Let $ ABCD$ be a convex quadrangle, $ P$ the intersection of lines $ AB$ and $ CD$, $ Q$ the intersection of lines $ AD$ and $ BC$ and $ O$ the intersection of diagonals $ AC$ and $ BD$. Show that if $ \angle POQ\equal{} 90^\circ$ then $ PO$ is the bisector of $ \angle AOD$ and $ OQ$ is the bisector of $ \angle AOB$.

Let $ABCD$ be cyclic quadrilateral and $E$ the midpoint of $AC$. The circumcircle of $\triangle CDE$ intersect the side $BC$ at $F$, which is different from $C$. If $B'$ is the reflection of $B$ across $F$, prove that $EF$ is tangent to the circumcircle of $\triangle B'DF$. [i]Proposed by Nikola Velov[/i]

A triangle $ABC$ is given. Find all the segments $XY$ that lie inside the triangle such that $XY$ and five of the segments $XA,XB, XC, YA,YB,YC$ divide the triangle $ABC$ into $5$ regions with equal areas. Furthermore, prove that all the segments $XY$ have a common point.

Find the locus of the projections of a given point on all planes containing another point $B$.

In triangle $ABC$, $\angle ACB=50^{\circ}$ and $\angle CBA=70^{\circ}$. Let $D$ be a foot of perpendicular from point $A$ to side $BC$, $O$ circumcenter of $ABC$ and $E$ antipode of $A$ in circumcircle $ABC$. Find $\angle DAE$

$C:(x-\arcsin a)(x-\arccos a)+(y-\arcsin a)(y+\arccos a)=0$. The length of string of $C$ cut by $l:x=\frac{\pi}{4}$ is $d$. When $a$ changes, the minumum value of $d$ is $\text{(A)}\frac{\pi}{4}\qquad\text{(B)}\frac{\pi}{3}\qquad\text{(C)}\frac{\pi}{2}\qquad\text{(D)}\pi$

[u]Round 1[/u] [b]p1.[/b] Solve the maze [img]https://cdn.artofproblemsolving.com/attachments/8/c/6439816a52b5f32c3cb415e2058556edb77c80.png[/img] [b]p2.[/b] Billiam can write a problem in $30$ minutes, Jerry can write a problem in $10$ minutes, and Evin can write a problem in $20$ minutes. Billiam begins writing problems alone at $3:00$ PM until Jerry joins himat $4:00$ PM, and Evin joins both of them at $4:30$ PM. Given that they write problems until the end of math team at $5:00$ PM, how many full problems have they written in total? [b]p3.[/b] How many pairs of positive integers $(n,k)$ are there such that ${n \choose k}= 6$? [u]Round 2 [/u] [b]p4.[/b] Find the sumof all integers $b > 1$ such that the expression of $143$ in base $b$ has an even number of digits and all digits are the same. [b]p5.[/b] Ιni thinks that $a \# b = a^2 - b$ and $a \& b = b^2 - a$, while Mimi thinks that $a \# b = b^2 - a$ and $a \& b = a^2 - b$. Both Ini and Mimi try to evaluate $6 \& (3 \# 4)$, each using what they think the operations $\&$ and $\#$ mean. What is the positive difference between their answers? [b]p6.[/b] A unit square sheet of paper lies on an infinite grid of unit squares. What is the maximum number of grid squares that the sheet of paper can partially cover at once? A grid square is partially covered if the area of the grid square under the sheet of paper is nonzero - i.e., lying on the edge only does not count. [u]Round 3[/u] [b]p7.[/b] Maya wants to buy lots of burgers. A burger without toppings costs $\$4$, and every added topping increases the price by 50 cents. There are 5 different toppings for Maya to choose from, and she can put any combination of toppings on each burger. How much would it cost forMaya to buy $1$ burger for each distinct set of toppings? Assume that the order in which the toppings are stacked onto the burger does not matter. [b]p8.[/b] Consider square $ABCD$ and right triangle $PQR$ in the plane. Given that both shapes have area $1$, $PQ =QR$, $PA = RB$, and $P$, $A$, $B$ and $R$ are collinear, find the area of the region inside both square $ABCD$ and $\vartriangle PQR$, given that it is not $0$. [b]p9.[/b] Find the sum of all $n$ such that $n$ is a $3$-digit perfect square that has the same tens digit as $\sqrt{n}$, but that has a different ones digit than $\sqrt{n}$. [u]Round 4[/u] [b]p10.[/b] Jeremy writes the string: $$LMTLMTLMTLMTLMTLMT$$ on a whiteboard (“$LMT$” written $6$ times). Find the number of ways to underline $3$ letters such that from left to right the underlined letters spell LMT. [b]p11.[/b] Compute the remainder when $12^{2022}$ is divided by $1331$. [b]p12.[/b] What is the greatest integer that cannot be expressed as the sum of $5$s, $23$s, and $29$s? [u]Round 5 [/u] [b]p13.[/b] Square $ABCD$ has point $E$ on side $BC$, and point $F$ on side $CD$, such that $\angle EAF = 45^o$. Let $BE = 3$, and $DF = 4$. Find the length of $FE$. [b]p14.[/b] Find the sum of all positive integers $k$ such that $\bullet$ $k$ is the power of some prime. $\bullet$ $k$ can be written as $5654_b$ for some $b > 6$. [b]p15.[/b] If $\sqrt[3]{x} + \sqrt[3]{y} = 2$ and $x + y = 20$, compute $\max \,(x, y)$. PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167372p28825861]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two circles meeting at points $A, B$ and a point $O$ lying outside them are given. Using a compass and a ruler construct a ray with origin $O$ meeting the first circle at point $C$ and the second one at point $D$ in such a way that the ratio $OC : OD$ be maximal.

[b]p1.[/b] An evil galactic empire is attacking the planet Naboo with numerous automatic drones. The fleet defending the planet consists of $101$ ships. By the decision of the commander of the fleet, some of these ships will be used as destroyers equipped with one rocket each or as rocket carriers that will supply destroyers with rockets. Destroyers can shoot rockets so that every rocket destroys one drone. During the attack each carrier will have enough time to provide each destroyer with one rocket but not more. How many destroyers and how many carriers should the commander assign to destroy the maximal number of drones and what is the maximal number of drones that the fleet can destroy? [b]p2.[/b] Solve the inequality: $\sqrt{x^2-3x+2} \le \sqrt{x+7}$ [b]p3.[/b] Find all positive real numbers $x$ and $y$ that satisfy the following system of equations: $$x^y = y^{x-y}$$ $$x^x = y^{12y}$$ [b]p4.[/b] A convex quadrilateral $ABCD$ with sides $AB = 2$, $BC = 8$, $CD = 6$, and $DA = 7$ is divided by a diagonal $AC$ into two triangles. A circle is inscribed in each of the obtained two triangles. These circles touch the diagonal at points $E$ and $F$. Find the distance between the points $E$ and $F$. [b]p5.[/b] Find all positive integer solutions $n$ and $k$ of the following equation: $$\underbrace{11... 1}_{n} \underbrace{00... 0}_{2n+3} + \underbrace{77...7}_{n+1} \underbrace{00...0}_{n+1}+\underbrace{11...1}_{n+2} = 3k^3.$$ [b]p6.[/b] The Royal Council of the planet Naboo consists of $12$ members. Some of these members mutually dislike each other. However, each member of the Council dislikes less than half of the members. The Council holds meetings around the round table. Queen Amidala knows about the relationship between the members so she tries to arrange their seats so that the members that dislike each other are not seated next to each other. But she does not know whether it is possible. Can you help the Queen in arranging the seats? Justify your answer. PS. You should use hide for answers.

[b]p1.[/b] The remainder of a number when divided by $7$ is $5$. If I multiply the number by $32$ and add $18$ to the product, what is the new remainder when divided by $7$? [b]p2.[/b] If a fair coin is flipped $15$ times, what is the probability that there are more heads than tails? [b]p3.[/b] Let $-\frac{\sqrt{p}}{q}$ be the smallest nonzero real number such that the reciprocal of the number is equal to the number minus the square root of the square of the number, where $p$ and $q$ are positive integers and $p$ is not divisible the square of any prime. Find $p + q$. [b]p4.[/b] Rachel likes to put fertilizers on her grass to help her grass grow. However, she has cows there as well, and they eat $3$ little fertilizer balls on average. If each ball is spherical with a radius of $4$, then the total volume that each cow consumes can be expressed in the form $a\pi$ where $a$ is an integer. What is $a$? [b]p5.[/b] One day, all $30$ students in Precalc class are bored, so they decide to play a game. Everyone enters into their calculators the expression $9 \diamondsuit 9 \diamondsuit 9 ... \diamondsuit 9$, where $9$ appears $2020$ times, and each $\diamondsuit$ is either a multiplication or division sign. Each student chooses the signs randomly, but they each choose one more multiplication sign than division sign. Then all $30$ students calculate their expression and take the class average. Find the expected value of the class average. [b]p6.[/b] NaNoWriMo, or National Novel Writing Month, is an event in November during which aspiring writers attempt to produce novel-length work - formally defined as $50,000$ words or more - within the span of $30$ days. Justin wants to participate in NaNoWriMo, but he's a busy high school student: after accounting for school, meals, showering, and other necessities, Justin only has six hours to do his homework and perhaps participate in NaNoWriMo on weekdays. On weekends, he has twelve hours on Saturday and only nine hours on Sunday, because he goes to church. Suppose Justin spends two hours on homework every single day, including the weekends. On Wednesdays, he has science team, which takes up another hour and a half of his time. On Fridays, he spends three hours in orchestra rehearsal. Assume that he spends all other time on writing. Then, if November $1$st is a Friday, let $w$ be the minimum number of words per minute that Justin must type to finish the novel. Round $w$ to the nearest whole number. [b]p7.[/b] Let positive reals $a$, $b$, $c$ be the side lengths of a triangle with area $2030$. Given $ab + bc + ca = 15000$ and $abc = 350000$, find the sum of the lengths of the altitudes of the triangle. [b]p8.[/b] Find the minimum possible area of a rectangle with integer sides such that a triangle with side lengths $3$, $4$, $5$, a triangle with side lengths $4$, $5$, $6$, and a triangle with side lengths $\frac94$, $4$, $4$ all fit inside the rectangle without overlapping. [b]p9.[/b] The base $16$ number $10111213...99_{16}$, which is a concatenation of all of the (base $10$) $2$-digit numbers, is written on the board. Then, the last $2n$ digits are erased such that the base $10$ value of remaining number is divisible by $51$. Find the smallest possible integer value of $n$. [b]p10.[/b] Consider sequences that consist entirely of $X$'s, $Y$ 's and $Z$'s where runs of consecutive $X$'s, $Y$ 's, and $Z$'s are at most length $3$. How many sequences with these properties of length $8$ are there? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many pairs of parallel edges, such as $\overline{AB}$ and $\overline{GH}$ or $\overline{EH}$ and $\overline{FG}$, does a cube have? $\textbf{(A) }6 \qquad\textbf{(B) }12 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 24 \qquad \textbf{(E) } 36$ [asy] import three; currentprojection=orthographic(1/2,-1,1/2); /* three - currentprojection, orthographic */ draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle); draw((0,0,0)--(0,0,1)); draw((0,1,0)--(0,1,1)); draw((1,1,0)--(1,1,1)); draw((1,0,0)--(1,0,1)); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle); label("$D$",(0,0,0),S); label("$A$",(0,0,1),N); label("$H$",(0,1,0),S); label("$E$",(0,1,1),N); label("$C$",(1,0,0),S); label("$B$",(1,0,1),N); label("$G$",(1,1,0),S); label("$F$",(1,1,1),N); [/asy]

Identical spherical tennis balls of radius 1 are placed inside a cylindrical container of radius 2 and height 19. Compute the maximum number of tennis balls that can fit entirely inside this container.

[b]p1.[/b] How many ways can we arrange the elements $\{1, 2, ..., n\}$ to a sequence $a_1, a_2, ..., a_n$ such that there is only exactly one $a_i$, $a_{i+1}$ such that $a_i > a_{i+1}$? [b]p2. [/b]How many distinct (non-congruent) triangles are there with integer side-lengths and perimeter $2012$? [b]p3.[/b] Let $\phi$ be the Euler totient function, and let $S = \{x| \frac{x}{\phi (x)} = 3\}$. What is $\sum_{x\in S} \frac{1}{x}$? [b]p4.[/b] Denote $f(N)$ as the largest odd divisor of $N$. Compute $f(1) + f(2) + f(3) +... + f(29) + f(30)$. [b]p5.[/b] Triangle $ABC$ has base $AC$ equal to $218$ and altitude $100$. Squares $s_1, s_2, s_3, ...$ are drawn such that $s_1$ has a side on $AC$ and has one point each touching $AB$ and $BC$, and square $s_k$ has a side on square $s_{k-}1$ and also touches $AB$ and $BC$ exactly once each. What is the sum of the area of these squares? [b]p6.[/b] Let $P$ be a parabola $6x^2 - 28x + 10$, and $F$ be the focus. A line $\ell$ passes through $F$ and intersects the parabola twice at points $P_1 = (2,-22)$, $P_2$. Tangents to the parabola with points at $P_1, P_2$ are then drawn, and intersect at a point $Q$. What is $m\angle P_1QP_2$? PS. You had better use hide for answers.

Let $ABC$ be at triangle with incircle $\Gamma$. Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$ be three circles inside $\triangle ABC$ each of which is tangent to $\Gamma$ and two sides of the triangle and their radii are $1,4,9$. Find the radius of $\Gamma$.

Let $AB$ be a diameter of a circle; let $t_1$ and $t_2$ be the tangents at $A$ and $B$, respectively; let $C$ be any point other than $A$ on $t_1$; and let $D_1D_2. E_1E_2$ be arcs on the circle determined by two lines through $C$. Prove that the lines $AD_1$ and $AD_2$ determine a segment on $t_2$ equal in length to that of the segment on $t_2$ determined by $AE_1$ and $AE_2.$

A triangle $ABC$ is considered, and let $D$ be the intersection point of the angle bisector corresponding to angle $A$ with side $BC$. Prove that the circumcircle that passes through $A$ and is tangent to line $BC$ at $D$, it is also tangent to the circle circumscribed around triangle $ABC$.

Let $ABC$ be isosceles triangle ($AB=BC$) and $K$ and $M$ be the midpoints of $AB$ and $AC,$ respectively.Let the circumcircle of $\triangle BKC$ meets the line $BM$ at $N$ other than $B.$ Let the line passing through $N$ and parallel to $AC$ intersects the circumcircle of $\triangle ABC$ at $A_1$ and $C_1.$ Prove that $\triangle A_1BC_1$ is equilateral.

Let $ABC$ be a triangle such that $AB$ is not equal to $AC$. Let $M$ be the midpoint of $BC$ and $H$ be the orthocenter of triangle $ABC$. Let $D$ be the midpoint of $AH$ and $O$ the circumcentre of triangle $BCH$. Prove that $DAMO$ is a parallelogram.

A tetrahedron $ABCD$ has all edges of length less than $100$, and contains two nonintersecting spheres of diameter $1$. Prove that it contains a sphere of diameter $1.01$.

Let $ABC$ be a triangle with $|AB|=18$, $|AC|=24$, and $m(\widehat{BAC}) = 150^\circ$. Let $D$, $E$, $F$ be points on sides $[AB]$, $[AC]$, $[BC]$, respectively, such that $|BD|=6$, $|CE|=8$, and $|CF|=2|BF|$. Let $H_1$, $H_2$, $H_3$ be the reflections of the orthocenter of triangle $ABC$ over the points $D$, $E$, $F$, respectively. What is the area of triangle $H_1H_2H_3$? $ \textbf{(A)}\ 70 \qquad\textbf{(B)}\ 72 \qquad\textbf{(C)}\ 84 \qquad\textbf{(D)}\ 96 \qquad\textbf{(E)}\ 108 $

Let $ R$ be a bounded domain of area $ t$ in the plane, and let $ C$ be its center of gravity. Denoting by $ T_{AB}$ the circle drawn with the diameter $ AB$, let $ K$ be a circle that contains each of the circles $ T_{AB} \;(A,B \in R)$. Is it true in general that $ K$ contains the circle of area $ 2t$ centered at $ C$? [i]J. Szucs[/i]

The $A$-excircle of the triangle $ABC$ touches the side $BC$ at point $K$. The circumcircles of triangles $AKB$ and $AKC$ intersect for the second time with the bisector of angle $A$ at points $X$ and $Y$ respectively. Let $M$ be the midpoint of $BC$. Prove that the circumcenter of triangle $XYM$ lies on $BC$.

* Three equal circles are tangent to each other externally and to the fourth circle internally. Tangent lines are drawn to the circles from an arbitrary point on the fourth circle. Prove that the sum of the lengths of two tangent lines equals the length of the third tangent.

Suppose that $\triangle ABC$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP = 1$, $BP = \sqrt{3}$, and $CP = 2$. What is $s?$ $\textbf{(A) } 1 + \sqrt{2} \qquad \textbf{(B) } \sqrt{7} \qquad \textbf{(C) } \frac{8}{3} \qquad \textbf{(D) } \sqrt{5 + \sqrt{5}} \qquad \textbf{(E) } 2\sqrt{2}$

Triangle $ABC$ is given. On its sides $AB$, $BC$ and $CA$, respectively, points $X, Y, Z$ are chosen so that $$AX : XB =BY : YC = CZ : ZA = 2:1.$$ It turned out that the triangle $XYZ$ is equilateral. Prove that the original triangle $ABC$ is also equilateral.