Let $ABCDEF GH$ be a unit cube such that $ABCD$ is one face of the cube and $\overline{AE}$, $\overline{BF}$, $\overline{CG}$, and $\overline{DH}$ are all edges of the cube. Points $I, J, K$, and $L$ are the respective midpoints of $\overline{AF}$, $\overline{BG}$, $\overline{CH}$, and $\overline{DE}$. The inscribed circle of $IJKL$ is the largest cross-section of some sphere. Compute the volume of this sphere.

p1. It is known that $m$ and $n$ are two positive integer numbers consisting of four digits and three digits respectively. Both numbers contain the number $4$ and the number $5$. The number $59$ is a prime factor of $m$. The remainder of the division of $n$ by $38$ is $ 1$. If the difference between $m$ and $n$ is not more than $2015$. determine all possible pairs of numbers $(m,n)$. p2. It is known that the equation $ax^2 + bx + c = $0 with $a> 0$ has two different real roots and the equation $ac^2x^4 + 2acdx^3 + (bc + ad^2) x^2 + bdx + c = 0$ has no real roots. Is it true that $ad^2 + 2ad^2 <4bc + 16c^3$ ? p3. A basketball competition consists of $6$ teams. Each team carries a team flag that is mounted on a pole located on the edge of the match field. There are four locations and each location has five poles in a row. Pairs of flags at each location starting from the far right pole in sequence. If not all poles in each location must be flagged, determine as many possible flag arrangements. p4. It is known that two intersecting circles $L_1$ and $L_2$ have centers at $M$ and $N$ respectively. The radii of the circles $L_1$ and $L_2$ are $5$ units and $6$ units respectively. The circle $L_1$ passes through the point $N$ and intersects the circle $L_2$ at point $P$ and at point $Q$. The point $U$ lies on the circle $L_2$ so that the line segment $PU$ is a diameter of the circle $L_2$. The point $T$ lies at the extension of the line segment $PQ$ such that the area of ​​the quadrilateral $QTUN$ is $792/25$ units of area. Determine the length of the $QT$. p5. An ice ball has an initial volume $V_0$. After $n$ seconds ($n$ is natural number), the volume of the ice ball becomes $V_n$ and its surface area is $L_n$. The ice ball melts with a change in volume per second proportional to its surface area, i.e. $V_n - V_{n+1} = a L_n$, for every n, where a is a positive constant. It is also known that the ratio between the volume changes and the change of the radius per second is proportional to the area of ​​the property, that is $\frac{V_n - V_{n+1}}{R_n - R_{n+1}}= k L_n$ , where $k$ is a positive constant. If $V_1=\frac{27}{64} V_0$ and the ice ball melts totally at exactly $h$ seconds, determine the value of $h$.

Let $\omega$ denote the circumscribed circle of triangle $ABC$, $I$ be its incenter, and $K$ be any point on arc $AC$ of $\omega$ not containing $B$. Point $P$ is symmetric to $I$ with respect to point $K$. Point $T$ on arc $AC$ of $\omega$ containing point $B$ is such that $\angle KCT = \angle PCI$. Show that the bisectors of angles $AKC$ and $ATC$ meet on line $CI$. [i](Proposed by Anton Trygub)[/i]

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

Given a triangle $ABC$, let $A',B',C'$ be the perpendicular feet dropped from the centroid $G$ of the triangle $ABC$ onto the sides $BC,CA,AB$ respectively. Reflect $A',B',C'$ through $G$ to $A'',B'',C''$ respectively. Prove that the lines $AA'',BB'',CC''$ are concurrent.

Let $ABC$ be an isosceles triangle $(AC = BC)$, $O$ be its circumcenter, $H$ be the orthocenter, and $P$ be a point inside the triangle such that $\angle APH = \angle BPO = \pi /2$. Prove that $\angle PAC = \angle PBA = \angle PCB$.

Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, how many of the four quantities listed below change? $\mathrm{(A)}\ \text{the length of the segment} MN$ $\mathrm{(B)}\ \text{the perimeter of }\triangle PAB$ $\mathrm{(C)}\ \text{ the area of }\triangle PAB$ $\mathrm{(D)}\ \text{ the area of trapezoid} ABNM$ [asy] draw((2,0)--(8,0)--(6,4)--cycle); draw((4,2)--(7,2)); draw((1,4)--(9,4),Arrows); label("$A$",(2,0),SW); label("$B$",(8,0),SE); label("$M$",(4,2),W); label("$N$",(7,2),E); label("$P$",(6,4),N);[/asy] $\mathrm{(A)}\ 0 \qquad\mathrm{(B)}\ 1 \qquad\mathrm{(C)}\ 2 \qquad\mathrm{(D)}\ 3 \qquad\mathrm{(E)}\ 4$

Through a point $P$ within a triangle $ABC$ the lines $l, m$, and $n$ perpendicular respectively to $AP,BP,CP$ are drawn. Prove that if $l$ intersects the line $BC$ in $Q$, $m$ intersects $AC$ in $R$, and $n$ intersects $AB$ in $S$, then the points $Q, R$, and $S$ are collinear.

Let $S$ be the locus of all points $(x,y)$ in the first quadrant such that $\dfrac{x}{t}+\dfrac{y}{1-t}=1$ for some $t$ with $0<t<1$. Find the area of $S$.

In the triangle $ABC , \angle A = \alpha, BC = a$. The inscribed circle touches the lines $AB$ and $AC$ at points $M$ and $P$. Find the length of the chord cut by the line $MP$ in a circle with diameter $BC$.

Given is a triangle $ABC$ with circumcenter $O$. Points $D, E$ are chosen on the angle bisector of $\angle ABC$ such that $EA=EB, DB=DC$. If $P, Q$ are the circumcenters of $(AOE), (COD)$, prove that either the line $PQ$ coincides with $AC$ or $PQCA$ is cyclic.

$ABCD$ is a tetrahedron with $\angle BAC = \angle ACD$ and $\angle ABD = \angle BDC$. Show that $AB = CD$.

[u]Round 5[/u] [b]5.1[/b] A circle with a radius of $1$ is inscribed in a regular hexagon. This hexagon is inscribed in a larger circle. If the area that is outside the hexagon but inside the larger circle can be expressed as $\frac{a\pi}{b} - c\sqrt{d}$, where $a, b, c, d$ are positive integers, $a, b$ are relatively prime, and no prime perfect square divides into $d$. find the value of $a + b + c + d$. [b]5.2[/b] At a dinner party, $10$ people are to be seated at a round table. If person A cannot be seated next to person $B$ and person $C$ must be next to person $D$, how many ways can the 10 people be seated? Consider rotations of a configuration identical. [b]5.3[/b] Let $N$ be the sum of all the positive integers that are less than $2022$ and relatively prime to $1011$. Find $\frac{N}{2022}$. [u]Round 6[/u] [b]6.1[/b] The line $y = m(x - 6)$ passes through the point $ A$ $(6, 0)$, and the line $y = 8 -\frac{x}{m}$ pass through point $B$ $(0,8)$. The two lines intersect at point $C$. What is the largest possible area of triangle $ABC$? [b]6.2[/b] Let $N$ be the number of ways there are to arrange the letters of the word MATHEMATICAL such that no two As can be adjacent. Find the last $3$ digits of $\frac{N}{100}$. [b]6.3[/b] Find the number of ordered triples of integers $(a, b, c)$ such that $|a|, |b|, |c| \le 100$ and $3abc = a^3 + b^3 + c^3$. [u]Round 7[/u] [b]7.1[/b] In a given plane, let $A, B$ be points such that $AB = 6$. Let $S$ be the set of points such that for any point $C$ in $S$, the circumradius of $\vartriangle ABC$ is at most $6$. Find $a + b + c$ if the area of $S$ can be expressed as $a\pi + b\sqrt{c}$ where $a, b, c$ are positive integers, and $c$ is not divisible by the square of any prime. [b]7.2[/b] Compute $\sum_{1\le a<b<c\le 7} abc$. [b]7.3[/b] Three identical circles are centered at points $A, B$, and $C$ respectively and are drawn inside a unit circle. The circles are internally tangent to the unit circle and externally tangent to each other. A circle centered at point $D$ is externally tangent to circles $A, B$, and $C$. If a circle centered at point $E$ is externally tangent to circles $A, B$, and $D$, what is the radius of circle $E$? The radius of circle $E$ can be expressed as $\frac{a\sqrt{b}-c}{d}$ where $a, b, c$, and d are all positive integers, gcd(a, c, d) = 1, and b is not divisible by the square of any prime. What is the sum of $a + b + c + d$? [u]Round 8[/u] [b]8.[/b] Let $A$ be the number of unused Algebra problems in our problem bank. Let $B$ be the number of times the letter ’b’ appears in our problem bank. Let M be the median speed round score. Finally, let $C$ be the number of correct answers to Speed Round $1$. Estimate $$A \cdot B + M \cdot C.$$ Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2826128p24988676]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A regular triangle $ABC$ is given. Points $K$ and $N$ lie in the segment $AB$, a point $L$ lies in the segment $AC$, and a point $M$ lies in the segment $BC$ so that $CL=AK$, $CM=BN$, $ML=KN$. Prove that $KL \parallel MN$.

On each side of triangle $ABC$, two distinct points are marked. It is known that these points are the feet of the altitudes and of the bisectors. a) Using only a ruler determine which points are the feet of the altitudes and which points are the feet of the bisectors. b) Solve p.a) drawing only three lines.

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 $XY$).

There exists a certain right triangle with the smallest area in the $2$D coordinate plane such that all of its vertices have integer coordinates but none of its sides are parallel to the $x$- or $y$-axis. Additionally, all of its sides have distinct, integer lengths. What is the area of this triangle?

Let $ABC$ be a triangle. A point $D$ lies on line $BC$ and points $E,F$ are taken on $AC,AB$ such that $DE \parallel AB$ and $DF\parallel AC$. Let $G = (AEF) \cap (ABC) \neq A$ and $I = (DEF) \cap BC\neq D$. Let $H$ and $O$ denote the orthocenter and the circumcenter of triangle $DEF$. Prove that $A,O,I$ are collinear if and only if $G,H,I$ are collinear. [i]Proposed by Kaan Bilge[/i]

A square is divided into several (greater than one) convex polygons with mutually different numbers of sides. Prove that one of these polygons is a triangle. (A.Zaslavsky)

Suppose $L_1$ is a circle with center $O$, and $L_2$ is a circle with center $O'$. The circles intersect at $ A$ and $ B$ such that $\angle OAO' = 90^o$. Suppose that point $X$ lies on the circumcircle of triangle $OAB$, but lies inside $L_2$. Let the extension of $OX$ intersect $L_1$ at $Y$ and $Z$. Let the extension of $O'X$ intersect $L_2$ at $W$ and $V$ . Prove that $\vartriangle XWZ$ is congruent with $\vartriangle XYV$.

A convex pentagon $APBCQ$ is given such that $AB < AC$. The circle $\omega$ centered at point $A{}$ passes through $P{}$ and $Q{}$ and touches the segment $BC$ at point $R{}$. Let the circle $\Gamma$ centered at the point $O{}$ be the circumcircle of the triangle $ABC$. It is known that $AO \perp P Q$ and $\angle BQR = \angle CP R$. Prove that the tangents at points $P{}$ and $Q{}$ to the circle $\omega$ intersect on $\Gamma$.

Let $ABCD$ be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at $E$. The midpoints of $AB$ and $CD$ are $F$ and $G$ respectively, and $\ell$ is the line through $G$ parallel to $AB$. The feet of the perpendiculars from E onto the lines $\ell$ and $CD$ are $H$ and $K$, respectively. Prove that the lines $EF$ and $HK$ are perpendicular.

Let $ABC$ be an equilateral triangle, and let $P$ stand for an arbitrary point inside the triangle. Is it true that $| \angle PAB - \angle PAC| \ge | \angle PBC - \angle PCB|$ ?

Let $P$ be an interior point of the triangle $ABC$ which is not on the median belonging to $BC$ and satisfying $\angle CAP = \angle BCP. \: BP \cap CA = \{B'\} \: , \: CP \cap AB = \{C'\}$ and $Q$ is the second point of intersection of $AP$ and the circumcircle of $ABC. \: B'Q$ intersects $CC'$ at $R$ and $B'Q$ intersects the line through $P$ parallel to $AC$ at $S.$ Let $T$ be the point of intersection of lines $B'C'$ and $QB$ and $T$ be on the other side of $AB$ with respect to $C.$ Prove that \[\angle BAT = \angle BB'Q \: \Longleftrightarrow \: |SQ|=|RB'| \]

Let a triangle $ABC$ be given. For any point $X$ of the triangle denote $m(X)=\min\{XA,XB,XC\}.$ Find all points $X$ (of triangle $ABC$) such that $m(X)$ is maximal.