Consider a trapezoid $ABCD$ with $BC||AD$ and $BC<AD$. Let the lines $AB$ and $CD$ meet at $X$. Let $\omega_1$ be the incircle of the triangle $XBC$, and let $\omega_2$ be the excircle of the triangle $XAD$ which is tangent to the segment $AD$ . Denote by $a$ and $d$ the lines tangent to $\omega_1$ , distinct from $AB$ and $CD$, and passing through $A$ and $D$, respectively. Denote by $b$ and $c$ the lines tangent to $\omega_2$ , distinct from $AB$ and $CD$, passing through $B$ and $C$ respectively. Assume that the lines $a,b,c$ and $d$ are distinct. Prove that they form a parallelogram.

All of the numbers $1, 2,3,...,1000000$ are initially colored black. On each move it is possible to choose the number $x$ (among the colored numbers) and change the color of $x$ and of all of the numbers that are not co-prime with $x$ (black into white, white into black). Is it possible to color all of the numbers white?

Define the set $P=\{a_1, a_2, a_3, \cdots, a_n\}$ and its arithmetic mean $$C_p = \frac{a_1+a_2+\cdots+a_m}m.$$ If we divide $S = \{1, 2, 3, \cdots, n\}$ into two disjoint subsets $A, B$, compute the greatest possible value of $|C_A-C_B|$. For how many $(A, B)$ is equality attained? [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 14)[/i]

Let $ ABC$ be an acute triangle and $ CL$ be the angle bisector of $ \angle ACB$. The point $ P$ lies on the segment $CL$ such that $ \angle APB\equal{}\pi\minus{}\frac{_1}{^2}\angle ACB$. Let $ k_1$ and $ k_2$ be the circumcircles of the triangles $ APC$ and $ BPC$. $ BP\cap k_1\equal{}Q, AP\cap k_2\equal{}R$. The tangents to $ k_1$ at $ Q$ and $ k_2$ at $ B$ intersect at $ S$ and the tangents to $ k_1$ at $ A$ and $ k_2$ at $ R$ intersect at $ T$. Prove that $ AS\equal{}BT.$

Given a triangle with $ \triangle ABC $, with: $ \angle C = 36^o$ and $ \angle A = \angle B $. Consider the points $ D $ on $ BC $, $ E $ on $ AD $, $ F $ on $ BE $, $ G $ on $ DF $ and $ H $ on $ EG $, so that the rays $ AD, BE, DF, EG, FH $ bisect the angles $ A, B, D, E, F $ respectively. It is known that $ FH = 1 $. Calculate $ AC$.

For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? $\textbf{(A) } 14 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 19 \qquad \textbf{(E) } 21$

$\textbf{(Self-Isogonal Cubics)}$ Let $ABC$ be a triangle with $AB = 2$, $AC = 3$, $BC = 4$. The $\emph{isogonal conjugate}$ of a point $P$, denoted $P^\ast$, is the point obtained by intersecting the reflection of lines $PA$, $PB$, $PC$ across the angle bisectors of $\angle A$, $\angle B$, and $\angle C$, respectively. Given a point $Q$, let $\mathfrak K(Q)$ denote the unique cubic plane curve which passes through all points $P$ such that line $PP^\ast$ contains $Q$. Consider: [list] [*] the M'Cay cubic $\mathfrak K(O)$, where $O$ is the circumcenter of $\triangle ABC$, [*] the Thomson cubic $\mathfrak K(G)$, where $G$ is the centroid of $\triangle ABC$, [*] the Napoleon-Feurerbach cubic $\mathfrak K(N)$, where $N$ is the nine-point center of $\triangle ABC$, [*] the Darboux cubic $\mathfrak K(L)$, where $L$ is the de Longchamps point (the reflection of the orthocenter across point $O$), [*] the Neuberg cubic $\mathfrak K(X_{30})$, where $X_{30}$ is the point at infinity along line $OG$, [*] the nine-point circle of $\triangle ABC$, [*] the incircle of $\triangle ABC$, and [*] the circumcircle of $\triangle ABC$. [/list] Estimate $N$, the number of points lying on at least two of these eight curves. An estimate of $E$ earns $\left\lfloor 20 \cdot 2^{-|N-E|/6} \right\rfloor$ points.

For a prime $ p\ge 5, $ determine the number of polynomials $ X^p+pX^k+pX^l+1 $ with $ 1<k<l<p, $ that are ireducible over the integers.

Suppose $2$ cars are going into a turn the shape of a half-circle. Car $ 1$ is traveling at $50$ meters per second and is hugging the inside of the turn, which has radius $200$ meters. Car $2$ is trying to pass Car $ 1$ going along the turn, but in order to do this, he has to move to the outside of the turn, which has radius $210$. Suppose that both cars come into the turn side by side, and that they also end the turn being side by side. What was the average speed of Car $2$, in meters per second, throughout the turn?

Compute $\sqrt{202 \times 3 - 20 \times 23 + 2 \times 23 - 23}$. [i]Proposed by Andy Xu[/i]

On the sides $BC$ and $AB$ of the triangle $ABC$ the points ${{A} _ {1}}$ and ${{C} _ {1}} $ are selected accordingly so that the segments $A {{A} _ {1}}$ and $C {{C} _ {1}}$ are equal and perpendicular. Prove that if $\angle ABC = 45 {} ^ \circ$, then $AC = A {{A} _ {1}} $. (Gogolev Andrew)

Let \(n\) be a positive integer. On a \(2 \times 3 n\) board, we mark some squares, so that any square (marked or not) is adjacent to at most two other distinct marked squares (two squares are adjacent when they are distinct and have at least one vertex in common, i.e. they are horizontal, vertical or diagonal neighbors; a square is not adjacent to itself). (a) What is the greatest possible number of marked square? (b) For this maximum number, in how many ways can we mark the squares? configurations that can be achieved through rotation or reflection are considered distinct.

The number of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is $990$. Find the greatest number of apples growing on any of the six trees.

Find the $2019$th strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$.

Let $p > 3$ be a prime. For each $k\in \{1,2, \ldots , p-1\}$, define $x_k$ to be the unique integer in $\{1, \ldots, p-1\}$ such that $kx_k\equiv 1 \pmod{p}$ and set $kx_k = 1+ pn_k$. Prove that : \[\sum_{k=1}^{p-1}kn_k \equiv \frac{p-1}{2} \pmod{p}\]

Find all pairs of integers $a, b$ for which equality holds $\frac{a^2+1}{2b^2-3}=\frac{a-1}{2b-1}$

A rectangle with perimeter $ 176$ is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles? [asy]defaultpen(linewidth(.8pt)); draw(origin--(0,3)--(4,3)--(4,0)--cycle); draw((0,1)--(4,1)); draw((2,0)--midpoint((0,1)--(4,1))); real r = 4/3; draw((r,3)--foot((r,3),(0,1),(4,1))); draw((2r,3)--foot((2r,3),(0,1),(4,1)));[/asy]$ \textbf{(A)}\ 35.2\qquad \textbf{(B)}\ 76\qquad \textbf{(C)}\ 80\qquad \textbf{(D)}\ 84\qquad \textbf{(E)}\ 86$

Determine all nonnegative integers $x, y$ satisfying the equation \begin{align*} \sqrt{xy}=\sqrt{x+y}+\sqrt{x}+\sqrt{y}. \end{align*}

Find the smallest number $k$, such that $ \frac{l_a+l_b}{a+b}<k$ for all triangles with sides $a$ and $b$ and bisectors $l_a$ and $l_b$ to them, respectively. [i]Proposed by Sava Grodzev, Svetlozar Doichev, Oleg Mushkarov and Nikolai Nikolov[/i]

Let $D$ be a point inside $\triangle ABC$ such that $m(\widehat{BAD})=20^{\circ}$, $m(\widehat{DAC})=80^{\circ}$, $m(\widehat{ACD})=20^{\circ}$, and $m(\widehat{DCB})=20^{\circ}$. $m(\widehat{ABD})= ?$ $ \textbf{(A)}\ 5^{\circ} \qquad \textbf{(B)}\ 10^{\circ} \qquad \textbf{(C)}\ 15^{\circ} \qquad \textbf{(D)}\ 20^{\circ} \qquad \textbf{(E)}\ 25^{\circ}$

Suppose $m$ and $n$ are relatively prime positive integers. A regular $m$-gon and a regular $n$-gon are inscribed in a circle. Let $d$ be the minimum distance in degrees (of the arc along the circle) between a vertex of the $m$-gon and a vertex of the $n$-gon. What is the maximum possible value of $d$?

$P_1$, $P_2$, $P_3$, $Q_1$, $Q_2$, $Q_3$ are distinct points in the plane. The distances $P_1Q_1$, $P_2Q_2$, $P_3Q_3$ are equal. $P_1P_2$ and $Q_2Q_1$ are parallel (not antiparallel), similarly $P_1P_3$ and $Q_3Q_1$, and $P_2P_3$ and $Q_3Q_2$. Show that $P_1Q_1$, $P_2Q_2$ and $P_3Q_3$ intersect in a point.

The incircle of $\triangle ABC$ tangents $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. The projections of $B$, $C$ to $AD$ are $U$, $V$, respectively; the projections of $C$, $A$ to $BE$ are $W$, $X$, respectively; and the projections of $A$, $B$ to $CF$ are $Y$, $Z$, respectively. Show that the circumcircle of the triangle formed by $UX$, $VY$, $WZ$ is tangent to the incircle of $\triangle ABC$.

The corner of a unit cube is chopped off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner. What is the height of the cube when the freshly-cut face is placed on a table?

Let $ABCD$ be a cyclic quadrilateral. Let $P$ be the intersection of $BC$ and $AD$. Line $AC$ intersects the circumcircle of triangle $BDP$ in points $S$ and $T$, with $S$ between $A$ and $C$. Line $BD$ intersects the circumcircle of triangle $ACP$ in points $U$ and $V$, with $U$ between $B$ and $D$. Prove that $PS$ = $PT$ = $PU$ = $PV$.