[b]p1.[/b] Give a fake proof that $0 = 1$ on the back of this page. The most convincing answer to this question at this test site will receive a point. [b]p2.[/b] It is often said that once you assume something false, anything can be derived from it. You may assume for this question that $0 = 1$, but you can only use other statements if they are generally accepted as true or if your prove them from this assumption and other generally acceptable mathematical statements. With this in mind, on the back of this page prove that every number is the same number. [b]p3.[/b] Suppose you write out all integers between $1$ and $1000$ inclusive. (The list would look something like $1$, $2$, $3$, $...$ , $10$, $11$, $...$ , $999$, $1000$.) Which digit occurs least frequently? [b]p4.[/b] Pick a real number between $0$ and $1$ inclusive. If your response is $r$ and the standard deviation of all responses at this site to this question is $\sigma$, you will receive $r(1 - (r - \sigma)^2)$ points. [b]p5.[/b] Find the sum of all possible values of $x$ that satisfy $243^{x+1} = 81^{x^2+2x}$. [b]p6.[/b] How many times during the day are the hour and minute hands of a clock aligned? [b]p7.[/b] A group of $N + 1$ students are at a math competition. All of them are wearing a single hat on their head. $N$ of the hats are red; one is blue. Anyone wearing a red hat can steal the blue hat, but in the process that person’s red hat disappears. In fact, someone can only steal the blue hat if they are wearing a red hat. After stealing it, they would wear the blue hat. Everyone prefers the blue hat over a red hat, but they would rather have a red hat than no hat at all. Assuming that everyone is perfectly rational, find the largest prime $N$ such that nobody will ever steal the blue hat. [b]p8.[/b] On the back of this page, prove there is no function f$(x)$ for which there exists a (finite degree) polynomial $p(x)$ such that $f(x) = p(x)(x + 3) + 8$ and $f(3x) = 2f(x)$. [b]p9.[/b] Given a cyclic quadrilateral $YALE$ with $Y A = 2$, $AL = 10$, $LE = 11$, $EY = 5$, what is the area of $YALE$? [b]p10.[/b] About how many pencils are made in the U.S. every year? If your answer to this question is $p$, and our (good) estimate is $\rho$, then you will receive $\max(0, 1 -\frac 12 | \log_{10}(p) - \log_{10}(\rho)|)$ points. [b]p11.[/b] The largest prime factor of $520, 302, 325$ has $5$ digits. What is this prime factor? [b]p12.[/b] The previous question was on the individual round from last year. It was one of the least frequently correctly answered questions. The first step to solving the problem and spotting the pattern is to divide $520, 302, 325$ by an appropriate integer. Unfortunately, when solving the problem many people divide it by $n$ instead, and then they fail to see the pattern. What is $n$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCDE$ be a convex pentagon such that $BC=DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB=TD,TC=TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P,B,A,Q$ occur on their line in that order. Let line $AE$ intersect $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R,E,A,S$ occur on their line in that order. Prove that the points $P,S,Q,R$ lie on a circle.

From a given triangle of unit area, we choose two points independetly with uniform distribution. The straight line connecting these points divides the triangle. with probability one, into a triangle and a quadrilateral. Calculate the expected values of the areas of these two regions. [A. Renyi]

A convex $2004$-sided polygon $P$ is given such that no four vertices are cyclic. We call a triangle whose vertices are vertices of $P$ thick if all other $2001$ vertices of $P$ lie inside the circumcircle of the triangle, and thin if they all lie outside its circumcircle. Prove that the number of thick triangles is equal to the number of thin triangles.

Convex quadrilaterals \(ABCD\), \(A_1B_1C_1D_1\), and \(A_2B_2C_2D_2\) are similar with vertices in order. Points \(A\), \(A_1\), \(B_2\), \(B\) are collinear in order, points \(B\), \(B_1\), \(C_2\), \(C\) are collinear in order, points \(C\), \(C_1\), \(D_2\), \(D\) are collinear in order, and points \(D\), \(D_1\), \(A_2\), \(A\) are collinear in order. Diagonals \(AC\) and \(BD\) intersect at \(P\), diagonals \(A_1C_1\) and \(B_1D_1\) intersect at \(P_1\), and diagonals \(A_2C_2\) and \(B_2D_2\) intersect at \(P_2\). Prove that points \(P\), \(P_1\), and \(P_2\) are collinear. [i]Proposed by Holden Mui[/i]

Let $ABC$ be a triangle for which the shortest side is $AC$. Its inscribed circle with center $I$ touches sides $AB$ and $BC$ in points $D$ and $E$ respectively. Point $M$ is the midpoint of $AC$. Points $F$ and $G$ lie on sides $BC$ and $AB$ respectively so that $FC=CA=AG$. The line through $I$ perpendicular to $MI$ intersects the line segments $AF$ and $CG$ in $P$ and $Q$ respectively. Prove that $AB=BC\Leftrightarrow PD=QE$.

On rectangular coordinates, point $A = (1,2)$, $B = (3,4)$. $P = (a, 0)$ is on $x$-axis. Given that $P$ is chosen such that $AP + PB$ is minimized, compute $60a$.

Circles ω[size=35]1[/size] and ω[size=35]2[/size] have no common point. Where is outerior tangents a and b, interior tangent c. Lines a, b and c touches circle ω[size=35]1[/size] respectively on points A[size=35]1[/size], B[size=35]1[/size] and C[size=35]1[/size], and circle ω[size=35]2[/size] – respectively on points A[size=35]2[/size], B[size=35]2[/size] and C[size=35]2[/size]. Prove that triangles A[size=35]1[/size]B[size=35]1[/size]C[size=35]1[/size] and A[size=35]2[/size]B[size=35]2[/size]C[size=35]2[/size] area ratio is the same as ratio of ω[size=35]1[/size] and ω[size=35]2[/size] radii.

Three 12 cm $\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in $\text{cm}^3$) of this polyhedron? [asy] defaultpen(fontsize(10)); size(250); draw(shift(0, sqrt(3)+1)*scale(2)*rotate(45)*polygon(4)); draw(shift(-sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(165)*polygon(4)); draw(shift(sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(285)*polygon(4)); filldraw(scale(2)*polygon(6), white, black); pair X=(2,0)+sqrt(2)*dir(75), Y=(-2,0)+sqrt(2)*dir(105), Z=(2*dir(300))+sqrt(2)*dir(225); pair[] roots={2*dir(0), 2*dir(60), 2*dir(120), 2*dir(180), 2*dir(240), 2*dir(300)}; draw(roots[0]--X--roots[1]); label("$B$", centroid(roots[0],X,roots[1])); draw(roots[2]--Y--roots[3]); label("$B$", centroid(roots[2],Y,roots[3])); draw(roots[4]--Z--roots[5]); label("$B$", centroid(roots[4],Z,roots[5])); label("$A$", (1+sqrt(3))*dir(90)); label("$A$", (1+sqrt(3))*dir(210)); label("$A$", (1+sqrt(3))*dir(330)); draw(shift(-10,0)*scale(2)*polygon(4)); draw((sqrt(2)-10,0)--(-10,sqrt(2))); label("$A$", (-10,0)); label("$B$", centroid((sqrt(2)-10,0),(-10,sqrt(2)),(sqrt(2)-10, sqrt(2))));[/asy]

Triangle $ABC$ is given. Circle $ \omega $ passes through $B$, touch $AC$ in $D$ and intersect sides $AB$ and $BC$ at $P$ and $Q$ respectively. Line $PQ$ intersect $BD$ and $AC$ at $M$ and $N$ respectively. Prove that $ \omega $, circumcircle of $DMN$ and circle, touching $PQ$ in $M$ and passes through B, intersects in one point.

On the sides of a triangle $XYZ$ to the outside construct similar triangles $YDZ, EXZ ,YXF$ with circumcenters $K, L ,M$ respectively. Here are $\angle ZDY = \angle ZXE = \angle FXY$ and $\angle YZD = \angle EZX = \angle YFX$. Show that the triangle $KLM$ is similar to the triangles . [img]https://cdn.artofproblemsolving.com/attachments/e/f/fe0d0d941015d32007b6c00b76b253e9b45ca5.png[/img]

Pete claims that he can draw $3$ segments of length $1$ and a circle of radius less than $\sqrt3 / 3$ on a piece of paper, such that all segments would lie inside the circle and there would be no line that intersects each of $3 $ segments. Is Pete right?

A square is inscribed into an acute-angled triangle: two vertices of this square lie on the same side of the triangle and two remaining vertices lies on two remaining sides. Two similar squares are constructed for the remaining sides. Prove that three segments congruent to the sides of these squares can be the sides of an acute-angled triangle.

A cylindrical hole of radius $r$ is bored through a cylinder of radiues $R$ ($r\leq R$) so that the axes intersect at right angles. i) Show that the area of the larger cylinder which is inside the smaller one can be expressed in the form $$S=8r^2\int_{0}^{1} \frac{1-v^{2}}{\sqrt{(1-v^2)(1-m^2 v^2)}}\;dv,\;\; \text{where} \;\; m=\frac{r}{R}.$$ ii) If $K=\int_{0}^{1} \frac{1}{\sqrt{(1-v^2)(1-m^2 v^2)}}\;dv$ and $E=\int_{0}^{1} \sqrt{\frac{1-m^2 v^2}{1-v^2 }}dv$. show that $$S=8[R^2 E - (R^2 - r^2 )K].$$

In triangle $ABC, AH$ is an altitude ($H$ is on $BC$) and $BE$ is a bisector ($E$ is on $AC$) . We are given that angle $BEA$ equals $45^o$ .Prove that angle $EHC$ equals $45^o$ . (I. Sharygin , Moscow)

The checkered plane is painted black and white, after a chessboard fashion. A polygon $\Pi$ of area $S$ and perimeter $P$ consists of some of these unit squares (i.e., its sides go along the borders of the squares). Prove the polygon $\Pi$ contains not more than $\dfrac {S} {2} + \dfrac {P} {8}$, and not less than $\dfrac {S} {2} - \dfrac {P} {8}$ squares of a same color. (Alexander Magazinov)

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. Prove that there exists a point $J$ such that for any point $X$ inside $ABC$ if $AX,BX,CX$ intersect $\omega$ in $A_1,B_1,C_1$ and $A_2,B_2,C_2$ be reflections of $A_1,B_1,C_1$ in midpoints of $BC,AC,AB$ respectively then $A_2,B_2,C_2,J$ lie on a circle.

[u]Round 1[/u] [b]p1.[/b] Alex is on a week-long mining quest. Each morning, she mines at least $1$ and at most $10$ diamonds and adds them to her treasure chest (which already contains some diamonds). Every night she counts the total number of diamonds in her collection and finds that it is divisible by either $22$ or $25$. Show that she miscounted. [b]p2.[/b] Hermione set out a row of $11$ Bertie Bott’s Every Flavor Beans for Ron to try. There are $5$ chocolateflavored beans that Ron likes and $6$ beans flavored like earwax, which he finds disgusting. All beans look the same, and Hermione tells Ron that a chocolate bean always has another chocolate bean next to it. What is the smallest number of beans that Ron must taste to guarantee he finds a chocolate one? [b]p3.[/b] There are $101$ pirates on a pirate ship: the captain and $100$ crew. Each pirate, including the captain, starts with $1$ gold coin. The captain makes proposals for redistributing the coins, and the crew vote on these proposals. The captain does not vote. For every proposal, each crew member greedily votes “yes” if he gains coins as a result of the proposal, “no” if he loses coins, and passes otherwise. If strictly more crew members vote “yes” than “no,” the proposal takes effect. The captain can make any number of proposals, one after the other. What is the largest number of coins the captain can accumulate? [b]p4.[/b] There are $100$ food trucks in a circle and $10$ gnomes who sample their menus. For the first course, all the gnomes eat at different trucks. For each course after the first, gnome #$1$ moves $1$ truck left or right and eats there; gnome #$2$ moves $2$ trucks left or right and eats there; ... gnome #$10$ moves $10$ trucks left or right and eats there. All gnomes move at the same time. After some number of courses, each food truck had served at least one gnome. Show that at least one gnome ate at some food truck twice. [b]p5.[/b] The town of Lumenville has $100$ houses and is preparing for the math festival. The Tesla wiring company lays lengths of power wire in straight lines between the houses so that power flows between any two houses, possibly by passing through other houses.The Edison lighting company hangs strings of lights in straight lines between pairs of houses so that each house is connected by a string to exactly one other. Show that however the houses are arranged, the Edison company can always hang their strings of lights so that the total length of the strings is no more than the total length of the power wires the Tesla company used. [img]https://cdn.artofproblemsolving.com/attachments/9/2/763de9f4138b4dc552247e9316175036c649b6.png[/img] [u]Round 2[/u] [b]p6.[/b] What is the largest number of zeros that could appear at the end of $1^n + 2^n + 3^n + 4^n$, where n can be any positive integer? [b]p7.[/b] A tennis academy has $2023$ members. For every group of 1011 people, there is a person outside of the group who played a match against everyone in it. Show there is someone who has played against all $2022$ other members. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[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].

Let $ABC$ be a triangle with $\angle BAC \neq 90^{\circ}.$ Let $O$ be the circumcenter of the triangle $ABC$ and $\Gamma$ be the circumcircle of the triangle $BOC.$ Suppose that $\Gamma$ intersects the line segment $AB$ at $P$ different from $B$, and the line segment $AC$ at $Q$ different from $C.$ Let $ON$ be the diameter of the circle $\Gamma.$ Prove that the quadrilateral $APNQ$ is a parallelogram.

Let $ABCD$ be a cyclic quadrilateral. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ such that $KLMN$ is a rhombus with $KL \parallel AC$ and $LM \parallel BD$. Let $\omega_A, \omega_B, \omega_C, \omega_D$ be the incircles of $\triangle ANK, \triangle BKL, \triangle CLM, \triangle DMN$. Prove that the common internal tangents to $\omega_A$, and $\omega_C$ and the common internal tangents to $\omega_B$ and $\omega_D$ are concurrent.

Let $\Omega$ be the circumcircle of a triangle $ABC$ with $\angle CAB = 90$. The medians through $B$ and $C$ meet $\Omega$ again at $D$ and $E$, respectively. The tangent to $\Omega$ at $D$ intersects the line $AC$ at $X$ and the tangent to $\Omega$ at $E$ intersects the line $AB$ at $Y$ . Prove that the line $XY$ is tangent to $\Omega$.

Draw two tangents to the circle from point $P$ outside a circle, touching the circle at $A$ and $B$, then draw a secant line passes $P$, intersecting the circle at points $C$ and $D$ ($C$ is between $P$ and $D$). $Q$ is a point on the chord $CD$ such that $\angle DAQ=\angle PBC$. Prove that $\angle DBQ=\angle PAC$.

Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible integral values for $BC$.

Two points $A$ and $B$ are given in the plane. A point $X$ is called their [i]preposterous midpoint[/i] if there is a Cartesian coordinate system in the plane such that the coordinates of $A$ and $B$ in this system are non-negative, the abscissa of $X$ is the geometric mean of the abscissae of $A$ and $B$, and the ordinate of $X$ is the geometric mean of the ordinates of $A$ and $B$. Find the locus of all the [i]preposterous midpoints[/i] of $A$ and $B$. (K. Tyschu)