A circle is divided by $2019$ points into equal parts. Two players delete these points in turns. A player loses, if after his turn it is possible to draw a diameter of the circle such that there are no undeleted points on one side of it. Which player has a winning strategy?

[b]p1.[/b] Evaluate $(2-0)^2 \cdot 3+ \frac{20}{2+3}$ . [b]p2.[/b] Let $x = 11 \cdot 99$ and $y = 9 \cdot 101$. Find the sumof the digits of $x \cdot y$. [b]p3.[/b] A rectangle is cut into two pieces. The ratio between the areas of the two pieces is$ 3 : 1$ and the positive difference between those areas is $20$. What’s the area of the rectangle? [b]p4.[/b] Edgeworth is scared of elevators. He is currently on floor $50$ of a building, and he wants to go down to floor $1$. Edgeworth can go down at most $4$ floors each time he uses the elevator. What’s the minimum number of times he needs to use the elevator to get to floor $1$? [b]p5.[/b] There are $20$ people at a party. Fifteen of those people are normal and $5$ are crazy. A normal person will shake hands once with every other normal person, while a crazy person will shake hands twice with every other crazy person. How many total handshakes occur at the party? [b]p6.[/b] Wam and Sang are chewing gum. Gum comes in packages, each package consisting of $14$ sticks of gum. Wam eats $6$ packs and $9$ individual sticks of gum. Sang wants to eat twice as much gum as Wam. How many packs of gum must Sang buy? [b]p7.[/b] At Lakeside Health School (LHS), $40\%$ of students are male and $60\%$ of the students are female. If half of the students at the school take biology, and the same number ofmale and female students take biology, to the nearest percent, what percent of female students take biology? [b]p8.[/b] Evin is bringing diluted raspberry iced tea to the annual LexingtonMath Team party. He has a cup with $10$ mL of iced tea and a $2000$ mL cup of water with $10\%$ raspberry iced tea. If he fills up the cup with $20$ more mL of $10\%$ raspberry iced tea water, what percent of the solution will be iced tea? [b]p9.[/b] Tree $1$ starts at height $220$ m and grows continuously at $3$ m per year. Tree $2$ starts at height $20$ m and grows at $5$ m during the first year, $7$ m per during the second year, $9$ m during the third year, and in general $(3+2n)$ m in the nth year. After which year is Tree $2$ taller than Tree $1$? [b]p10.[/b] Leo and Chris are playing a game in which Chris flips a coin. The coin lands on heads with probability $\frac{499}{999}$ , tails with probability $\frac{499}{999}$ , and it lands on its side with probability $\frac{1}{999}$ . For each flip of the coin, Leo agrees to give Chris $4$ dollars if it lands on heads, nothing if it lands on tails, and $2$ dollars if it lands on its side. What’s the expected value of the number of dollars Chris gets after flipping the coin $17$ times? [b]p11.[/b] Ephram has a pile of balls, which he tries to divide into piles. If he divides the balls into piles of $7$, there are $5$ balls that don’t get divided into any pile. If he divides the balls into piles of $11$, there are $9$ balls that aren’t in any pile. If he divides the balls into piles of $13$, there are $11$ balls that aren’t in any pile. What is the minimumnumber of balls Ephram has? [b]p12.[/b] Let $\vartriangle ABC$ be a triangle such that $AB = 3$, $BC = 4$, and $C A = 5$. Let $F$ be the midpoint of $AB$. Let $E$ be the point on $AC$ such that $EF \parallel BC$. Let CF and $BE$ intersect at $D$. Find $AD$. [b]p13.[/b] Compute the sum of all even positive integers $n \le 1000$ such that: $$lcm(1,2, 3, ..., (n -1)) \ne lcm(1,2, 3,, ...,n)$$. [b]p14.[/b] Find the sum of all palindromes with $6$ digits in binary, including those written with leading zeroes. [b]p15.[/b] What is the side length of the smallest square that can entirely contain $3$ non-overlapping unit circles? [b]p16.[/b] Find the sum of the digits in the base $7$ representation of $6250000$. Express your answer in base $10$. [b]p17.[/b] A number $n$ is called sus if $n^4$ is one more than a multiple of $59$. Compute the largest sus number less than $2023$. [b]p18.[/b] Michael chooses real numbers $a$ and $b$ independently and randomly from $(0, 1)$. Given that $a$ and $b$ differ by at most $\frac14$, what is the probability $a$ and $b$ are both greater than $\frac12$ ? [b]p19.[/b] In quadrilateral $ABCD$, $AB = 7$ and $DA = 5$, $BC =CD$, $\angle BAD = 135^o$ and $\angle BCD = 45^o$. Find the area of $ABCD$. [b]p20.[/b] Find the value of $$\sum_{i |210} \sum_{j |i} \left \lfloor \frac{i +1}{j} \right \rfloor$$ [b]p21.[/b] Let $a_n$ be the number of words of length $n$ with letters $\{A,B,C,D\}$ that contain an odd number of $A$s. Evaluate $a_6$. [b]p22.[/b] Detective Hooa is investigating a case where a criminal stole someone’s pizza. There are $69$ people involved in the case, among whom one is the criminal and another is a witness of the crime. Every day, Hooa is allowed to invite any of the people involved in the case to his rather large house for questioning. If on some given day, the witness is present and the criminal is not, the witness will reveal who the criminal is. What is the minimum number of days of questioning required such that Hooa is guaranteed to learn who the criminal is? [b]p23.[/b] Find $$\sum^{\infty}_{n=2} \frac{2n +10}{n^3 +4n^2 +n -6}.$$ [b]p24.[/b] Let $\vartriangle ABC$ be a triangle with circumcircle $\omega$ such that $AB = 1$, $\angle B = 75^o$, and $BC =\sqrt2$. Let lines $\ell_1$ and $\ell_2$ be tangent to $\omega$ at $A$ and $C$ respectively. Let $D$ be the intersection of $\ell_1$ and $\ell_2$. Find $\angle ABD$ (in degrees). [b]p25.[/b] Find the sum of the prime factors of $14^6 +27$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Triangle $ABC$. Let $B_1$ and $C_1$ be such points, that $AB= BB_1, AC=CC_1$ and $B_1, C_1$ lie on the circumscribed circle $\Gamma$ of $\vartriangle ABC$. Perpendiculars drawn from from points $B_1$ and $C_1$ on the lines $AB$ and $AC$ intersect $\Gamma$ at points $B_2$ and $C_2$ respectively, these points lie on smaller arcs $AB$ and $AC$ of circle $\Gamma$ respectively, Prove that $BB_2 \parallel CC_2$.

Let $ABC$ be a right-angled triangle in $A$ such that $AB<AC$. Let $M$ be the midpoint of $BC$ and let $D$ be the intersection of $AC$ with the perpendicular line to $BC$ which passes through $M$. Let $E$ be the intersection point of the parallel line to $AC$ which passes through $M$ with the perpendicular line to $BD$ which passes through $B$. Prove that triangles $AEM$ and $MCA$ are similar if and only if $\angle ABC=60^{\circ}$.

Which of the following polynomials does not divide $x^{60} - 1$? $ \textbf{a)}\ x^2+x+1 \qquad\textbf{b)}\ x^4-1 \qquad\textbf{c)}\ x^5-1 \qquad\textbf{d)}\ x^{15}-1 \qquad\textbf{e)}\ \text{None of above} $

For which values of $n$ does there exist a convex polyhedron with $n$ edges?

In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such that $\frac{F_1}{F_2}<m$ holds for any $\triangle{ABC}$.

Let $ABC$ be a triangle with incenter $I$, and let the tangency points of its incircle with its sides $AB$, $BC$, $CA$ be $C'$, $A'$ and $B'$ respectively. Prove that the circumcenters of $AIA'$, $BIB'$, and $CIC'$ are collinear.

$ABCD$ is a trapezoid with $AB || CD$. There are two circles $\omega_1$ and $\omega_2$ is the trapezoid such that $\omega_1$ is tangent to $DA$, $AB$, $BC$ and $\omega_2$ is tangent to $BC$, $CD$, $DA$. Let $l_1$ be a line passing through $A$ and tangent to $\omega_2$(other than $AD$), Let $l_2$ be a line passing through $C$ and tangent to $\omega_1$ (other than $CB$). Prove that $l_1 || l_2$.

An acute-angled $ABC \ (AB<AC)$ is inscribed into a circle $\omega$. Let $M$ be the centroid of $ABC$, and let $AH$ be an altitude of this triangle. A ray $MH$ meets $\omega$ at $A'$. Prove that the circumcircle of the triangle $A'HB$ is tangent to $AB$. [i](A.I. Golovanov , A.Yakubov)[/i]

Let $A'$ be the result of reflection of vertex $A$ of triangle ABC through line $BC$ and let $B'$ be the result of reflection of vertex $B$ through line $AC$. Given that $\angle BA' C = \angle BB'C$, can the largest angle of triangle $ABC$ be located: a) At vertex $A$, b) At vertex $B$, c) At vertex $C$?

A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$ \[|l(z)| \leq M \rho,\] where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$

On sides $AB$ and $BC$ of a square $ABCD$ the respective points $E$ and $F$ have been chosen so that $BE = BF$. Let $BN$ be the altitude in triangle $BCE$. Prove that $\angle DNF = 90$.

Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$. $(a)$ Prove that the tangent to the second circle at $R$ is parallel to $AK$. $(b)$ Prove that these two tangents meet on $KM$.

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

Let $ABC$ be an isosceles triangle with base $AB$. We choose a point $P$ inside the triangle on altitude through $C$. The circle with diameter $CP$ intersects the straight line through $B$ and $P$ again at the point $D_P$ and the Straight through $A$ and $C$ one more time at point $E_P$. Prove that there is a point $F$ such that for any choice of $P$ the points $D_P , E_P$ and $F$ lie on a straight line. [i](Walther Janous)[/i]

Let $\Gamma_1, \Gamma_2$ be circles that touch at a point $A$, and $\Gamma_2$ is inside $\Gamma_1$. Let $B$ be on $\Gamma_2$, and let $AB$ intersect $\Gamma_1$ on $C$. Let $D$ be on $\Gamma_1$ and $P$ be on the line $CD$ (may be outside of the segment $CD$). $BP$ intersects $\Gamma_2$ at $Q$. Prove that $A,D,P,Q$ lie on a circle.

In triangle $ ABC$, $ AC \equal{} 13, BC \equal{} 14,$ and $ AB\equal{}15$. Points $ M$ and $ D$ lie on $ AC$ with $ AM\equal{}MC$ and $ \angle ABD \equal{} \angle DBC$. Points $ N$ and $ E$ lie on $ AB$ with $ AN\equal{}NB$ and $ \angle ACE \equal{} \angle ECB$. Let $ P$ be the point, other than $ A$, of intersection of the circumcircles of $ \triangle AMN$ and $ \triangle ADE$. Ray $ AP$ meets $ BC$ at $ Q$. The ratio $ \frac{BQ}{CQ}$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\minus{}n$.

Let $ABC$ be a scalene triangle with circumcenter $O$ and orthocenter $H$. Let $AYZ$ be another triangle sharing the vertex $A$ such that its circumcenter is $H$ and its orthocenter is $O$. Show that if $Z$ is on $BC$, then $A,H,O,Y$ are concyclic. [i]Proposed by usjl[/i]

The mapping $f$ assigns a circle to every triangle in the plane so that the following conditions hold. (We consider all nondegenerate triangles and circles of nonzero radius.) [b](a)[/b] Let $\sigma$ be any similarity in the plane and let $\sigma$ map triangle $\Delta_1$ onto triangle $\Delta_2$. Then $\sigma$ also maps circle $f(\Delta_1)$ onto circle $f(\Delta_2)$. [b](b)[/b] Let $A,B,C$ and $D$ be any four points in general position. Then circles $f(ABC),f(BCD),f(CDA)$ and $f(DAB)$ have a common point. Prove that for any triangle $\Delta$, the circle $f(\Delta)$ is the Euler circle of $\Delta$.

Let $ABC$ be a triangle. Let its altitudes $AD$, $BE$ and $CF$ concur at $H$. Let $K, L$ and $M$ be the midpoints of $BC$, $CA$ and $AB$, respectively. Prove that, if $\angle BAC = 60^o$, then the midpoints of the segments $AH$, $DK$, $EL$, $FM$ are concyclic.

A boy and his father are standing on a seashore. If the boy stands on his tiptoes, his eyes are at a height of $1$ m above sea-level, and if he seats on father’s shoulders, they are at a height of $2$ m. What is the ratio of distances visible for him in two eases? (Find the answer to $0,1$, assuming that the radius of Earth equals $6000$ km.)

Let $ABCD$ be a rectangle with $AB = 30$ and $BC = 28$. Point $P$ and $Q$ lie on $\overline{BC}$ and $\overline{CD}$ respectively so that all sides of $\triangle{ABP}, \triangle{PCQ},$ and $\triangle{QDA}$ have integer lengths. What is the perimeter of $\triangle{APQ}$? (A) 84 (B) 86 (C) 88 (D)90 (E)92

Given triangle $ABC, BC$ is extended beyond $B$ to the point $D$ such that $BD = BA$. The bisectors of the exterior angles at vertices $B$ and $C$ intersect at the point $M$. Prove that quadrilateral $ADMC$ is cyclic. (4)

Let $ABC$ be a triangle with altitude $AD$ , angle bisectors $AE$ and $BZ$ that intersecting at point $I$. From point $I$ let $IT$ be a perpendicular on $AC$. Also let line $(e)$ be perpendicular on $AC$ at point $A$. Extension of $ET$ intersects line $(e)$ at point $K$. Prove that $AK=AD$.