Given isosceles tetrahedron $PABC$ (faces are equal triangles). Let $A_0$, $B_0$ and $C_0$ be the touchpoints of the circle inscribed in the triangle $ABC$ with sides $BC$, $AC$ and $AB$ respectively, $A_1$, $B_1$ and $C_1$ are the touchpoints of the excircles of triangles $PCA$, $PAB$ and $PBC$ with extensions of sides $PA$, $PB$ and $PC$, respectively (beyond points $A$, $B$, $C$). Prove that the lines $A_0A_1$, $B_0B_1$ and $C_0C_1$ intersect at one point.

Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$. Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.

Let $G$ be the centroid of triangle $ABC$. Find the biggest $\alpha$ such that there exists a triangle for which there are at least three angles among $\angle GAB, \angle GAC, \angle GBA, \angle GBC, \angle GCA, \angle GCB$ which are $\geq \alpha$.

In the triangle $ABC, I$ is the center of the inscribed circle, point $M$ lies on the side of $BC$, with $\angle BIM = 90^o$. Prove that the distance from point $M$ to line $AB$ is equal to the diameter of the circle inscribed in triangle $ABC$

Triangle $ABC$ has sides $AB = 14$, $BC = 13$, and $CA = 15$. It is inscribed in circle $\Gamma$, which has center $O$. Let $M$ be the midpoint of $AB$, let $B'$ be the point on $\Gamma$ diametrically opposite $B$, and let $X$ be the intersection of $AO$ and $MB'$. Find the length of $AX$.

[b]p1.[/b] At a certain point in time, $20\%$ of seniors, $30\%$ of juniors, and $50\%$ of sophomores at a school had a cold. If the number of sick students was the same for each grade, the fraction of sick students across all three grades can be written as $\frac{a}{b}$ , where a and b are relatively prime positive integers. Find $a + b$. [b]p2.[/b] The average score on Mr. Feng’s recent test is a $63$ out of $100$. After two students drop out of the class, the average score of the remaining students on that test is now a $72$. What is the maximum number of students that could initially have been in Mr. Feng’s class? (All of the scores on the test are integers between $0$ and $100$, inclusive.) [b]p3.[/b] Madeline is climbing Celeste Mountain. She starts at $(0, 0)$ on the coordinate plane and wants to reach the summit at $(7, 4)$. Every hour, she moves either $1$ unit up or $1$ unit to the right. A strawberry is located at each of $(1, 1)$ and $(4, 3)$. How many paths can Madeline take so that she encounters exactly one strawberry? [b]p4.[/b] Let $E$ be a point on side $AD$ of rectangle $ABCD$. Given that $AB = 3$, $AE = 4$, and $\angle BEC = \angle CED$, the length of segment $CE$ can be written as $\sqrt{a}$ for some positive integer $a$. Find $a$. [b]p5.[/b] Lucy has some spare change. If she were to convert it into quarters and pennies, the minimum number of coins she would need is $66$. If she were to convert it into dimes and pennies, the minimum number of coins she would need is $147$. How much money, in cents, does Lucy have? [b]p6.[/b] For how many positive integers $x$ does there exist a triangle with altitudes of length $20$, $22$, and $x$? [b]p7.[/b] Compute the number of positive integers $x$ for which $\frac{x^{20}}{x+22}$ is an integer. [b]p8.[/b] Vincent the Bug is crawling along an octagonal prism. He starts on a fixed vertex $A$, visits all other vertices exactly once by traveling along the edges, and returns to $A$. Find the number of paths Vincent could have taken. [b]p9.[/b] Point $U$ is chosen inside square $ALEX$ so that $\angle AUL = 90^o$. Given that $UL = 56$ and $UE = 65$, what is the sum of all possible values for the area of square $ALEX$? [b]p10.[/b] Miranda has prepared $8$ outfits, no two of which are the same quality. She asks her intern Andrea to order these outfits for the new runway show. Andrea first randomly orders the outfits in a list. She then starts removing outfits according to the following method: she chooses a random outfit which is both immediately preceded and immediately succeeded by a better outfit and then removes it. Andrea repeats this process until there are no outfits that can be removed. Given that the expected number of outfits in the final routine can be written as $\frac{a}{b}$ for some relatively prime positive integers $a$ and $b$, find $a + b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose, medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that: (a) The medians of the triangle correspond to the sides of a right-angled triangle. (b) If $a,b,c$ are the side-lengths of the triangle, then, the following inequality holds:\[5(a^2+b^2-c^2)\geq 8ab\]

Consider the parallelogram $ABCD$ with obtuse angle $A$. Let $H$ be the feet of perpendicular from $A$ to the side $BC$. The median from $C$ in triangle $ABC$ meets the circumcircle of triangle $ABC$ at the point $K$. Prove that points $K,H,C,D$ lie on the same circle.

We consider dissections of regular $n$-gons into $n - 2$ triangles by $n - 3$ diagonals which do not intersect inside the $n$-gon. A [i]bicoloured triangulation[/i] is such a dissection of an $n$-gon in which each triangle is coloured black or white and any two triangles which share an edge have different colours. We call a positive integer $n \ge 4$ [i]triangulable[/i] if every regular $n$-gon has a bicoloured triangulation such that for each vertex $A$ of the $n$-gon the number of black triangles of which $A$ is a vertex is greater than the number of white triangles of which $A$ is a vertex. Find all triangulable numbers.

Given is an acute angled triangle $ABC$ with orthocenter $H$ and circumcircle $k$. Let $\omega$ be the circle with diameter $AH$ and $P$ be the point of intersection of $\omega$ and $k$ other than $A$. Assume that $BP$ and $CP$ intersect $\omega$ for the second time at points $Q$ and $R$, respectively. If $D$ is the foot of the altitude from $A$ to $BC$ and $S$ is the point of the intersection of $\omega$ and $QD$, prove that $HR = HS$.

Inside the acute-angled triangle $ABC$, mark the point $O$ so that $\angle AOB=90^o$, a point $M$ on the side $BC$ such that $\angle COM=90^o$, and a point $N$ on the segment $BO$ such that $\angle OMN = 90^o$. Let $P$ be the point of intersection of the lines $AM$ and $CN$, and let $Q$ be a point on the side $AB$ that such $\angle POQ = 90^o$. Prove that the lines $AN, CO$ and $MQ$ intersect at one point.

Square $ABCD$ is shown in the diagram below. Points $E$, $F$, and $G$ are on sides $\overline{AB}$, $\overline{BC}$ and $\overline{DA}$, respectively, so that lengths $\overline{BE}$, $\overline{BF}$, and $\overline{DG}$ are equal. Points $H$ and $I$ are the midpoints of segments $\overline{EF}$ and $\overline{CG}$, respectively. Segment $\overline{GJ}$ is the perpendicular bisector of segment $\overline{HI}$. The ratio of the areas of pentagon $AEHJG$ and quadrilateral $CIHF$ can be written as $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [asy] draw((0,0)--(50,0)--(50,50)--(0,50)--cycle); label("$A$",(0,50),NW); label("$B$",(50,50),NE); label("$C$",(50,0),SE); label("$D$",(0,0),SW); label("$E$",(0,100/3-1),W); label("$F$",(100/3-1,0),S); label("$G$",(20,50),N); label("$H$",((100/3-1)/2,(100/3-1)/2),SW); label("$I$",(35,25),NE); label("$J$",(((100/3-1)/2+35)/2,((100/3-1)/2+25)/2),S); draw((0,100/3-1)--(100/3-1,0)); draw((20,50)--(50,0)); draw((100/6-1/2,100/6-1/2)--(35,25)); draw((((100/3-1)/2+35)/2,((100/3-1)/2+25)/2)--(20,50)); [/asy]

In a convex hexagon $ABCDEF$ triangles $ABC , CDE , EFA$ are similar. Find conditions on these triangles under which triangle $ACE$ is equilateral if and only if so is $BDF.$

Assume there are $ n\ge3$ points in the plane, Prove that there exist three points $ A,B,C$ satisfying $ 1\le\frac{AB}{AC}\le\frac{n\plus{}1}{n\minus{}1}.$

We are given a triangle $ABC$. Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$, with the arrangment of points $D - A - B - E$. The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$, and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$. Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$.

Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.

Let $ABC$ be a non-isosceles,aqute triangle with $AB<AC$ inscribed in circle $c(O,R)$.The circle $c_{1}(B,AB)$ crosses $AC$ at $K$ and $c$ at $E$. $KE$ crosses $c$ at $F$ and $BO$ crosses $KE$ at $L$ and $AC$ at $M$ while $AE$ crosses $BF$ at $D$.Prove that: i)$D,L,M,F$ are concyclic. ii)$B,D,K,M,E$ are concyclic.

On side $AC$ of triangle $ABC$ an arbitrary point is selected $D$. The tangent in $D$ to the circumcircle of triangle $BDC$ meets $AB$ in point $C_{1}$; point $A_{1}$ is defined similarly. Prove that $A_{1}C_{1}\parallel AC$.

Given two acute-angled triangles $ABC$ and $A'B'C'$ with the points $O$ and $O'$ inside. Three pairs of the perpendiculars are drawn: $[OA_1]$ to the side $[BC]$, $[O'A'_1]$ to the side $[B'C']$, $[OB_1]$ to the side $[AC]$, $[O'B'_1]$ to the side $[A'C']$, $[OC_1] $ to the side $[AB]$, $[O'C'_1]$ to the side $[A'B']$; It is known that $$[OA_1] \parallel [O'A'], [OB_1] \parallel [O'B'], [OC_1] \parallel [O'C'] $$ and $$|OA_1|\cdot|O'A'| = |OB_1|\cdot |O'B'| = |OC_1|\cdot |O'C'|$$ Prove that $$[O'A'_1] \parallel [OA], [O'B'_1] \parallel[OB], [O'C'_1] \parallel[OC]$$ and $$|O'A'_1|\cdot|OA| = |O'B'_1|\cdot|OB| = |O'C'_1|\cdot|OC|$$

Let $ABC$ and $DEF$ be acute-angled triangles. Write $d = EF, e = FD, f = DE.$ Show that there exists a point $P$ in the interior of $ABC$ for which the value of the expression $X=d \cdot AP +e \cdot BP +f \cdot CP$ attains a minimum.

p1. Given $f(x)=\frac{1+x}{1-x}$ , for $x \ne 1$ . Defined $p @ q = \frac{p+q}{1+pq}$ for all positive rational numbers $p$ and $q$. Note the sequence with $a_1,a_2,a_3,...$ with $a_1=2 @3$, $a_{n}=a_{n-1}@ (n+2)$ for $n \ge 2$. Determine $f(a_{233})$ and $a_{233}$ p2. It is known that $ a$ and $ b$ are positive integers with $a > b > 2$. Is $\frac{2^a+1}{2^b-1}$ an integer? Write down your reasons. p3. Given a cube $ABCD.EFGH$ with side length $ 1$ dm. There is a square $PQRS$ on the diagonal plane $ABGH$ with points $P$ on $HG$ and $Q$ on $AH$ as shown in the figure below. Point $T$ is the center point of the square $PQRS$. The line $HT$ is extended so that it intersects the diagonal line $BG$ at $N$. Point $M$ is the projection of $N$ on $BC$. Determine the volume of the truncated prism $DCM.HGN$. [img]https://cdn.artofproblemsolving.com/attachments/f/6/22c26f2c7c66293ad7065a3c8ce3ac2ffd938b.png[/img] 4. Nine pairs of husband and wife want to take pictures in a three-line position with the background of the Palembang Ampera Bridge. There are $4$ people in the front row, $6$ people in the middle row, and $ 8$ people in the back row. They agreed that every married couple must be in the same row, and every two people next to each other must be a married couple or of the same sex. Specify the number of different possible arrangements of positions. p5. p5. A hotel provides four types of rooms with capacity, rate, and number of rooms as presented in the following table. [b] type of room, capacity of persons/ room, day / rate (Rp.), / number of rooms [/b][img]https://cdn.artofproblemsolving.com/attachments/3/c/e9e1ed86887e692f9d66349a82eaaffc730b46.jpg[/img] A group of four families wanted to stay overnight at the hotel. Each family consists of husband and wife and their unmarried children. The number of family members by gender is presented in the following table. [b]family / man / woman/ total[/b] [img]https://cdn.artofproblemsolving.com/attachments/4/6/5961b130c13723dc9fa4e34b43be30c31ee635.jpg[/img] The group leader enforces the following provisions. I. Each husband and wife must share a room and may not share a room with other married couples. II. Men and women may not share the same room unless they are from the same family. III. At least one room is occupied by all family representatives (“representative room”) IV. Each family occupies at most $3$ types of rooms. V. No rooms are occupied by more than one family except representative rooms. You are asked to arrange a room for the group so that the total cost of lodging is as low as possible. Provide two possible alternative room arrangements for each family and determine the total cost.

We have an acute-angled triangle which is not isosceles. We denote the orthocenter, the circumcenter and the incenter of it by $H$, $O$, $I$ respectively. Prove that if a vertex of the triangle lies on the circle $HOI$, then there must be another vertex on this circle as well.

The circle touches the square and goes through its two vertices as shown in the figure. Find the area of the square. (Distance in the picture is measured horizontally from the midpoint of the side of the square.) [img]https://cdn.artofproblemsolving.com/attachments/7/5/ab4b5f3f4fb4b70013e6226ce5189f3dc2e5be.png[/img]

Suppose that $A_1A_2A_3$ is a triangle with $A_1A_2 = 16$ and $A_1A_3 = A_2A_3 = 10$. For each integer $n \ge 4$, set An to be the circumcenter of triangle $A_{n-1}A_{n-2}A_{n-3}$. There exists a unique point $Z$ lying in the interiors of the circumcircles of triangles $A_kA_{k+1}A_{k+2}$ for all integers $k \ge 1$. If $ZA^2_1+ ZA^2_2+ ZA^2_3+ ZA^2_4$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, find $a + b$.

Given quadrilateral $ABCD$. Its sidelines$ AB$ and $CD$ intersect in point $K$. It's diagonals intersect in point $L$. It is known that line $KL$ pass through the centroid of $ABCD$. Prove that $ABCD$ is trapezoid. (F.Nilov)