a win. 2. For all sufficiently large $k$, there exists an integer $n \ge (1.99)^k$ such that $B$ cannot guarantee a win. [i]Proposed by David Arthur, Canada[/i]

Let $AA_1, BB_1, CC_1$ be the heights of triangle $ABC$ and $H$ be its orthocenter. Liune $\ell$ parallel to $AC$, intersects straight lines $AA_1$ and $CC_1$ at points $A_2$ and $C_2$, respectively. Suppose that point $B_1$ lies outside the circumscribed circle of triangle $A_2 HC_2$. Let $B_1P$ and $B_1T$ be tangent to of this circle. Prove that points $A_1, C_1, P$, and $T$ are cyclic.

Let Q+ and Z denote the set of positive rationals and the set of inte- gers, respectively. Find all functions f : Q+ → Z satisfying the following conditions: (i) f(1999) = 1; (ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+; (iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.

For every positive integer $n$ show that \[[\sqrt{4n + 1}] = [\sqrt{4n + 2}] = [\sqrt{4n + 3}] = [\sqrt{n} + \sqrt{n + 1}]\] where $[x]$ is the greatest integer less than or equal to $x$ (for example $[2.3] = 2$, $[\pi] = 3$, $[5] = 5$).

Let $ ABCD$ be a cyclic quadrilateral with $ AB>CD$ and $ BC>AD$. Take points $ X$ and $ Y$ on the sides $ AB$ and $ BC$, respectively, so that $ AX\equal{}CD$ and $ AD\equal{}CY$. Let $ M$ be the midpoint of $ XY$. Prove that $ AMC$ is a right angle.

For a positive integer $n$, consider the set \[S = \{0, 1, 1 + 2, 1 + 2 + 3, \ldots, 1 + 2 + 3 +\ldots + (n - 1)\}\] Prove that the elements of $S$ are mutually incongruent modulo $n$ if and only if $n$ is a power of $2$.

A five-digit positive integer is called [i]$k$-phobic[/i] if no matter how one chooses to alter at most four of the digits, the resulting number (after disregarding any leading zeroes) will not be a multiple of $k$. Find the smallest positive integer value of $k$ such that there exists a $k$-phobic number. [i]Proposed by Yannick Yao[/i]

Let $n\ge 3$ be an integer such that for every prime factor $q$ of $n-1$ exists an integer $a > 1$ such that $a^{n-1} \equiv 1 \,(\mod n \, )$ and $a^{\frac{n-1} {q}}\not\equiv 1 \,(\mod n \, )$. Prove that $n$ is not prime.

The sum $$\frac{1^2-2}{1!} + \frac{2^2-2}{2!} + \frac{3^2-2}{3!} + \cdots + \frac{2021^2 - 2}{2021!}$$ $ $ \\ can be expressed as a rational number $N$. Find the last 3 digits of $2021! \cdot N$.

Let $ABCDEF$ be a convex hexagon such that $AB = BC, CD = DE, EF = FA$. Prove that $\frac{BC}{BE} +\frac{DE}{DA} +\frac{FA}{FC} \ge \frac{3}{2}$ . When does equality occur?

Is it possible to divide the plane into polygons so that each polygon is transformed into itself under some rotation by $360/7$ degrees about some point? All sides of these polygons must be greater than $1$ cm. (A polygon is the part of a plane bounded by one non-self-intersect-ing closed broken line, not necessarily convex.) (A. Andjans, Riga)

$ A_{1}B_{1}B_{2}A_{2}$ is a convex quadrilateral, and $ A_{1}B_{1}\neq A_{2}B_{2}$. Show that there exists a point $ M$ such that \[\frac{A_{1}B_{1}}{A_{2}B_{2}}\equal{}\frac{MA_{1}}{MA_{2}}\equal{}\frac{MB_{1}}{MB_{2}}\]

Reimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each vertex of the polyhedron along a plane through the trisection point on each adjacent edge that is closer to the vertex. For example, the polyhedron after the first step has six octagonal faces and eight equilateral triangular faces. How many faces are on the polyhedron after the fifth step?

An eel is a polyomino formed by a path of unit squares which makes two turns in opposite directions (note that this means the smallest eel has four cells). For example, the polyomino shown below is an eel. What is the maximum area of a $1000 \times 1000$ grid of unit squares that can be covered by eels without overlap? [asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(4cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -3.5240093012803015, xmax = 11.281247367575242, ymin = -2.6545617728840334, ymax = 7.1662584841234755; /* image dimensions */ draw((0,0)--(5,0)--(5,3)--(7,3)--(7,4)--(4,4)--(4,1)--(0,1)--cycle, linewidth(2)); /* draw figures */ draw((0,0)--(5,0), linewidth(2)); draw((5,0)--(5,3), linewidth(2)); draw((5,3)--(7,3), linewidth(2)); draw((7,3)--(7,4), linewidth(2)); draw((7,4)--(4,4), linewidth(2)); draw((4,4)--(4,1), linewidth(2)); draw((4,1)--(0,1), linewidth(2)); draw((0,1)--(0,0), linewidth(2)); draw((6,3)--(6,4), linewidth(2)); draw((5,4)--(5,3), linewidth(2)); draw((5,3)--(4,3), linewidth(2)); draw((4,2)--(5,2), linewidth(2)); draw((5,1)--(4,1), linewidth(2)); draw((4,0)--(4,1), linewidth(2)); draw((3,0)--(3,1), linewidth(2)); draw((2,0)--(2,1), linewidth(2)); draw((1,1)--(1,0), linewidth(2)); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from $1$ to $n$ and let $a_i$ be the number of friends of the $i^{\text{th}}$ resident. Suppose that \[ \sum_{i=1}^{n}a_i^2=n^2-n \] Let $k$ be the smallest number of residents (at least three) who can be seated at a round table in such a way that any two neighbors are friends. Determine all possible values of $k.$

One fair die has faces $ 1$, $ 1$, $ 2$, $ 2$, $ 3$, $ 3$ and another has faces $ 4$, $ 4$, $ 5$, $ 5$, $ 6$, $ 6$. The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd? $ \textbf{(A)}\ \frac{1}{3}\qquad \textbf{(B)}\ \frac{4}{9}\qquad \textbf{(C)}\ \frac{1}{2}\qquad \textbf{(D)}\ \frac{5}{9}\qquad \textbf{(E)}\ \frac{2}{3}$

Prove that any positive integer can be represented as a sum of Fibonacci numbers, no two of which are consecutive.

Show that in a tournament of $799$ teams (every team plays with every other team for a win or loss), there exist $14$ teams such that the first seven teams have each defeated the remaining teams.

For a set $S$ we denote its cardinality by $|S|$. Let $e_1,e_2,\ldots,e_k$ be non-negative integers. Let $A_k$ (respectively $B_k$) be the set of all $k$-tuples $(f_1,f_2,\ldots,f_k)$ of integers such that $0\leq f_i\leq e_i$ for all $i$ and $\sum_{i=1}^k f_i$ is even (respectively odd). Show that $|A_k|-|B_k|=0 \textrm{ or } 1$.

The bisectors $AA_1$ and $CC_1$ of triangle $ABC$ intersect at point $I$. The circumscribed circles of triangles $AIC_1$ and $CIA_1$ intersect the arcs $AC$ and $BC$ (not containing points $B$ and $A$ respectively) of the circumscribed circle of triangle $ABC$ at points $C_2$ and $A_2$, respectively. Prove that lines $A_1A_2$ and $C_1C_2$ intersect on the circumscribed circle of triangle $ABC$.

Assume that it is possible to color more than half of the surfaces of a given polyhedron so that no two colored surfaces have a common edge. (a) Describe one polyhedron with the above property. (b) Prove that one cannot inscribe a sphere touching all the surfaces of a polyhedron with the above property.

Determine all real solutions $(x,y)$ of the following system of equations: \begin{align*} x&=3x^2y-y^3,\\ y &= x^3-3xy^2 \end{align*}

The area of this figure is $ 100\text{ cm}^{2} $. Its perimeter is [asy] draw((0,2)--(2,2)--(2,1)--(3,1)--(3,0)--(1,0)--(1,1)--(0,1)--cycle,linewidth(1)); draw((1,2)--(1,1)--(2,1)--(2,0),dashed);[/asy] $ \text{(A)}\ \text{20 cm}\qquad\text{(B)}\ \text{25 cm}\qquad\text{(C)}\ \text{30 cm}\qquad\text{(D)}\ \text{40 cm}\qquad\text{(E)}\ \text{50 cm} $

Two congruent rectangles are located as shown in the figure. Find the area of the shaded part. [img]https://cdn.artofproblemsolving.com/attachments/2/e/10b164535ab5b3a3b98ce1a0b84892cd11d76f.png[/img]