Tetrahedron $ABCD$ in which $AB=CD$ is given. Sphere inscribed in it is tangent to faces $ABC$ and $ABD$ respectively in $K$ and $L$. Prove that if points $K$ and $L$ are centroids of faces $ABC$ and $ABD$ then tetrahedron $ABCD$ is regular.

Let $ABC$ be an acute scalene triangle with incenter $I$. Show that the circumcircle of $BIC$ intersects the Euler line of $ABC$ in two distinct points. (Recall that the [i]Euler line[/i] of a scalene triangle is the line that passes through its circumcenter, centroid, orthocenter, and the nine-point center.) [i]Andrew Gu[/i]

Let $T$ be an arbitrary tetrahedron satisfying the following conditions: [b]I.[/b] Each its side has length not greater than 1, [b]II.[/b] Each of its faces is a right triangle. Let $s(T) = S^2_{ABC} + S^2_{BCD} + S^2_{CDA} + S^2_{DAB}$. Find the maximal possible value of $s(T)$.

Let $ABC$ be an acute triangle, with $AB \ne AC$. Let $D$ be the midpoint of the line segment $BC$, and let $E$ and $F$ be the projections of $D$ onto the sides $AB$ and $AC$, respectively. If $M$ is the midpoint of the line segment $EF$, and $O$ is the circumcenter of triangle $ABC$, prove that the lines $DM$ and $AO$ are parallel. [hide=PS] As source was given [url=https://artofproblemsolving.com/community/c629086_caucasus_mathematical_olympiad]Caucasus MO[/url], but I was unable to find this problem in the contest collections [/hide]

Let $P$ be a point inside the acute triangle $ABC,$ and let $Q$ be the isogonal conjugate of $P.$ Let $L,M$ and $N$ be the midpoints of the shorter arcs $BC,CA$ and $AB$ of the circumcircle of $ABC,$ respectively. Let $X_A$ be the intersection of ray $LQ$ and circle $(PBC),$ let $X_B$ be the intersection of ray $MQ$ and circle $PCA,$ and let $X_C$ be the intersection of ray $NQ$ and circle $(PAB).$ Prove that $P,X_A,X_B$ and $X_C$ are concyclic or coincide. [i]Proposed by Gustavo Cruz (São Paulo)[/i]

Let $ABC$ be a triangle in which $AB =AC$ and $\angle CAB = 90^{\circ}$. Suppose that $M$ and $N$ are points on the hypotenuse $BC$ such that $BM^2 + CN^2 = MN^2$. Prove that $\angle MAN = 45^{\circ}$.

A regular dodecagon $A_1A_2...A_{12}$ is inscribed in a circle with a diameter of $20$ cm . Calculate the perimeter of the pentagon $A_1A_3A_6A_8A_{11}$. (Alexey Panasenko)

In a triangle, the area is numerically equal to the perimeter. What is the radius of the inscribed circle? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }6$

A chord $AB$ is drawn in a circle, with center $M$ and radius $r$, that the two diameters which divide the largest arc $AB$ into three equal parts also divide the chord $AB$ into three equal parts. Express the length of that chord in terms of $r$.

[u]Fermi Questions[/u] [b]p1.[/b] What is $\sin (1000)$? (note: that's $1000$ radians, not degrees) [b]p2.[/b] In liters, what is the volume of $10$ million US dollars' worth of gold? [b]p3.[/b] How many trees are there on Earth? [b]p4.[/b] How many prime numbers are there between $10^8$ and $10^9$? [b]p5.[/b] What is the total amount of time spent by humans in spaceflight? [b]p6.[/b] What is the global domestic product (total monetary value of all goods and services produced in a country's borders in a year) of Bangladesh in US dollars? [b]p7.[/b] How much time does the average American spend eating during their lifetime, in hours? [b]p8.[/b] How many CHMMC-related emails did the directors receive or send in the last month? [u]Suspiciously Familiar. . .[/u] [b]p9.[/b] Suppose a farmer learns that he will die at the end of the year (day $365$, where today is day $0$) and that he has $100$ sheep. He decides to sell all his sheep on one day, and that his utility is given by $ab$ where $a$ is the money he makes by selling the sheep (which always have a fixed price) and $b$ is the number of days he has left to enjoy the profit; i.e., $365 - k$ where $k$ is the day number. If every day his sheep breed and multiply their numbers by $(421 + b)/421$ (yes, there are small, fractional sheep), on which day should he sell out? [b]p10.[/b] Suppose in your sock drawer of $14$ socks there are $5$ different colors and $3$ different lengths present. One day, you decide you want to wear two socks that have either different colors or different lengths but not both. Given only this information, what is the maximum number of choices you might have? [u]I'm So Meta Even This Acronym[/u] [b]p11.[/b] Let $\frac{s}{t}$ be the answer of problem $13$, written in lowest terms. Let $\frac{p}{q}$ be the answer of problem $12$, written in lowest terms. If player $1$ wins in problem $11$, let $n = q$. Otherwise, let $n = p$. Two players play a game on a connected graph with $n$ vertices and $t$ edges. On each player's turn, they remove one edge of the graph, and lose if this causes the graph to become disconnected. Which player (first or second) wins? [b]p12.[/b] Let $\frac{s}{t}$ be the answer of problem $13$, written in lowest terms. If player $1$ wins in problem $11$, let $n = t$. Otherwise, let $n = s$. Find the maximum value of $$\frac{x^n}{1 + \frac12 x + \frac14 x^2 + ...+ \frac{1}{2^{2n}} x^{2n}}$$ for $x > 0$. [b]p13.[/b] Let $\frac{p}{q}$ be the answer of problem $12$, written in lowest terms. Let $y$ be the largest integer such that $2^y$ divides $p$. If player $1$ wins in problem $11$, let $z = q$. Otherwise, let $z = p$. Suppose that $a_1 = 1$ and $$a_{n+1} = a_n -\frac{z}{n + 2}+\frac{2z}{n + 1}-\frac{z}{n}$$ What is $a_y$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $I$ be the center of a circle inscribed in triangle $ABC$. The circle circumscribed about the triangle $BIC$ intersects lines $AB$ and $AC$ at points $E$ and $F$, respectively. Prove that the line $EF$ touches the circle inscribed in the triangle $ABC$.

Construct a triangle $ABC$ if $AC=b$, $AB=c$ and $\angle AMB=w$, where $M$ is the midpoint of the segment $BC$ and $w<90$. Prove that a solution exists if and only if \[ b \tan{\dfrac{w}{2}} \leq c <b \] In what case does the equality hold?

A point-sized cue ball is fired in a straight path from the center of a regular hexagonal billiards table of side length $1$. If it is not launched directly into a pocket but travels an integer distance before falling into one of the pockets (located in the corners), find the minimum distance that it could have traveled.

It is known that four of these sticks can be assembled into a quadrilateral. Is it always true that you can make a triangle out of three of them?

A set $M$ of points in the plane will be called obtuse, if any 3 points from $M$ are the vertices of an obtuse triangle. a.) Prove: For each finite obtuse set $M$ there is a point in the plane with the following property: $P$ is no element from $M$ and $M \cup \{P\}$ is also obtuse. b.) Determine whether the statement from a.) will remain valid, if it is replaced by infinite.

Let $ABC$ be an acute-angled triangle and $M$ be the midpoint of the side $BC$. Let $N$ be a point in the interior of the triangle $ABC$ such that $\angle NBA=\angle BAM$ and $\angle NCA=\angle CAM$. Prove that $\angle NAB=\angle MAC$. [i]Gabriel Nagy[/i]

The sides of a triangle are $25,39,$ and $40$. The diameter of the circumscribed circle is: $ \textbf{(A)}\ \frac{133}{3}\qquad\textbf{(B)}\ \frac{125}{3}\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 41\qquad\textbf{(E)}\ 40 $

Triangle $\vartriangle ABC$ is inscribed in circle $O$ and has sides $AB = 47$, $BC = 69$, and $CA = 34$. Let $E$ be the point on $O$ such that $\overline{AE}$ and $\overline{BC}$ intersect inside $O$, $8$ units away from $B$. Let $P$ and $Q$ be the points on $\overleftrightarrow{BE}$ and $\overleftrightarrow{CE}$, respectively, such that $\angle EPA$ and $\angle EQA$ are right angles. Suppose lines $\overleftrightarrow{AP}$ and $\overleftrightarrow{AQ}$ respectively intersect $O$ again at $X$ and $Y$ . Compute the distance $XY$.

Given two equilateral triangles $A_1B_1C_1$ and $A_2B_2C_2$ in the plane. (The vertices are mentioned counterclockwise.) We draw vectors $\overrightarrow{OA}, \overrightarrow{OB}, \overrightarrow{OC}$, from the arbitrary point $O$, equal to $\overrightarrow{A_1A_2}, \overrightarrow{B_1B_2}, \overrightarrow{C_1C_2}$ respectively. Prove that the triangle $ABC$ is equilateral.

Let $\omega$ be the circumcircle of triangle $ABC$ with center $O$, and the $A$ inmixtilinear circle is tangent to $AB, AC, \omega$ at $D,E,T$ respectively. $P$ is the intersection of $TO$ and $DE$ and $X$ is the intersection of $AP$ and $\omega$. Prove that the isogonal conjugate of $P$ lies on the line passing through the midpoint of $BC$ and $X$.

In the Triangle ABC AB>AC, the extension of the altitude AD with D lying inside BC intersects the circum-circle of the Triangle ABC at P. The circle through P and tangent to BC at D intersects the circum-circle of Triangle ABC at Q distinct from P with PQ=DQ. Prove that AD=BD-DC

The point $E$ lies on the base $[AD]$ of the trapezoid $ABCD$. The perimeters of the triangles $ABE, BCE$ and $CDE$ are equal. Prove that $|BC| = |AD|/2$

The incircle $ \omega $ of a quadrilateral $ ABCD $ touches $ AB, BC, CD, DA $ at $ E, F, G, H $, respectively. Choose an arbitrary point $ X$ on the segment $ AC $ inside $ \omega $. The segments $ XB, XD $ meet $ \omega $ at $ I, J $ respectively. Prove that $ FJ, IG, AC $ are concurrent.

[b]p1.[/b] We define $a \oplus b = \frac{ab}{a+b}$. Compute $(3 \oplus 5) \oplus (5 \oplus 4)$. [b]p2.[/b] Let $ABCD$ be a quadrilateral with $\angle A = 45^o$ and $\angle B = 45^o$. If $BC = 5\sqrt2$, $AD = 6\sqrt2$, and $AB = 18$, find the length of side $CD$. [b]p3.[/b] A positive real number $x$ satisfies the equation $x^2 + x + 1 + \frac{1}{x }+\frac{1}{x^2} = 10$. Find the sum of all possible values of $x + 1 + \frac{1}{x}$. [b]p4.[/b] David writes $6$ positive integers on the board (not necessarily distinct) from least to greatest. The mean of the first three numbers is $3$, the median of the first four numbers is $4$, the unique mode of the first five numbers is $5$, and the range of all 6 numbers is $6$. Find the maximum possible value of the product of David’s $6$ integers. [b]p5.[/b] Let $ABCD$ be a convex quadrilateral such that $\angle A = \angle B = 120^o$ and $\angle C = \angle D = 60^o$. There exists a circle with center $I$ which is tangent to all four sides of $ABCD$. If $IA \cdot IB \cdot IC \cdot ID = 240$, find the area of quadrilateral $ABCD$. [b]p6.[/b] The letters $EXETERMATH$ are placed into cells on an annulus as shown below. How many ways are there to color each cell of the annulus with red, blue, green, or yellow such that each letter is always colored the same color and adjacent cells are always colored differently? [img]https://cdn.artofproblemsolving.com/attachments/3/5/b470a771a5279a7746c06996f2bb5487c33ecc.png[/img] [b]p7.[/b] Let $ABCD$ be a square, and let $\omega$ be a quarter circle centered at $A$ passing through points $B$ and $D$. Points $E$ and $F$ lie on sides $BC$ and $CD$ respectively. Line $EF$ intersects $\omega$ at two points, $G$ and $H$. Given that $EG = 2$, $GH = 16$ and $HF = 9$, find the length of side $AB$. [b]p8.[/b] Let x be equal to $\frac{2022! + 2021!}{2020! + 2019! + 2018!}$ . Find the closest integer to $2\sqrt{x}$. [b]p9.[/b] For how many ordered pairs of positive integers $(m, n)$ is the absolute difference between $lcm(m, n)$ and $gcd(m, n)$ equal to $2023$? [b]p10.[/b] There are $2023$ distinguishable frogs sitting on a number line with one frog sitting on $i$ for all integers $i$ between $-1011$ and $1011$, inclusive. Each minute, every frog randomly jumps either one unit left or one unit right with equal probability. After $1011$ minutes, over all possible arrangements of the frogs, what is the average number of frogs sitting on the number $0$? [b]p11.[/b] Albert has a calculator initially displaying $0$ with two buttons: the first button increases the number on the display by one, and the second button returns the square root of the number on the display. Each second, he presses one of the two buttons at random with equal probability. What is the probability that Albert’s calculator will display the number $6$ at some point? [b]p12.[/b] For a positive integer $k \ge 2$, let $f(k)$ be the number of positive integers $n$ such that n divides $(n-1)!+k$. Find $$f(2) + f(3) + f(4) + f(5) + ... + f(100).$$ [b]p13.[/b] Mr. Atf has nine towers shaped like rectangular prisms. Each tower has a $1$ by $1$ base. The first tower as height $1$, the next has height $2$, up until the ninth tower, which has height $9$. Mr. Atf randomly arranges these $9$ towers on his table so that their square bases form a $3$ by $3$ square on the surface of his table. Over all possible solids Mr. Atf could make, what is the average surface area of the solid? [b]p14.[/b] Let $ABCD$ be a cyclic quadrilateral whose diagonals are perpendicular. Let $E$ be the intersection of $AC$ and $BD$, and let the feet of the altitudes from $E$ to the sides $AB$, $BC$, $CD$, $DA$ be $W, X, Y , Z$ respectively. Given that $EW = 2EY$ and $EW \cdot EX \cdot EY \cdot EZ = 36$, find the minimum possible value of $\frac{1}{[EAB]} +\frac{1}{[EBC]}+\frac{1}{[ECD]} +\frac{1}{[EDA]}$. The notation $[XY Z]$ denotes the area of triangle $XY Z$. [b]p15.[/b] Given that $x^2 - xy + y^2 = (x + y)^3$, $y^2 - yz + z^2 = (y + z)^3$, and $z^2 - zx + x^2 = (z + x)^3$ for complex numbers $x, y, z$, find the product of all distinct possible nonzero values of $x + y + z$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In pentagon $ABCDE$, the altitudes of triangle $ABE$ meet at point $H$. Suppose that $BCDE$ is a rectangle, and that $B$, $C$, $D$, $E$, and $H$ lie on a single circle. Prove that triangles $ABE$ and $HCD$ are congruent. [i]Alan Cheng[/i]