The incircle of right-angled triangle $ABC$ ($\angle B = 90^o$) touches $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. Points $A_2, C_2$ are the reflections of $B_1$ in lines $BC, AB$ respectively. Prove that lines $A_1A_2$ and $C_1C_2$ meet on the median of triangle $ABC$.

In regular hexagon $ABCDEF$ with side length $2$, let $P$, $Q$, $R$, and $S$ be the feet of the altitudes from $A$ to $BC$, $EF$, $CF$, and $BE$, respectively. Find the area of quadrilateral $PQRS$.

Let $H$ be the orthocenter of an acute-angled triangle $ABC$ and $P$ a point on the circumcenter of triangle $ABC$. Prove that the Simson line of $P$ bisects the segment $[P H]$.

Four perpendiculars are drawn from the vertices of a convex quadrangle to its diagonals. Prove that their bases make a quadrangle similar to the given one.

In a convex quadrilateral $ABCD$ ,$\angle ABC$ and $\angle BCD$ are $\geq 120^o$. Prove that $|AC|$ + $|BD| \geq |AB|+|BC|+|CD|$. (With $|XY|$ we understand the length of the segment $XY$).

Find all functions $\displaystyle f : \mathbb N^\ast \to \mathbb N^\ast$ ($\displaystyle N^\ast = \{ 1,2,3,\ldots \}$) with the property that, for all $\displaystyle n \geq 1$, \[ f(1) + f(2) + \ldots + f(n) \] is a perfect cube $\leq n^3$. [i]Dinu Teodorescu[/i]

Let $A_1, A_2, ..., A_m$ and $B_2 , B_3,..., B_n$ be the points on a circle such that $A_1A_2... A_n$ is a regular $m$-gon and $A_1B_2...B_n$ is a regular $n$-gon whereby $n > m$ and the point $B_2$ lies between $A_1$ and $A_2$. Find $\angle B_2A_1A_2$.

Let $ABCD$ be a quadrilateral with parallel sides $AD$ and $BC$, $M$ and $N$ are the midpoints of its sides $AB$ and $CD$, respectively. The straight line $MN$ bisects the segment connecting the centers of the circumcircles of triangles $ABC$ and $ADC$. Prove that $ABCD$ is a parallelogram.

A circle $\omega$ and a point $P$ not lying on it are given. Let $ABC$ be an arbitrary equilateral triangle inscribed into $\omega$ and $A', B', C'$ be the projections of $P$ to $BC, CA, AB$. Find the locus of centroids of triangles $A' B'C'$.

Colorado and Wyoming are both defined to be $4$ degrees tall in latitude and $7$ degree wide in longitude. In particular, Colorado is defined to be at $37^o N$ to $41^o N$, and $102^o03' W$ to $109^o03' W$, whereas Wyoming is defined to be $41^o N$ to $45^o N$, and $104^o 03' W$ to $111^o 03' W$. Assuming Earth is a perfect sphere with radius $R$, what is the ratio of the areas of Wyoming to Colorado, in terms of $R$?

Let $ABCD$ be a square. Let $E, Z$ be points on the sides $AB, CD$ of the square respectively, such that $DE\parallel BZ$. Assume that the triangles $\triangle EAD, \triangle ZCB$ and the parallelogram $BEDZ$ have the same area. If the distance between the parallel lines $DE$ and $BZ$ is equal to $1$, determine the area of the square.

Let $AX$, $AY$ be tangents to circle $\omega$ from point $A$. Le $B$, $C$ be points inside $AX$ and $AY$ respectively, such that perimeter of $\triangle ABC$ is equal to length of $AX$. $D$ is reflection of $A$ over $BC$. Prove that circumcircle $\triangle BDC$ and $\omega$ are tangent to each other.

Let $ABC$ be a triangle such that $\angle{ABC} = 2\angle{BCA}$ and $\angle{CAB}>90^\circ$. Let $M$ be the midpoint of $BC$. The line perpendicular to $AC$ that passes through $C$ cuts the line $AB$ at point $D$. Show that $\angle{AMB} = \angle{DMC}$.

Consider $ \rho$ a semicircle of diameter $ AB$. A parallel to $ AB$ cuts the semicircle at $ C, D$ such that $ AD$ separates $ B, C$. The parallel at $ AD$ through $ C$ intersects the semicircle the second time at $ E$. Let $ F$ be the intersection point of the lines $ BE$ and $ CD$. The parallel through $ F$ at $ AD$ cuts $ AB$ in $ P$. Prove that $ PC$ is tangent to $ \rho$. [i]Author: Cosmin Pohoata[/i]

$A,B,C,D,E$ are points on a circle $\omega$, satisfying $AB=BD$, $BC=CE$. $AC$ meets $BE$ at $P$. $Q$ is on $DE$ such that $BE//AQ$. Suppose $\odot(APQ)$ intersects $\omega$ again at $T$. $A'$ is the reflection of $A$ wrt $BC$. Prove that $A'BPT$ lies on the same circle.

The incentre of the triangle $ABC$ is $I.$ The lines $AI, BI$ and $CI$ meet the circumcircle of the triangle $ABC$ also at points $D, E$ and $F,$ respectively. Prove that $AD$ and $EF$ are perpendicular.

The consecutive angles of a trapezoid form an arithmetic sequence. If the smallest angle is $75^\circ$, then the largest angle is $\textbf{(A) }95^\circ\qquad \textbf{(B) }100^\circ\qquad \textbf{(C) }105^\circ\qquad \textbf{(D) }110^\circ\qquad \textbf{(E) }115^\circ$

[u]Round 6 [/u] [b]p16.[/b] Triangle $ABC$ with $AB < AC$ is inscribed in a circle. Point $D$ lies on the circle and point $E$ lies on side $AC$ such that $ABDE$ is a rhombus. Given that $CD = 4$ and $CE = 3$, compute $AD^2$. [b]p17.[/b] Wam and Sang are walking on the coordinate plane. Both start at the origin. Sang walks to the right at a constant rate of $1$ m/s. At any positive time $t$ (in seconds),Wam walks with a speed of $1$ m/s with a direction of $t$ radians clockwise of the positive $x$-axis. Evaluate the square of the distance betweenWamand Sang in meters after exactly $5\pi$ seconds. [b]p18.[/b] Mawile is playing a game against Salamance. Every turn,Mawile chooses one of two moves: Sucker Punch or IronHead, and Salamance chooses one of two moves: Dragon Dance or Earthquake. Mawile wins if the moves used are Sucker Punch and Earthquake, or Iron Head and Dragon Dance. Salamance wins if the moves used are Iron Head and Earthquake. If the moves used are Sucker Punch and Dragon Dance, nothing happens and a new turn begins. Mawile can only use Sucker Punch up to $8$ times. All other moves can be used indefinitely. Assuming bothMawile and Salamance play optimally, find the probability thatMawile wins. [u]Round 7 [/u] [b]p19.[/b] Ephram is attempting to organize what rounds everyone is doing for the NEAML competition. There are $4$ rounds, of which everyone must attend exactly $2$. Additionally, of the 6 people on the team(Ephram,Wam, Billiam, Hacooba,Matata, and Derke), exactly $3$ must attend every round. In how many different ways can Ephram organize the teams like this? [b]p20.[/b] For some $4$th degree polynomial $f (x)$, the following is true: $\bullet$ $f (-1) = 1$. $\bullet$ $f (0) = 2$. $\bullet$ $f (1) = 4$. $\bullet$ $f (-2) = f (2) = f (3)$. Find $f (4)$. [b]p21.[/b] Find the minimum value of the expression $\sqrt{5x^2-16x +16}+\sqrt{5x^2-18x +29}$ over all real $x$. [u]Round 8 [/u] [b]p22.[/b] Let $O$ and $I$ be the circumcenter and incenter, respectively, of $\vartriangle ABC$ with $AB = 15$, $BC = 17$, and $C A = 16$. Let $X \ne A$ be the intersection of line $AI$ and the circumcircle of $\vartriangle ABC$. Find the area of $\vartriangle IOX$. [b]p23.[/b] Find the sum of all integers $x$ such that there exist integers $y$ and $z$ such that $$x^2 + y^2 = 3(2016^z )+77.$$ [b]p24.[/b] Evaluate $$ \left \lfloor \sum^{2022}_{i=1} \frac{1}{\sqrt{i}} \right \rfloor = \left \lfloor \frac{1}{\sqrt{1}} +\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+ \frac{1}{\sqrt{2022}}\right \rfloor$$ [u]Round 9[/u] [b]p25.[/b]Either: 1. Submit $-2$ as your answer and you’ll be rewarded with two points OR 2. Estimate the number of teams that choose the first option. If your answer is within $1$ of the correct answer, you’ll be rewarded with three points, and if you are correct, you’ll receive ten points. [b]p26.[/b] Jeff is playing a turn-based game that starts with a positive integer $n$. Each turn, if the current number is $n$, Jeff must choose one of the following: 1. The number becomes the nearest perfect square to $n$ 2. The number becomes $n-a$, where $a$ is the largest digit in $n$ Let $S(k)$ be the least number of turns Jeff needs to get from the starting number $k$ to $0$. Estimate $$\sum^{2023}_{k=1}S(k).$$ If your estimation is $E$ and the actual answer is $A$, you will receive $\max \left( \left \lfloor 10 - \left| \frac{E-A}{6000} \right| \right \rfloor , 0 \right)$ points. [b]p27.[/b] Estimate the smallest positive integer n such that if $N$ is the area of the $n$-sided regular polygon with circumradius $100$, $10000\pi -N < 1$ is true. If your estimation is $E$ and the actual answer is $A$, you will receive $ \max \left \lfloor \left( 10 - \left| 10 \cdot \log_3 \left( \frac{A}{E}\right) \right|\right| ,0\right \rfloor.$ points. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167360p28825713]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a triangle $ABC$ , $D$ is a point on $BC$ such that $AD$ is the internal bisector of $\angle A$ . Now Suppose $\angle B$=$2\angle C$ and $CD=AB$ . Prove that $\angle A=72^0$.

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

The diagonals of a convex quadrilateral $ABCD$ are perpendicular to each other and intersect at the point $O$. The sum of the inradii of triangles $AOB$ and $COD$ is equal to the sum of the inradii of triangles $BOC$ and $DOA$. $(a)$ Prove that $ABCD$ has an incircle. $(b)$ Prove that $ABCD$ is symmetric about one of its diagonals.

Let $ABC$ be an acute triangle with incenter $I$ and circumcircle $\omega$. Suppose a circle $\omega_B$ is tangent to $BA,BC$, and internally tangent to $\omega$ at $B_1$, while a circle $\omega_C$ is tangent to $CA, CB$, and internally tangent to $\omega$ at $C_1$. If $B_2, C_2$ are the points opposite to $B,C$ on $\omega$, respectively, and $X$ denotes the intersection of $B_1C_2, B_2C_1$, prove that $XA=XI$. [i]Proposed by Vincent Huang and Nathan Weckwerth

Prove that we can draw a line (by a ruler and a compass) from a vertice of a convex quadrilateral such that, the line divides the quadrilateral to two equal areas.

Let $ABC$ be an isosceles triangle, $AB=AC$ inscribed in a circle $\omega$. The $B$-symmedian intersects $\omega$ again at $D$. The circle through $C,D$ and tangent to $BC$ and the circle through $A,D$ and tangent to $CD$ intersect at points $D,X$. The incenter of $ABC$ is denoted $I$. Prove that $B,C,I,X$ are concyclic.

A line segment of length 1 is given on the plane. Show that a line segment of length $\sqrt{2010}$ can be constructed using only a straightedge and a compass.