Let $H$ be the orthocenter of an acute triangle $ABC$. (The orthocenter is the point at the intersection of the three altitudes. An acute triangle has all angles less than $90^o$.) Draw three circles: one passing through $A, B$, and $H$, another passing through $B, C$, and $H$, and finally, one passing through $C, A$, and $H$. Prove that the triangle whose vertices are the centers of those three circles is congruent to triangle $ABC$.

In triangle $ABC$ we have $AC = CB$. On side $AB$ is a point $D$ such that the radius of the incircle of triangle $ACD$ is equal to the radius of the circle tangent to the segment $DB$ and to the extensions of the lines $CD$ and $CB$. Prove that this radius equals a quarter of either of the two equal altitudes of triangle $ABC$.

Given $k$ parallel lines $l_1, \ldots, l_k$ and $n_i$ points on the line $l_i, i = 1, 2, \ldots, k$, find the maximum possible number of triangles with vertices at these points.

In triangle ABC,X,Y are on the angle bisector of ∠BAC and ∠ABX=∠ACY.BX intersects CY at P and circles (BYP) and (CXP) intersect at Q different from P. Prove that A,P,Q are on a line.

Let $ABCD$ be the regular tetrahedron, and $M, N$ points in space. Prove that: $AM \cdot AN + BM \cdot BN + CM \cdot CN \geq DM \cdot DN$

$ABC$ is a triangle where $AB = 10$, $BC = 14$, and $AC = 16$. Let $DEF$ be the triangle with smallest area so that $DE$ is parallel to $AB$, $EF$ is parallel to $BC$, $DF$ is parallel to $AC$, and the circumcircle of $ABC$ is $DEF$’s inscribed circle. Line $DA$ meets the circumcircle of $ABC$ again at a point $X$. Find $AX^2$ .

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

The bisectors of angles $A$, $B$, and $C$ of triangle $ABC$ meet for the second time its circumcircle at points $A_1$, $B_1$, $C_1$ respectively. Let $A_2$, $B_2$, $C_2$ be the midpoints of segments $AA_1$, $BB_1$, $CC_1$ respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.

[b]p1.[/b] There is a string of numbers $1234567891023...910134 ...91012...$ that concatenates the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, then $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $1$, then $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $1$, $2$, and so on. After $10$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, the string will be concatenated with $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$ again. What is the $2021$st digit? [b]p2.[/b] Bob really likes eating rice. Bob starts eating at the rate of $1$ bowl of rice per minute. Every minute, the number of bowls of rice Bob eats per minute increases by $1$. Given there are $78$ bowls of rice, find number of minutes Bob needs to finish all the rice. [b]p3.[/b] Suppose John has $4$ fair coins, one red, one blue, one yellow, one green. If John flips all $4$ coins at once, the probability he will land exactly $3$ heads and land heads on both the blue and red coins can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, Find $a + b$. [b]p4.[/b] Three of the sides of an isosceles trapezoid have lengths $1$, $10$, $20$ Find the sum of all possible values of the fourth side. [b]p5.[/b] An number two-three-delightful if and only if it can be expressed as the product of $2$ consecutive integers larger than $1$ and as the product of $3$ consecutive integers larger than $1$. What is the smallest two-three-delightful number? [b]p6.[/b] There are $3$ students total in Justin's online chemistry class. On a $100$ point test, Justin's two classmates scored $4$ and $7$ points. The teacher notices that the class median score is equal to $gcd(x, 42)$, where the positive integer $x$ is Justin's score. Find the sum of all possible values of Justin's score. [b]p7.[/b] Eddie's gym class of $10$ students decides to play ping pong. However, there are only $4$ tables and only $2$ people can play at a table. If $8$ students are randomly selected to play and randomly assigned a partner to play against at a table, the probability that Eddie plays against Allen is $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, Find $a + b$. [b]p8.[/b] Let $S$ be the set of integers $k$ consisting of nonzero digits, such that $300 < k < 400$ and $k - 300$ is not divisible by $11$. For each $k$ in $S$, let $A(k)$ denote the set of integers in $S$ not equal to $k$ that can be formed by permuting the digits of $k$. Find the number of integers $k$ in $S$ such that $k$ is relatively prime to all elements of $A(k)$. [b]p9.[/b] In $\vartriangle ABC$, $AB = 6$ and $BC = 5$. Point $D$ is on side $AC$ such that $BD$ bisects angle $\angle ABC$. Let $E$ be the foot of the altitude from $D$ to $AB$. Given $BE = 4$, find $AC^2$. [b]p10.[/b] For each integer $1 \le n \le 10$, Abe writes the number $2^n + 1$ on a blackboard. Each minute, he takes two numbers $a$ and $b$, erases them, and writes $\frac{ab-1}{a+b-2}$ instead. After $9$ minutes, there is one number $C$ left on the board. The minimum possible value of $C$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$. [b]p11.[/b] Estimation (Tiebreaker) Let $A$ and $B$ be the proportions of contestants that correctly answered Questions $9$ and $10$ of this round, respectively. Estimate $\left \lfloor \dfrac{1}{(AB)^2} \right \rfloor$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given three circumferences of the same radius in a plane. a) All three are crossing in one point $K$. Consider three arcs $AK,CK,EK$ : the $A,C,E$ are the points of the circumferences intersection and the arcs are taken in the clockwise direction. Every arc is inside one circle, outside the second and on the border of the third one. Prove that the sum of the arcs is $180$ degrees. b) Consider the case, when the three circles give a curvilinear triangle $BDF$ as their intersection (instead of one point $K$). The arcs are taken in the clockwise direction. Every arc is inside one circle, outside the second and on the border of the third one. Prove that the sum of the $AB, CD$ and $EF$ arcs is $180$ degrees.

Let $P$ be an interior point of acute triangle $\Delta ABC$, which is different from the orthocenter. Let $D$ and $E$ be the feet of altitudes from $A$ to $BP$ and $CP$, and let $F$ and $G$ be the feet of the altitudes from $P$ to sides $AB$ and $AC$. Denote by $X$ the midpoint of $[AP]$, and let the second intersection of the circumcircles of triangles $\Delta DFX$ and $\Delta EGX$ lie on $BC$. Prove that $AP$ is perpendicular to $BC$ or $\angle PBA = \angle PCA$.

The circumferences $C_1$ and $C_2$ cut each other at different points $A$ and $K$. The common tangent to $C_1$ and $C_2$ nearer to $K$ touches $C_1$ at $B$ and $C_2$ at $C$. Let $P$ be the foot of the perpendicular from $B$ to $AC$, and let $Q$ be the foot of the perpendicular from $C$ to $AB$. If $E$ and $F$ are the symmetric points of $K$ with respect to the lines $PQ$ and $BC$, respectively, prove that $A, E$ and $F$ are collinear.

[u]Round 1[/u] [b]p1.[/b] If $2m = 200 cm$ and $m \ne 0$, find $c$. [b]p2.[/b] A right triangle has two sides of lengths $3$ and $4$. Find the smallest possible length of the third side. [b]p3.[/b] Given that $20(x + 17) = 17(x + 20)$, determine the value of $x$. [u]Round 2[/u] [b]p4.[/b] According to the Egyptian Metropolitan Culinary Community, food service is delayed on $\frac23$ of flights departing from Cairo airport. On average, if flights with delayed food service have twice as many passengers per flight as those without, what is the probability that a passenger departing from Cairo airport experiences delayed food service? [b]p5.[/b] In a positive geometric sequence $\{a_n\}$, $a_1 = 9$, $a_9 = 25$. Find the integer $k$ such that $a_k = 15$ [b]p6.[/b] In the Delicate, Elegant, and Exotic Music Organization, pianist Hans is selling two types of owers with different prices (per ower): magnolias and myosotis. His friend Alice originally plans to buy a bunch containing $50\%$ more magnolias than myosotis for $\$50$, but then she realizes that if she buys $50\%$ less magnolias and $50\%$ more myosotis than her original plan, she would still need to pay the same amount of money. If instead she buys $50\%$ more magnolias and $50\%$ less myosotis than her original plan, then how much, in dollars, would she need to pay? [u]Round 3[/u] [b]p7.[/b] In square $ABCD$, point $P$ lies on side $AB$ such that $AP = 3$,$BP = 7$. Points $Q,R, S$ lie on sides $BC,CD,DA$ respectively such that $PQ = PR = PS = AB$. Find the area of quadrilateral $PQRS$. [b]p8.[/b] Kristy is thinking of a number $n < 10^4$ and she says that $143$ is one of its divisors. What is the smallest number greater than $143$ that could divide $n$? [b]p9.[/b] A positive integer $n$ is called [i]special [/i] if the product of the $n$ smallest prime numbers is divisible by the sum of the $n$ smallest prime numbers. Find the sum of the three smallest special numbers. [u]Round 4[/u] [b]p10.[/b] In the diagram below, all adjacent points connected with a segment are unit distance apart. Find the number of squares whose vertices are among the points in the diagram and whose sides coincide with the drawn segments. [img]https://cdn.artofproblemsolving.com/attachments/b/a/923e4d2d44e436ccec90661648967908306fea.png[/img] [b]p11.[/b] Geyang tells Junze that he is thinking of a positive integer. Geyang gives Junze the following clues: $\bullet$ My number has three distinct odd digits. $\bullet$ It is divisible by each of its three digits, as well as their sum. What is the sum of all possible values of Geyang's number? [b]p12.[/b] Regular octagon $ABCDEFGH$ has center $O$ and side length $2$. A circle passes through $A,B$, and $O$. What is the area of the part of the circle that lies outside of the octagon? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2936505p26278645]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let triangle $ABC$ have altitudes $BE$ and $CF$ which meet at $H$. The reflection of $A$ over $BC$ is $A'$. Let $(ABC)$ meet $(AA'E)$ at $P$ and $(AA'F)$ at $Q$. Let $BC$ meet $PQ$ at $R$. Prove that $EF \parallel HR$. [i]Proposed by Daniel Hu[/i]

Determine the ratio of the sides of the rectangle circumscribed around a corner of five cells (see figure). (M. Evdokimov) [img]https://cdn.artofproblemsolving.com/attachments/f/f/9c3e345f33cabbbd83f65d7240aac29a163b19.png[/img]

$\vartriangle ABC$ has side lengths $AB = 4$ and $AC = 9$. Angle bisector $AD$ bisects angle $A$ and intersects $BC$ at $D$. Let $k$ be the ratio $\frac{BD}{AB}$ . Given that the length $AD$ is an integer, find the sum of all possible $k^2$ .

Given is a square $ABCD$ with $E \in BC$, arbitrarily. On $CD$ lies the point $F$ is such that $\angle EAF = 45^o$. Prove that $EF$ is tangent to the circle with center $A$ and radius $AB$.

Let $ABC$ be an acute triangle, let $H$ and $O$ be its orthocentre and circumcentre, respectively, and let $S$ and $T$ be the feet of the altitudes from $B$ to $AC$ and from $C$ to $AB$, respectively. Let $M$ be the midpoint of the segment $ST$, and let $N$ be the midpoint of the segment $AH$. The line through $O$, parallel to $BC$, crosses the sides $AC$ and $AB$ at $F$ and $G$, respectively. The line $NG$ meets the circle $BGO$ again at $K$, and the line $NF$ meets the circle $CFO$ again at $L$. Prove that the triangles $BCM$ and $KLN$ are similar.

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. A circle $\Gamma$ is internally tangent to $\omega$ at $A$ and also tangent to $BC$ at $D$. Let $AB$ and $AC$ intersect $\Gamma$ at $P$ and $Q$ respectively. Let $M$ and $N$ be points on line $BC$ such that $B$ is the midpoint of $DM$ and $C$ is the midpoint of $DN$. Lines $MP$ and $NQ$ meet at $K$ and intersect $\Gamma$ again at $I$ and $J$ respectively. The ray $KA$ meets the circumcircle of triangle $IJK$ again at $X\neq K$. Prove that $\angle BXP = \angle CXQ$. [i]Kian Moshiri, United Kingdom[/i]

Through the center of gravity of the acute-angled triangle $ ABC $, lines are drawn perpendicular to the sides $ BC $, $ CA $, $ AB $, intersecting them at the points $ P $, $ Q $, $ R $, respectively. Prove that if $ |BP|\cdot |CQ| \cdot |AR| = |PC| \cdot |QA| \cdot |RB| $, then the triangle $ ABC $ is isosceles. Note: According to Ceva's theorem, the assumed equality of products is equivalent to the fact that the lines $ AP $, $ BQ $, $ CR $ have a common point.

The diagram below shows the regular hexagon $BCEGHJ$ surrounded by the rectangle $ADFI$. Let $\theta$ be the measure of the acute angle between the side $\overline{EG}$ of the hexagon and the diagonal of the rectangle $\overline{AF}$. There are relatively prime positive integers $m$ and $n$ so that $\sin^2\theta  = \tfrac{m}{n}$. Find $m + n$. [asy] import graph; size(3.2cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((-1,3)--(-1,2)--(-0.13,1.5)--(0.73,2)--(0.73,3)--(-0.13,3.5)--cycle); draw((-1,3)--(-1,2)); draw((-1,2)--(-0.13,1.5)); draw((-0.13,1.5)--(0.73,2)); draw((0.73,2)--(0.73,3)); draw((0.73,3)--(-0.13,3.5)); draw((-0.13,3.5)--(-1,3)); draw((-1,3.5)--(0.73,3.5)); draw((0.73,3.5)--(0.73,1.5)); draw((-1,1.5)--(0.73,1.5)); draw((-1,3.5)--(-1,1.5)); label("$ A $",(-1.4,3.9),SE*labelscalefactor); label("$ B $",(-1.4,3.28),SE*labelscalefactor); label("$ C $",(-1.4,2.29),SE*labelscalefactor); label("$ D $",(-1.4,1.45),SE*labelscalefactor); label("$ E $",(-0.3,1.4),SE*labelscalefactor); label("$ F $",(0.8,1.45),SE*labelscalefactor); label("$ G $",(0.8,2.24),SE*labelscalefactor); label("$ H $",(0.8,3.26),SE*labelscalefactor); label("$ I $",(0.8,3.9),SE*labelscalefactor); label("$ J $",(-0.25,3.9),SE*labelscalefactor); [/asy]

Let $L$ be a subset in the coordinate plane defined by $L = \{(41x + 2y, 59x + 15y) | x, y \in \mathbb Z \}$, where $\mathbb Z$ is set of integers. Prove that for any parallelogram with center in the origin of coordinate and area $1990$, there exist at least two points of $L$ located in it.

Let $\omega$ be the circumcircle of $ABC$, and $KL$ be the diameter of $\omega$ passing through $M$ midpoint of $AB$ ($K,C$ lies on different sides of $AB$). A circle passing through $L$ and $M$ meets $CK$ at points $P$ and $Q$ ($Q$ lies on $KP$). Let $LQ$ meet the circumcircle of $KMQ$ again at $R$. Prove that $APBR$ is cyclic.

$ABCD$ is a parallelogram satisfying $AB = 7$, $BC = 2$, and $\angle DAB = 120^o$. Parallelogram $ECFA$ is contained in $ABCD$ and is similar to it. Find the ratio of the area of $ECFA$ to the area of $ABCD$.

Is it possible for an isosceles triangle with all its sides of positive integer lengths to have an angle of $36^o$? [i] (Adapted from Archimedes 2011, Traian Preda)[/i]