$n$ points are given on the surface of a sphere. Show that the surface can be divided into $n$ congruent regions such that each of them contains exactly one of the given points.

For three points $A,B,C$ in the plane, we define $m(ABC)$ to be the smallest length of the three heights of the triangle $ABC$, where in the case $A$, $B$, $C$ are collinear, we set $m(ABC) = 0$. Let $A$, $B$, $C$ be given points in the plane. Prove that for any point $X$ in the plane, \[ m(ABC) \leq m(ABX) + m(AXC) + m(XBC). \]

[u]Round 1[/u] [b]1.1[/b] If the sum of two non-zero integers is $28$, then find the largest possible ratio of these integers. [b]1.2[/b] If Tom rolls a eight-sided die where the numbers $1$ − $8$ are all on a side, let $\frac{m}{n}$ be the probability that the number is a factor of $16$ where $m, n$ are relatively prime positive integers. Find $m + n$. [b]1.3[/b] The average score of $35$ second graders on an IQ test was $180$ while the average score of $70$ adults was $90$. What was the total average IQ score of the adults and kids combined? [u]Round 2[/u] [b]2.1[/b] So far this year, Bob has gotten a $95$ and a 98 in Term $1$ and Term $2$. How many different pairs of Term $3$ and Term $4$ grades can Bob get such that he finishes with an average of $97$ for the whole year? Bob can only get integer grades between $0$ and $100$, inclusive. [b]2.2[/b] If a complement of an angle $M$ is one-third the measure of its supplement, then what would be the measure (in degrees) of the third angle of an isosceles triangle in which two of its angles were equal to the measure of angle $M$? [b]2.3[/b] The distinct symbols $\heartsuit, \diamondsuit, \clubsuit$ and $\spadesuit$ each correlate to one of $+, -, \times , \div$, not necessarily in that given order. Given that $$((((72 \,\, \,\, \diamondsuit \,\, \,\,36) \,\, \,\,\spadesuit \,\, \,\,0 ) \,\, \,\, \diamondsuit \,\, \,\, 32) \,\, \,\, \clubsuit \,\, \,\, 3)\,\, \,\, \heartsuit \,\, \,\, 2 = \,\, \,\, 6,$$ what is the value of $$(((((64 \,\, \,\, \spadesuit \,\, \,\, 8) \heartsuit \,\, \,\, 6) \,\, \,\, \spadesuit \,\, \,\, 5) \,\, \,\, \heartsuit \,\, \,\, 1) \,\, \,\, \clubsuit \,\, \,\, 7) \,\, \,\, \diamondsuit \,\, \,\, 1?$$ [u]Round 3[/u] [b]3.1[/b] How many ways can $5$ bunnies be chosen from $7$ male bunnies and $9$ female bunnies if a majority of female bunnies is required? All bunnies are distinct from each other. [b]3.2[/b] If the product of the LCM and GCD of two positive integers is $2021$, what is the product of the two positive integers? [b]3.3[/b] The month of April in ABMC-land is $50$ days long. In this month, on $44\%$ of the days it rained, and on $28\%$ of the days it was sunny. On half of the days it was sunny, it rained as well. The rest of the days were cloudy. How many days were cloudy in April in ABMC-land? [u]Round 4[/u] [b]4.1[/b] In how many ways can $4$ distinct dice be rolled such that a sum of $10$ is produced? [b]4.2[/b] If $p, q, r$ are positive integers such that $p^3\sqrt{q}r^2 = 50$, find the sum of all possible values of $pqr$. [b]4.3[/b] Given that numbers $a, b, c$ satisfy $a + b + c = 0$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}= 10$, and $ab + bc + ac \ne 0$, compute the value of $\frac{-a^2 - b^2 - a^2}{ab + bc + ac}$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2826137p24988781]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The midpoint of triangle's side and the base of the altitude to this side are symmetric wrt the touching point of this side with the incircle. Prove that this side equals one third of triangle's perimeter. (A.Blinkov, Y.Blinkov)

In an acute-angled triangle $ABC$, in which $AB<AC$, the point $M$ is the midpoint of the side $BC, K$ is the midpoint of the broken line segment $BAC$ . Prove that $\sqrt2 KM > AB$. (George Naumenko)

In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE = 6,$ and $AF = 2$. Which of the following is closest to the area of the rectangle $ABCD$? [asy] size(140); pair A, B, C, D, E, F, X, Y; real length = 12.5; real width = 10; A = origin; B = (length, 0); C = (length, width); D = (0, width); X = rotate(330, C)*B; E = extension(C, X, A, B); Y = rotate(30, C)*D; F = extension(C, Y, A, D); draw(E--C--F); label("$2$", A--F, dir(180)); label("$6$", E--B, dir(270)); draw(A--B--C--D--cycle); dot(A);dot(B);dot(C);dot(D);dot(E);dot(F); label("$A$", A, dir(225)); label("$B$", B, dir(315)); label("$C$", C, dir(45)); label("$D$", D, dir(135)); label("$E$", E, dir(270)); label("$F$", F, dir(180)); [/asy] $\textbf{(A)} \ 110 \qquad \textbf{(B)} \ 120 \qquad \textbf{(C)} \ 130 \qquad \textbf{(D)} \ 140 \qquad \textbf{(E)} \ 150$

Prove that it is impossible to divide a scalene triangle into two equal triangles.

Incircle $\omega$ of triangle $ABC$ is tangent to sides $CB$ and $CA$ at $D$ and $E$, respectively. Point $X$ is the reflection of $D$ with respect to $B$. Suppose that the line $DE$ is tangent to the $A$-excircle at $Z$. Let the circumcircle of triangle $XZE$ intersect $\omega$ for the second time at $K$. Prove that the intersection of $BK$ and $AZ$ lies on $\omega$. Proposed by Mahdi Etesamifard and Alireza Dadgarnia

$BL $ is the bisector of an isosceles triangle $ABC $. A point $D $ is chosen on the Base $BC $ and a point $E $ is chosen on the lateral side $AB $ so that $AE=\frac {1}{2}AL=CD $. Prove that $LE=LD $. Tuymaada 2017 Q5 Juniors

$(SWE 1)$ Six points $P_1, . . . , P_6$ are given in $3-$dimensional space such that no four of them lie in the same plane. Each of the line segments $P_jP_k$ is colored black or white. Prove that there exists one triangle $P_jP_kP_l$ whose edges are of the same color.

Given triangle $\triangle ABC $ with $AC\neq BC $,and let $D $ be a point inside triangle such that $\measuredangle ADB=90^{\circ} + \frac {1}{2}\measuredangle ACB $.Tangents from $C $ to the circumcircles of $\triangle ABC $ and $\triangle ADC $ intersect $AB $ and $AD $ at $P $ and $Q $ , respectively.Prove that $PQ $ bisects the angle $\measuredangle BPC $.

The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to $ \textbf{(A)} \ 50 \qquad \textbf{(B)} \ 52 \qquad \textbf{(C)} \ 54 \qquad \textbf{(D)} \ 56 \qquad \textbf{(E)} \ 58 \qquad$ [asy]unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; for(int i=0; i<3; ++i) { for(int j=0; j<3; ++j) { draw(shift(3*i,3*j)*p); } }[/asy]

Let $ABC$ be an acute triangle, and let $M$ and $N$ be two points on the line $AC$ such that the vectors $MN$ and $AC$ are identical. Let $X$ be the orthogonal projection of $M$ on $BC$, and let $Y$ be the orthogonal projection of $N$ on $AB$. Finally, let $H$ be the orthocenter of triangle $ABC$. Show that the points $B$, $X$, $H$, $Y$ lie on one circle.

Triangle $DEF$ is acute. Circle $c_1$ is drawn with $DF$ as its diameter and circle $c_2$ is drawn with $DE$ as its diameter. Points $Y$ and $Z$ are on $DF$ and $DE$ respectively so that $EY$ and $FZ$ are altitudes of triangle $DEF$ . $EY$ intersects $c_1$ at $P$, and $FZ$ intersects $c_2$ at $Q$. $EY$ extended intersects $c_1$ at $R$, and $FZ$ extended intersects $c_2$ at $S$. Prove that $P$, $Q$, $R$, and $S$ are concyclic points.

Kati cut two equal regular $ n\minus{}gons$ out of paper. To the vertices of both $ n\minus{}gons$, she wrote the numbers 1 to $ n$ in some order. Then she stabbed a needle through the centres of these $ n\minus{}gons$ so that they could be rotated with respect to each other. Kati noticed that there is a position where the numbers at each pair of aligned vertices are different. Prove that the $ n\minus{}gons$ can be rotated to a position where at least two pairs of aligned vertices contain equal numbers.

Determine the poles of the inversions that transform four collienar points $A,B, C, D$, aligned in this order, at four points $A' $, $B' $, $C'$ , $D'$ that are vertices of a rectangle, and such that $A'$ and $C'$ are opposite vertices.

The angles subtended by a tower at distances $100$, $200$ and $300$ from its foot sum to $90^o$. What is its height?

Let $ABC$ to be an acute triangle. Also, let $K$ and $L$ to be the two intersections of the perpendicular from $B$ with respect to side $AC$ with the circle of diameter $AC$, with $K$ closer to $B$ than $L$. Analogously, $X$ and $Y$ are the two intersections of the perpendicular from $C$ with respect to side $AB$ with the circle of diamter $AB$, with $X$ closer to $C$ than $Y$. Prove that the intersection of $XL$ and $KY$ lies on $BC$.

Let $P$ be a point on the circumcircle of acute triangle $ABC$. Let $D,E,F$ be the reflections of $P$ in the $A$-midline, $B$-midline, and $C$-midline. Let $\omega$ be the circumcircle of the triangle formed by the perpendicular bisectors of $AD, BE, CF$. Show that the circumcircles of $\triangle ADP, \triangle BEP, \triangle CFP,$ and $\omega$ share a common point.

$ABC$ is a triangle. $A' , B' , C'$ are points on the segments $BC, CA, AB$ respectively. $\angle B' A' C' = \angle A$ , $\frac{AC'}{C'B} = \frac{BA' }{A' C} = \frac{CB'}{B'A}$. Show that $ABC$ and $A'B'C'$ are similar.

Clyde is making a Pacman sticker to put on his laptop. A Pacman sticker is a circular sticker of radius $3$ inches with a sector of $120^o$ cut out. What is the perimeter of the Pacman sticker in inches?

Let $ABC$ be a triangle, and let $a$, $b$, and $c$ be the lengths of the sides opposite vertices $A$, $B$, and $C$, respectively. Let $R$ be its circumradius and $r$ its inradius. Suppose that $b + c = 2a$ and $R = 3r$. The excircle relative to vertex $A$ intersects the circumcircle of $ABC$ at points $P$ and $Q$. Let $U$ be the midpoint of side $BC$, and let $I$ be the incenter of $ABC$. Prove that $U$ is the centroid of triangle $QIP$.

In a plane the circles $\mathcal K_1$ and $\mathcal K_2$ with centers $I_1$ and $I_2$, respectively, intersect in two points $A$ and $B$. Assume that $\angle I_1AI_2$ is obtuse. The tangent to $\mathcal K_1$ in $A$ intersects $\mathcal K_2$ again in $C$ and the tangent to $\mathcal K_2$ in $A$ intersects $\mathcal K_1$ again in $D$. Let $\mathcal K_3$ be the circumcircle of the triangle $BCD$. Let $E$ be the midpoint of that arc $CD$ of $\mathcal K_3$ that contains $B$. The lines $AC$ and $AD$ intersect $\mathcal K_3$ again in $K$ and $L$, respectively. Prove that the line $AE$ is perpendicular to $KL$.

Three equilateral triangles $ABC, CDE, EHK$ (the vertices are mentioned counterclockwise) are lying in the plane so, that the vectors $\overrightarrow{AD}$ and $\overrightarrow{DK}$ are equal. Prove that the triangle $BHD$ is also equilateral

Let $L$ be a circle with center $O$ and tangent to sides $AB$ and $AC$ of a triangle $ABC$ in points $E$ and $F$, respectively. Let the perpendicular from $O$ to $BC$ meet $EF$ at $D$. Prove that $A,D$ and $M$ are collinear, where $M$ is the midpoint of $BC$.