In a circle, two non-intersecting chords $AB,CD$ are drawn.On the chord $AB$,a point $E$ (different from $A$,$B$) is taken Consider the arc $AB$ that does not contain the points $C,D$. With a compass and a straighthedge, find all possible point $F$ on that arc such that $\dfrac{PE}{EQ}=\dfrac{1}{2}$, where $P$ and $Q$ are the points in which the chord $AB$ meets the segment $FC$ and $FD$. [i](tatari/nightmare)[/i]

Circles $c_1, c_2$ with centers $O_1, O_2$, respectively, intersect at points $P$ and $Q$ and touch circle c internally at points $A_1$ and $A_2$, respectively. Line $PQ$ intersects circle c at points $B$ and $D$. Lines $A_1B$ and $A_1D$ intersect circle $c_1$ the second time at points $E_1$ and $F_1$, respectively, and lines $A_2B$ and $A_2D$ intersect circle $c_2$ the second time at points $ E_2$ and $F_2$, respectively. Prove that $E_1, E_2, F_1, F_2$ lie on a circle whose center coincides with the midpoint of line segment $O_1O_2$.

The points $A$ and $P$ are marked on the plane. Consider all such points $B, C $ of this plane that $\angle ABP = \angle MAB$ and $\angle ACP = \angle MAC $, where $M$ is the midpoint of the segment $BC$. Prove that all the circumscribed circles around the triangle $ABC$ for different points $B$ and $C$ pass through some fixed point other than the point $A$. (Alexei Klurman)

In triangle $\triangle ABC$ , $I$ is the incenter and $M$ is the midpoint of arc $(BC)$ in the circumcircle of $(ABC)$not containing $A$. Let $X$ be an arbitrary point on the external angle bisector of $A$. Let $BX \cap (BIC) = T$. $Y$ lies on $(AXC)$ , different from $A$ , st $MA=MY$ . Prove that $TC || AY$ (Assume that $X$ is not on $(ABC)$ or $BC$)

Let $A$ and $B$ be points with $AB=12$. A point $P$ in the plane of $A$ and $B$ is $\textit{special}$ if there exist points $X, Y$ such that [list] [*]$P$ lies on segment $XY$, [*]$PX : PY = 4 : 7$, and [*]the circumcircles of $AXY$ and $BXY$ are both tangent to line $AB$. [/list] A point $P$ that is not special is called $\textit{boring}$. Compute the smallest integer $n$ such that any two boring points have distance less than $\sqrt{n/10}$ from each other. [i]Proposed by Michael Ren[/i]

Two circles $\Gamma$ and $\Omega$ intersect at two distinct points $A$ and $B$. Let $P$ be a point on $\Gamma$. The tangent at $P$ to $\Gamma$ meets $\Omega$ at the points $C$ and $D$, where $D$ lies between $P$ and $C$, and $ABCD$ is a convex quadrilateral. The lines $CA$ and $CB$ meet $\Gamma$ again at $E$ and $F$ respectively. The lines $DA$ and $DB$ meet $\Gamma$ again at $S$ and $T$ respectively. Suppose the points $P,E,S,F,B,T,A$ lie on $\Gamma$ in this order. Prove that $PC,ET,SF$ are parallel.

Let $ABCD$ be a parallelogram. A line $\ell$ intersects lines $AB,~ BC,~ CD, ~DA$ at four different points $E,~ F,~ G,~ H,$ respectively. The circumcircles of triangles $AEF$ and $AGH$ intersect again at $P$. The circumcircles of triangles $CEF$ and $CGH$ intersect again at $Q$. Prove that the line $P Q$ bisects the diagonal $BD$.

Suppose there are $n$ distinct points on plane. There is circle with radius $r$ and center $O$ on the plane. At least one of the points are in the circle. We do the following instructions. At each step we move $O$ to the baricenter of the point in the circle. Prove that location of $O$ is constant after some steps.

$C$, $C'$ are concentric circles with radii $R$, $R'$. A rectangle has two adjacent vertices on $C$ and the other two vertices on $C'$. Find its sides if its area is as large as possible.

Let $PQRS$ be a quadrilateral with $| P Q | = | QR | = | RS |$, $\angle Q= 110^o$ and $\angle R = 130^o$ . Determine $\angle P$ and $\angle S$ .

The right triangle $ABC$ with a right angle at $C$. From all the rectangles $CA_1MB_1$, where $A_1\in BC, M\in AB$ and $B_1\in AC$ which one has the biggest area?

Let $\vartriangle ABC$ be a right triangle with $AB = 4, BC = 5$, and hypotenuse $AC$. Let I be the incenter of $\vartriangle ABC$ and $E$ be the excenter of $\vartriangle ABC$ opposite $A$ (the center of the circle tangent to $BC$ and the extensions of segments $AB$ and $AC$). Suppose the circle with diameter $IE$ intersects line $AB$ beyond $B$ at $D$. If $BD =\sqrt{a}- b$, where a and b are positive integers. Find $a + b$.

A non-isosceles triangle is given. Prove that one of the circles touching internally its incircle and circumcircle and externally one of its excircles passes through a vertex of the triangle.

The quadrilateral $ABCD$ is known to have $BC = CD = AC$, and the angle $\angle ABC= 70^o$. Calculate the degree measure of the angle $\angle ADB$. (Alexey Panasenko)

[b]p1.[/b] A large square is cut into four smaller, congruent squares. If each of the smaller squares has perimeter $4$, what was the perimeter of the original square? [b]p2.[/b] Pie loves to bake apples so much that he spends $24$ hours a day baking them. If Pie bakes a dozen apples in one day, how many minutes does it take Pie to bake one apple, on average? [b]p3.[/b] Bames Jond is sent to spy on James Pond. One day, Bames sees James type in his $4$-digit phone password. Bames remembers that James used the digits $0$, $5$, and $9$, and no other digits, but he does not remember the order. How many possible phone passwords satisfy this condition? [b]p4.[/b] What do you get if you square the answer to this question, add $256$ to it, and then divide by $32$? [b]p5.[/b] Chloe the Horse and Flower the Chicken are best friends. When Chloe gets sad for any reason, she calls Flower, so Chloe must remember Flower's $3$ digit phone number, which can consist of any digits $0-5$. Given that the phone number's digits are unique and add to $5$, the number does not start with $0$, and the $3$ digit number is prime, what is the sum of all possible phone numbers? [b]p6.[/b] Anuj has a circular pizza with diameter $A$ inches, which is cut into $B$ congruent slices, where $A$,$B$ are positive integers. If one of Anuj's pizza slices has a perimeter of $3\pi + 30$ inches, find $A + B$. [b]p7.[/b] Bob really likes to study math. Unfortunately, he gets easily distracted by messages sent by friends. At the beginning of every minute, there is an $\frac{6}{10}$ chance that he will get a message from a friend. If Bob does get a message from a friend, there is a $\frac{9}{10}$ chance that he will look at the message, causing him to waste $30$ seconds before resuming his studying. If Bob doesn't get a message from a friend, there is a $\frac{3}{10}$ chance Bob will still check his messages hoping for a message from his friends, wasting $10$ seconds before he resumes his studying. What is the expected number of minutes in $100$ minutes for which Bob will be studying math? [b]p8.[/b] Suppose there is a positive integer $n$ with $225$ distinct positive integer divisors. What is the minimum possible number of divisors of n that are perfect squares? [b]p9.[/b] Let $a, b, c$ be positive integers. $a$ has $12$ divisors, $b$ has $8$ divisors, $c$ has $6$ divisors, and $lcm(a, b, c) = abc$. Let $d$ be the number of divisors of $a^2bc$. Find the sum of all possible values of $d$. [b]p10.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 17$, $BC = 28$, $AC = 25$. Let the altitude from $A$ to $BC$ and the angle bisector of angle $B$ meet at $P$. Given the length of $BP$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, $c$ where $gcd(a, c) = 1$ and $b$ is not divisible by the square of any prime, find $a + b + c$. [b]p11.[/b] Let $a$, $b$, and $c$ be the roots of the cubic equation $x^3-5x+3 = 0$. Let $S = a^4b+ab^4+a^4c+ac^4+b^4c+bc^4$. Find $|S|$. [b]p12.[/b] Call a number palindromeish if changing a single digit of the number into a different digit results in a new six-digit palindrome. For example, the number $110012$ is a palindromeish number since you can change the last digit into a $1$, which results in the palindrome $110011$. Find the number of $6$ digit palindromeish numbers. [b]p13.[/b] Let $P(x)$ be a polynomial of degree $3$ with real coecients and leading coecient $1$. Let the roots of $P(x)$ be $a$, $b$, $c$. Given that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}= 4$ and $a^2 + b^2 + c^2 = 36$, the coefficient of $x^2$ is negative, and $P(1) = 2$, let the $S$ be the sum of possible values of $P(0)$. Then $|S|$ can be expressed as $\frac{a + b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ such that $gcd(a, b, d) = 1$ and $c$ is not divisible by the square of any prime. Find $a + b + c + d$. [b]p14.[/b] Let $ABC$ be a triangle with side lengths $AB = 7$, $BC = 8$, $AC = 9$. Draw a circle tangent to $AB$ at $B$ and passing through $C$. Let the center of the circle be $O$. The length of $AO$ can be expressed as $\frac{a\sqrt{b}}{c\sqrt{d}}$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a, c) = gcd(b, d) = 1$ and $b$,$ d$ are not divisible by the square of any prime. Find $a + b + c + d$. [b]p15.[/b] Many students in Mr. Noeth's BC Calculus class missed their first test, and to avoid taking a makeup, have decided to never leave their houses again. As a result, Mr. Noeth decides that he will have to visit their houses to deliver the makeup tests. Conveniently, the $17$ absent students in his class live in consecutive houses on the same street. Mr. Noeth chooses at least three of every four people in consecutive houses to take a makeup. How many ways can Mr. Noeth select students to take makeups? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Points $P$ and $Q$ lie in a plane with $PQ=8$. How many locations for point $R$ in this plane are there such that the triangle with vertices $P,$ $Q,$ and $R$ is a right triangle with area $12$ square units? $\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) }8 \qquad\textbf{(E) } 12$

[b]p1.[/b] Thirty players participate in a chess tournament. Every player plays one game with every other player. What maximal number of players can get exactly $5$ points? (any game adds $1$ point to the winner’s score, $0$ points to a loser’s score, in the case of a draw each player obtains $1/2$ point.) [b]p2.[/b] A father and his son returned from a fishing trip. To make their catches equal the father gave to his son some of his fish. If, instead, the son had given his father the same number of fish, then father would have had twice as many fish as his son. What percent more is the father's catch more than his son's? [b]p3.[/b] What is the maximal number of pieces of two shapes, [img]https://cdn.artofproblemsolving.com/attachments/a/5/6c567cf6a04b0aa9e998dbae3803b6eeb24a35.png[/img] and [img]https://cdn.artofproblemsolving.com/attachments/8/a/7a7754d0f2517c93c5bb931fb7b5ae8f5e3217.png[/img], that can be used to tile a $7\times 7$ square? [b]p4.[/b] Six shooters participate in a shooting competition. Every participant has $5$ shots. Each shot adds from 1 to $10$ points to shooter’s score. Every person can score totally for all five shots from $5$ to $50$ points. Each participant gets $7$ points for at least one of his shots. The scores of all participants are different. We enumerate the shooters $1$ to $6$ according to their scores, the person with maximal score obtains number $1$, the next one obtains number $2$, the person with minimal score obtains number $6$. What score does obtain the participant number 3? The total number of all obtained points is $264$. [b]p5.[/b] There are $2014$ stones in a pile. Two players play the following game. First, player $A$ takes some number of stones (from $1$ to $30$) from the pile, then player B takes $1$ or $2$ stones, then player $A$ takes $2$ or $3$ stones, then player $B$ takes $3$ or $4$ stones, then player A takes $4$ or $5$ stones, etc. The player who gets the last stone is the winner. If no player gets the last stone (there is at least one stone in the pile but the next move is not allowed) then the game results in a draw. Who wins the game using the right strategy? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the following three lines in the Cartesian plane: $$\begin{cases} \ell_1: & 2x - y = 7\\ \ell_2: & 5x + y = 42\\ \ell_3: & x + y = 14 \end{cases}$$ and let $f_i(P)$ correspond to the reflection of the point $P$ across $\ell_i$. Suppose $X$ and $Y$ are points on the $x$ and $y$ axes, respectively, such that $f_1(f_2(f_3(X)))= Y$. Let $t$ be the length of segment $XY$; what is the sum of all possible values of $t^2$?

A line in the plane is called [i]strange[/i] if it passes through $(a,0)$ and $(0,10-a)$ for some $a$ in the interval $[0,10]$. A point in the plane is called [i]charming[/i] if it lies in the first quadrant and also lies [b]below[/b] some strange line. What is the area of the set of all charming points?

Two circles $(O)$ and $(O')$ intersect at $A$ and $B$. Take two points $P,Q$ on $(O)$ and $(O')$, respectively, such that $AP=AQ$. The line $PQ$ intersects $(O)$ and $(O')$ respectively at $M,N$. Let $E,F$ respectively be the centers of the two arcs $BP$ and $BQ$ (which don't contains $A$). Prove that $MNEF$ is a cyclic quadrilateral.

A circle of radius 1 has four circles $\omega_1, \omega_2, \omega_3$, and $\omega_4$ of equal radius internally tangent to it, so that $\omega_1$ is tangent to $\omega_2$, which is tangent to $\omega_3$, which is tangent to $\omega_4$, which is tangent to $\omega_1$, as shown. The radius of the circle externally tangent to $\omega_1, \omega_2, \omega_3$, and $\omega_4$ has radius r. If $r = a -\sqrt{b}$ for positive integers $a$ and $b$, compute $a + b$. [img]https://cdn.artofproblemsolving.com/attachments/e/3/c23f66333c0b4c0bf31b704cec665e50816149.png[/img]

The incircle of triangle $\vartriangle ABC$ is tangent to side $AB$ at point $P$ and to side $BC$ at point $Q$. The circle passing through points $A,P,Q$ intersects line $BC$ a second time at $M$ and the circle passes through the points $C,P,Q$ and cuts the line $AB$ a second time at point$ N$. Prove that $NM$ is tangent to the incircle of $ABC$.

Given a $\triangle ABC$, determine which is the circle with the smallest area that contains it.

Given an octagon such that all its interior angles are equal, and all its sides have integer lengths. Prove that any pair of opposite sides have equal lengths.

[u]Round 1[/u] [b]p1.[/b] A party is attended by ten people (men and women). Among them is Pat, who always lies to people of the opposite gender and tells the truth to people of the same gender. Pat tells five of the guests: “There are more men than women at the party.” Pat tells four of the guests: “There are more women than men at the party.” Is Pat a man or a woman? [b]p2.[/b] Once upon a time in a land far, far away there lived $100$ knights, $99$ princesses, and $101$ dragons. Over time, knights beheaded dragons, dragons ate princesses, and princesses poisoned knights. But they always obeyed an ancient law that prohibits killing any creature who has killed an odd number of others. Eventually only one creature remained alive. Could it have been a knight? A dragon? A princess? [b]p3.[/b] The numbers $1 \circ 2 \circ 3 \circ 4 \circ 5 \circ 6 \circ 7 \circ 8 \circ 9 \circ 10$ are written in a line. Alex and Vicky play a game, taking turns inserting either an addition or a multiplication symbol between adjacent numbers. The last player to place a symbol wins if the resulting expression is odd and loses if it is even. Alex moves first. Who wins? (Remember that multiplication is performed before addition.) [b]p4.[/b] A chess tournament had $8$ participants. Each participant played each other participant once. The winner of a game got $1$ point, the loser $0$ points, and in the case of a draw each got $1/2$ a point. Each participant scored a different number of points, and the person who got $2$nd prize scored the same number of points as the $5$th, $6$th, $7$th and $8$th place participants combined. Can you determine the result of the game between the $3$rd place player and the $5$th place player? [b]p5.[/b] One hundred gnomes sit in a circle. Each gnome gets a card with a number written on one side and a different number written on the other side. Prove that it is possible for all the gnomes to lay down their cards so that no two neighbors have the same numbers facing up. [u]Round 2[/u] [b]p6.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png[/img] [b]p7.[/b] Each of the $100$ residents of Pleasantville has at least $30$ friends in town. A resident votes in the mayoral election only if one of her friends is a candidate. Prove that it is possible to nominate two candidates for mayor so that at least half of the residents will vote. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].