Let there be given three circles $K_1,K_2,K_3$ with centers $O_1,O_2,O_3$ respectively, which meet at a common point $P$. Also, let $K_1 \cap K_2 = \{P,A\}, K_2 \cap K_3 = \{P,B\}, K_3 \cap K_1 = \{P,C\}$. Given an arbitrary point $X$ on $K_1$, join $X$ to $A$ to meet $K_2$ again in $Y$ , and join $X$ to $C$ to meet $K_3$ again in $Z.$ [b](a)[/b] Show that the points $Z,B, Y$ are collinear. [b](b)[/b] Show that the area of triangle $XY Z$ is less than or equal to $4$ times the area of triangle $O_1O_2O_3.$

Let $ABC$ be an acute-angled scalene triangle. Let $P$ be a point on the extension of $AB$ past $B$, and $Q$ a point on the extension of $AC$ past $C$ such that $BPQC$ is a cyclic quadrilateral. Let $N$ be the foot of the perpendicular from $A$ to $BC$. If $NP = NQ$ then prove that $N$ is also the centre of the circumcircle of $APQ$.

A cube of edge $3$ cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then $N=$ $\text{(A)}\ 4 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 20$

Prove that the bisectors of the angles of a rectangle, extended to their mutual intersection, form a square.

If $a_1, a_2, \cdots, a_7$ are not necessarily distinct real numbers such that $1 < a_i < 13$ for all $i$, then show that we can choose three of them such that they are the lengths of the sides of a triangle.

Let $ABCDE$ be a pentagon with area $2017$ such that four of its sides $AB, BC, CD$, and $EA$ have integer length. Suppose that $\angle A = \angle B = \angle C = 90^o$, $AB = BC$, and $CD = EA$. The maximum possible perimeter of $ABCDE$ is $a + b \sqrt{c}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a + b + c$.

Two non-intersecting circles, not lying inside each other, are drawn in the plane. Two lines pass through a point P which lies outside each circle. The first line intersects the first circle at A and A′ and the second circle at B and B′; here A and B are closer to P than A′ and B′, respectively, and P lies on segment AB. Analogously, the second line intersects the first circle at C and C′ and the second circle at D and D′. Prove that the points A, B, C, D are concyclic if and only if the points A′, B′, C′, D′ are concyclic.

Let $ABCDE$ be a convex pentagon. Let $P$ be the intersection of the lines $CE$ and $BD$. Assume that $\angle PAD = \angle ACB$ and $\angle CAP = \angle EDA$. Prove that the circumcentres of the triangles $ABC$ and $ADE$ are collinear with $P$.

Let $ABC$ be an acute triangle with $AB\neq AC$. Let $D, E, F$ be the midpoints of the sides $BC$, $CA$, and $AB$ respectively, and $M, N$ be the midpoints of minor arc $BC$ not containing $A$ and major arc $BAC$ respectively. Suppose $W, X, Y, Z$ are the incenter, $D$-excenter, $E$-excenter, and $F$-excenter of triangle $DEF$ respectively. Prove that the circumcircles of the triangles $ABC$, $WNX$, $YMZ$ meet at a common point. [i]Proposed by Ivan Chan Kai Chin[/i]

[b]p1.[/b] In the following $3$ by $3$ grid, $a, b, c$ are numbers such that the sum of each row is listed at the right and the sum of each column is written below it: [center][img]https://cdn.artofproblemsolving.com/attachments/d/9/4f6fd2bc959c25e49add58e6e09a7b7eed9346.png[/img][/center] What is $n$? [b]p2.[/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 both different colors and different lengths. Given only this information, what is the maximum number of choices you might have? [b]p3.[/b] The population of Arveymuddica is $2014$, which is divided into some number of equal groups. During an election, each person votes for one of two candidates, and the person who was voted for by $2/3$ or more of the group wins. When neither candidate gets $2/3$ of the vote, no one wins the group. The person who wins the most groups wins the election. What should the size of the groups be if we want to minimize the minimum total number of votes required to win an election? [b]p4.[/b] A farmer learns that he will die at the end of the year (day $365$, where today is day $0$) and that he has a number of sheep. He decides that his utility is given by ab where a is the money he makes by selling his 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. If every day his sheep breed and multiply their numbers by $103/101$ (yes, there are small, fractional sheep), on which day should he sell them all? [b]p5.[/b] Line segments $\overline{AB}$ and $\overline{AC}$ are tangent to a convex arc $BC$ and $\angle BAC = \frac{\pi}{3}$ . If $\overline{AB} = \overline{AC} = 3\sqrt3$, find the length of arc $BC$. [b]p6.[/b] Suppose that you start with the number $8$ and always have two legal moves: $\bullet$ Square the number $\bullet$ Add one if the number is divisible by $8$ or multiply by $4$ otherwise How many sequences of $4$ moves are there that return to a multiple of $8$? [b]p7.[/b] A robot is shuffling a $9$ card deck. Being very well machined, it does every shuffle in exactly the same way: it splits the deck into two piles, one containing the $5$ cards from the bottom of the deck and the other with the $4$ cards from the top. It then interleaves the cards from the two piles, starting with a card from the bottom of the larger pile at the bottom of the new deck, and then alternating cards from the two piles while maintaining the relative order of each pile. The top card of the new deck will be the top card of the bottom pile. The robot repeats this shuffling procedure a total of n times, and notices that the cards are in the same order as they were when it started shuffling. What is the smallest possible value of $n$? [b]p8.[/b] A secant line incident to a circle at points $A$ and $C$ intersects the circle's diameter at point $B$ with a $45^o$ angle. If the length of $AB$ is $1$ and the length of $BC$ is $7$, then what is the circle's radius? [b]p9.[/b] If a complex number $z$ satisfies $z + 1/z = 1$, then what is $z^{96} + 1/z^{96}$? [b]p10.[/b] Let $a, b$ be two acute angles where $\tan a = 5 \tan b$. Find the maximum possible value of $\sin (a - b)$. [b]p11.[/b] A pyramid, represented by $SABCD$ has parallelogram $ABCD$ as base ($A$ is across from $C$) and vertex $S$. Let the midpoint of edge $SC$ be $P$. Consider plane $AMPN$ where$ M$ is on edge $SB$ and $N$ is on edge $SD$. Find the minimum value $r_1$ and maximum value $r_2$ of $\frac{V_1}{V_2}$ where $V_1$ is the volume of pyramid $SAMPN$ and $V_2$ is the volume of pyramid $SABCD$. Express your answer as an ordered pair $(r_1, r_2)$. [b]p12.[/b] A $5 \times 5$ grid is missing one of its main diagonals. In how many ways can we place $5$ pieces on the grid such that no two pieces share a row or column? [b]p13.[/b] There are $20$ cities in a country, some of which have highways connecting them. Each highway goes from one city to another, both ways. There is no way to start in a city, drive along the highways of the country such that you travel through each city exactly once, and return to the same city you started in. What is the maximum number of roads this country could have? [b]p14.[/b] Find the area of the cyclic quadrilateral with side lengths given by the solutions to $$x^4-10x^3+34x^2- 45x + 19 = 0.$$ [b]p15.[/b] Suppose that we know $u_{0,m} = m^2 + m$ and $u_{1,m} = m^2 + 3m$ for all integers $m$, and that $$u_{n-1,m} + u_{n+1,m} = u_{n,m-1} + u_{n,m+1}$$ Find $u_{30,-5}$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the maximum length of a broken line on the surface of a unit cube, such that its links are the cube’s edges and diagonals of faces, the line does not intersect itself and passes no more than once through any vertex of the cube, and its endpoints are in two opposite vertices of the cube.

Let $ABC$ be a triangle with $AB > AC$. The internal angle bisector of $\angle BAC$ intersects the side $BC$ at $D$. The circles with diameters $BD$ and $CD$ intersect the circumcircle of $\triangle ABC$ a second time at $P \not= B$ and $Q \not= C$, respectively. The lines $PQ$ and $BC$ intersect at $X$. Prove that $AX$ is tangent to the circumcircle of $\triangle ABC$.

As in the picture below, the rectangle on the left hand side has been divided into four parts by line segments which are parallel to a side of the rectangle. The areas of the small rectangles are $A,B,C$ and $D$. Similarly, the small rectangles on the right hand side have areas $A^\prime,B^\prime,C^\prime$ and $D^\prime$. It is known that $A\leq A^\prime$, $B\leq B^\prime$, $C\leq C^\prime$ but $D\leq B^\prime$. [asy] import graph; size(12cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.3,xmax=12.32,ymin=-10.68,ymax=6.3; draw((0,3)--(0,0)); draw((3,0)--(0,0)); draw((3,0)--(3,3)); draw((0,3)--(3,3)); draw((2,0)--(2,3)); draw((0,2)--(3,2)); label("$A$",(0.86,2.72),SE*lsf); label("$B$",(2.38,2.7),SE*lsf); label("$C$",(2.3,1.1),SE*lsf); label("$D$",(0.82,1.14),SE*lsf); draw((5,2)--(11,2)); draw((5,2)--(5,0)); draw((11,0)--(5,0)); draw((11,2)--(11,0)); draw((8,0)--(8,2)); draw((5,1)--(11,1)); label("$A'$",(6.28,1.8),SE*lsf); label("$B'$",(9.44,1.82),SE*lsf); label("$C'$",(9.4,0.8),SE*lsf); label("$D'$",(6.3,0.86),SE*lsf); dot((0,3),linewidth(1pt)+ds); dot((0,0),linewidth(1pt)+ds); dot((3,0),linewidth(1pt)+ds); dot((3,3),linewidth(1pt)+ds); dot((2,0),linewidth(1pt)+ds); dot((2,3),linewidth(1pt)+ds); dot((0,2),linewidth(1pt)+ds); dot((3,2),linewidth(1pt)+ds); dot((5,0),linewidth(1pt)+ds); dot((5,2),linewidth(1pt)+ds); dot((11,0),linewidth(1pt)+ds); dot((11,2),linewidth(1pt)+ds); dot((8,0),linewidth(1pt)+ds); dot((8,2),linewidth(1pt)+ds); dot((5,1),linewidth(1pt)+ds); dot((11,1),linewidth(1pt)+ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy] Prove that the big rectangle on the left hand side has area smaller or equal to the area of the big rectangle on the right hand side, i.e. $A+B+C+D\leq A^\prime+B^\prime+C^\prime+D^\prime$.

It is known that the bisector of the angle $\angle ADC$ of the inscribed quadrilateral $ABCD$ passes through the center of the circle inscribed in the triangle $ABC$. Let $M$ be an arbitrary point of the arc $ABC$ of the circle circumscribed around $ABCD$. Denote by $P$ and $Q$ the centers of the circles inscribed in the triangles $ABM$ and $BCM$. Prove that all triangles $DPQ$ obtained by moving point $M$ are similar to each other. Find the angle $\angle PDQ$ and ratio $BP : PQ$ if $\angle BAC = \alpha$, $\angle BCA = \beta$

Triangle $ABC$ has circumcircle $\omega$. The angle bisectors of $\angle A$ and $\angle B$ intersect $\omega$ at points $D$ and $E$ respectively. $DE$ intersects $BC$ and $AC$ at $X$ and $Y$ respectively. Given $DX = 7,$ $XY = 8$ and $YE = 9,$ the area of $\triangle ABC$ can be written as $\frac{a\sqrt{b}}{c}$ where $a, b, c$ are positive integers, $\gcd(a,c) = 1,$ and $b$ is square free. Find $a+b+c.$

In the following, the point of intersection of two lines $ g$ and $ h$ will be abbreviated as $ g\cap h$. Suppose $ ABC$ is a triangle in which $ \angle A \equal{} 90^{\circ}$ and $ \angle B > \angle C$. Let $ O$ be the circumcircle of the triangle $ ABC$. Let $ l_{A}$ and $ l_{B}$ be the tangents to the circle $ O$ at $ A$ and $ B$, respectively. Let $ BC \cap l_{A} \equal{} S$ and $ AC \cap l_{B} \equal{} D$. Furthermore, let $ AB \cap DS \equal{} E$, and let $ CE \cap l_{A} \equal{} T$. Denote by $ P$ the foot of the perpendicular from $ E$ on $ l_{A}$. Denote by $ Q$ the point of intersection of the line $ CP$ with the circle $ O$ (different from $ C$). Denote by $ R$ be the point of intersection of the line $ QT$ with the circle $ O$ (different from $ Q$). Finally, define $ U \equal{} BR \cap l_{A}$. Prove that \[ \frac {SU \cdot SP}{TU \cdot TP} \equal{} \frac {SA^{2}}{TA^{2}}. \]

In a convex set $S$ that contains the origin it is possible to draw $n$ disjoint unit circles such that viewing from the origin non of the unit circles blocks out a part of another (or a complete) unit circle. Prove that the area of $S$ is at least $\frac{n^2}{100}$.

Let $ABCD$ be a square, and $C$ the circle whose diameter is $AB.$ Let $Q$ be an arbitrary point on the segment $CD.$ We know that $QA$ meets $C$ on $E$ and $QB$ meets it on $F.$ Also $CF$ and $DE$ intersect in $M.$ show that $M$ belongs to $C.$

Circle centred at point $O$ intersects sides $AC, AB$ of triangle $\triangle ABC$ at points $B_1$ and $C_1$ respectively and passes through points $B,C$. It is known that lines $AO, CC_1, BB_1 $ are concurrent. Prove that $\triangle ABC$ is isosceles.

$ABC$ is a triangle, with inradius $r$ and circumradius $R.$ Show that: \[ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{B}{2} \right) + \sin \left( \frac{B}{2} \right) \cdot \sin \left( \frac{C}{2} \right) + \sin \left( \frac{C}{2} \right) \cdot \sin \left( \frac{A}{2} \right) \leq \frac{5}{8} + \frac{r}{4 \cdot R}. \]

The altitudes of triangle $ABC$ intersect at a point $H$.Find $\angle ACB$ if it is known that $AB = CH$.

Let $\vartriangle ABC$ be an acute triangle with $AB < AC$. Let $H$ be its orthocentre and $M$ be the midpoint of arc $BAC$ on the circumcircle. It is given that $B$, $H$, $M$ are collinear, the length of the altitude from $M$ to $AB$ is $1$, and the length of the altitude from $M$ to $BC$ is $6$. Determine all possible areas for $\vartriangle ABC$ .

Let $\Delta ABC$ be an isosceles triangle with $BC = AC$ and $\angle ACB = 40^{\circ}$. Contruct the circle with diameter $\overline{BC}$, and let $D$ and $E$ be the other intersection points of the circle with the sides $\overline{AC}$ and $\overline{AB}$, respectively. Let $F$ be the intersection of the diagonals of the quadrilateral $BCDE$. What is the degree measure of $\angle BFC ?$ $\textbf{(A) } 90 \qquad\textbf{(B) } 100 \qquad\textbf{(C) } 105 \qquad\textbf{(D) } 110 \qquad\textbf{(E) } 120$

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Evaluate $20 \times 21 + 2021$. [b]p2.[/b] Let points $A$, $B$, $C$, and $D$ lie on a line in that order. Given that $AB = 5CD$ and $BD = 2BC$, compute $\frac{AC}{BD}$. [b]p3.[/b] There are $18$ students in Vincent the Bug's math class. Given that $11$ of the students take U.S. History, $15$ of the students take English, and $2$ of the students take neither, how many students take both U.S. History and English? [b]p4.[/b] Among all pairs of positive integers $(x, y)$ such that $xy = 12$, what is the least possible value of $x + y$? [b]p5.[/b] What is the smallest positive integer $n$ such that $n! + 1$ is composite? [b]p6.[/b] How many ordered triples of positive integers $(a, b,c)$ are there such that $a + b + c = 6$? [b]p7.[/b] Thomas orders some pizzas and splits each into $8$ slices. Hungry Yunseo eats one slice and then finds that she is able to distribute all the remaining slices equally among the $29$ other math club students. What is the fewest number of pizzas that Thomas could have ordered? [b]p8.[/b] Stephanie has two distinct prime numbers $a$ and $b$ such that $a^2-9b^2$ is also a prime. Compute $a + b$. [b]p9.[/b] Let $ABCD$ be a unit square and $E$ be a point on diagonal $AC$ such that $AE = 1$. Compute $\angle BED$, in degrees. [b]p10.[/b] Sheldon wants to trace each edge of a cube exactly once with a pen. What is the fewest number of continuous strokes that he needs to make? A continuous stroke is one that goes along the edges and does not leave the surface of the cube. [b]p11.[/b] In base $b$, $130_b$ is equal to $3n$ in base ten, and $1300_b$ is equal to $n^2$ in base ten. What is the value of $n$, expressed in base ten? [b]p12.[/b] Lin is writing a book with $n$ pages, numbered $1,2,..., n$. Given that $n > 20$, what is the least value of $n$ such that the average number of digits of the page numbers is an integer? [b]p13.[/b] Max is playing bingo on a $5\times 5$ board. He needs to fill in four of the twelve rows, columns, and main diagonals of his bingo board to win. What is the minimum number of boxes he needs to fill in to win? [b]p14.[/b] Given that $x$ and $y$ are distinct real numbers such that $x^2 + y = y^2 + x = 211$, compute the value of $|x - y|$. [b]p15.[/b] How many ways are there to place 8 indistinguishable pieces on a $4\times 4$ checkerboard such that there are two pieces in each row and two pieces in each column? [b]p16.[/b] The Manhattan distance between two points $(a, b)$ and $(c, d)$ in the plane is defined to be $|a - c| + |b - d|$. Suppose Neil, Neel, and Nail are at the points $(5, 3)$, $(-2,-2)$ and $(6, 0)$, respectively, and wish to meet at a point $(x, y)$ such that their Manhattan distances to$ (x, y)$ are equal. Find $10x + y$. [b]p17.[/b] How many positive integers that have a composite number of divisors are there between $1$ and $100$, inclusive? [b]p18.[/b] Find the number of distinct roots of the polynomial $$(x - 1)(x - 2) ... (x - 90)(x^2 - 1)(x^2 - 2) ... (x^2 - 90)(x^4 - 1)(x^4 - 2)...(x^4 - 90)$$. [b]p19.[/b] In triangle $ABC$, let $D$ be the foot of the altitude from $ A$ to $BC$. Let $P,Q$ be points on $AB$, $AC$, respectively, such that $PQ$ is parallel to $BC$ and $\angle PDQ = 90^o$. Given that $AD = 25$, $BD = 9$, and $CD = 16$, compute $111 \times PQ$. [b]p20.[/b] The simplified fraction with numerator less than $1000$ that is closest but not equal to $\frac{47}{18}$ is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

From a point $A$ construct tangents to a circle centered at point $O$, intersecting the circle at $P$ and $Q$ respectively. Let $M$ be the midpoint of $PQ$. If $K$ and $L$ are points on circle $O$ such that $K, L$, and $A$ are collinear, prove $\angle MKO = \angle MLO$.