$\triangle ABC$ is inscribed to unit circle. Bisector of $\angle A,\angle B,\angle C$ intersect the circle at $A_1,B_1,C_1$ respectively. The value of $\frac{\displaystyle AA_1\cdot\cos\frac{A}{2}+BB_1\cdot\cos\frac{B}{2}+CC_1\cdot\cos\frac{C}{2}}{\sin A+\sin B+\sin C}$ is $\text{(A)}2\qquad\text{(B)}4\qquad\text{(C)}6\qquad\text{(D)}8$

The medians $AL, BM$, and $CN$ are drawn in the triangle $ABC$. Prove that $\angle ANC = \angle ALB$ if and only if $\angle ABM =\angle LAC$. (Veklich Bogdan)

Let $ \vartriangle ABC $ be an acute triangle and scalene, with $ BC $ its smallest side. Let $ P, Q $ points on $ AB, AC $ respectively, such that $ BQ = CP = BC $. Let $ O_1, O_2 $ be the centers of the circles circumscribed to $ \vartriangle AQB, \vartriangle APC $, respectively. Sean $ H, O $ the orthocenter and circumcenter of $ \vartriangle ABC $ a) Show that $ O_1O_2 = BC $. b) Show that $ BO_2, CO_1 $ and $ HO $ are concurrent

An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\times11$ rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.)

[u]Round 1[/u] [b]p1.[/b] Three two-digit numbers are written on a board. One starts with $5$, another with $6$, and the last one with $7$. Annie added the first and the second numbers; Benny added the second and the third numbers; Denny added the third and the first numbers. Could it be that one of these sums is equal to $148$, and the two other sums are three-digit numbers that both start with $12$? [b]p2.[/b] Three rocks, three seashells, and one pearl are placed in identical boxes on a circular plate in the order shown. The lids of the boxes are then closed, and the plate is secretly rotated. You can open one box at a time. What is the smallest number of boxes you need to open to know where the pearl is, no matter how the plate was rotated? [img]https://cdn.artofproblemsolving.com/attachments/0/2/6bb3a2a27f417a84ab9a64100b90b8768f7978.png[/img] [b]p3.[/b] Two detectives, Holmes and Watson, are hunting the thief Raffles in a library, which has the floorplan exactly as shown in the diagram. Holmes and Watson start from the center room marked $D$. Show that no matter where Raffles is or how he moves, Holmes and Watson can find him. Holmes and Watson do not need to stay together. A detective sees Raffles only if they are in the same room. A detective cannot stand in a doorway to see two rooms at the same time. [img]https://cdn.artofproblemsolving.com/attachments/c/1/6812f615e60a36aea922f145a1ffc470d0f1bc.png[/img] [b]p4.[/b] A museum has a $4\times 4$ grid of rooms. Every two rooms that share a wall are connected by a door. Each room contains some paintings. The total number of paintings along any path of $7$ rooms from the lower left to the upper right room is always the same. Furthermore, the total number of paintings along any path of $7$ rooms from the lower right to the upper left room is always the same. The guide states that the museum has exactly $500$ paintings. Show that the guide is mistaken. [img]https://cdn.artofproblemsolving.com/attachments/4/6/bf0185e142cd3f653d4a9c0882d818c55c64e4.png[/img] [b]p5.[/b] The numbers $1–14$ are placed around a circle in some order. You can swap two neighbors if they differ by more than $1$. Is it always possible to rearrange the numbers using swaps so they are ordered clockwise from $1$ to $14$? [u]Round 2[/u] [b]p6.[/b] A triangulation of a regular polygon is a way of drawing line segments between its vertices so that no two segments cross, and the interior of the polygon is divided into triangles. A flip move erases a line segment between two triangles, creating a quadrilateral, and replaces it with the opposite diagonal through that quadrilateral. This results in a new triangulation. [img]https://cdn.artofproblemsolving.com/attachments/a/a/657a7cf2382bab4d03046075c6e128374c72d4.png[/img] Given any two triangulations of a polygon, is it always possible to find a sequence of flip moves that transforms the first one into the second one? [img]https://cdn.artofproblemsolving.com/attachments/0/9/d09a3be9a01610ffc85010d2ac2f5b93fab46a.png[/img] [b]p7.[/b] Is it possible to place the numbers from $1$ to $121$ in an $11\times 11$ table so that numbers that differ by $1$ are in horizontally or vertically adjacent cells and all the perfect squares $(1, 4, 9,..., 121)$ are in one column? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

An acute triangle $ABC$ in which $AB<AC$ is given. The bisector of $\angle BAC$ crosses $BC$ at $D$. Point $M$ is the midpoint of $BC$. Prove that the line though centers of circles escribed on triangles $ABC$ and $ADM$ is parallel to $AD$.

Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$. [i]Proposed by C.R. Pranesachar, India [/i]

Radii of five concentric circles $\omega_0,\omega_1,\omega_2,\omega_3,\omega_4$ form a geometric progression with common ratio $q$ in this order. What is the maximal value of $q$ for which it's possible to draw a broken line $A_0A_1A_2A_3A_4$ consisting of four equal segments such that $A_i$ lies on $\omega_i$ for every $i=\overline{0,4}$? [hide=thanks ]Thanks to the user Vlados021 for translating the problem.[/hide]

[u]Round 1[/u] [b]p1.[/b] Is it possible to draw some number of diagonals in a convex hexagon so that every diagonal crosses EXACTLY three others in the interior of the hexagon? (Diagonals that touch at one of the corners of the hexagon DO NOT count as crossing.) [b]p2.[/b] A $ 3\times 3$ square grid is filled with positive numbers so that (a) the product of the numbers in every row is $1$, (b) the product of the numbers in every column is $1$, (c) the product of the numbers in any of the four $2\times 2$ squares is $2$. What is the middle number in the grid? Find all possible answers and show that there are no others. [b]p3.[/b] Each letter in $HAGRID$'s name represents a distinct digit between $0$ and $9$. Show that $$HAGRID \times H \times A\times G\times R\times I\times D$$ is divisible by $3$. (For example, if $H=1$, $A=2$, $G=3$, $R = 4$, $I = 5$, $D = 64$, then $HAGRID \times H \times A\times G\times R\times I\times D= 123456\times 1\times2\times3\times4\times5\times 6$). [b]p4.[/b] You walk into a room and find five boxes sitting on a table. Each box contains some number of coins, and you can see how many coins are in each box. In the corner of the room, there is a large pile of coins. You can take two coins at a time from the pile and place them in different boxes. If you can add coins to boxes in this way as many times as you like, can you guarantee that each box on the table will eventually contain the same number of coins? [b]p5.[/b] Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of $10$ games, Bob played $15$ games, and Chad played $17$ games. Who lost the second game? [u]Round 2[/u] [b]p6.[/b] After going for a swim in his vault of gold coins, Scrooge McDuck decides he wants to try to arrange some of his gold coins on a table so that every coin he places on the table touches exactly three others. Can he possibly do this? You need to justify your answer. (Assume the gold coins are circular, and that they all have the same size. Coins must be laid at on the table, and no two of them can overlap.) [b]p7.[/b] You have a deck of $50$ cards, each of which is labeled with a number between $1$ and $25$. In the deck, there are exactly two cards with each label. The cards are shuffled and dealt to $25$ students who are sitting at a round table, and each student receives two cards. The students will now play a game. On every move of the game, each student takes the card with the smaller number out of his or her hand and passes it to the person on his/her right. Each student makes this move at the same time so that everyone always has exactly two cards. The game continues until some student has a pair of cards with the same number. Show that this game will eventually end. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be an acute triangle with $AB<AC$ and $\angle BAC=60^{\circ}$. Denote its altitudes by $AD,BE,CF$ and its orthocenter by $H$. Let $K,L,M$ be the midpoints of sides $BC,CA,AB$, respectively. Prove that the midpoints of segments $AH, DK, EL, FM$ lie on a single circle.

Let $ABC$ be a triangle, and let $P$ be the midpoint of $AC$. A circle intersects $AP, CP, BC, AB$ sequentially at their inner points $K, L, M, N$. Let $S$ be the midpoint of $KL$. Let also $2 \cdot | AN |\cdot |AB |\cdot |CL | = 2 \cdot | CM | \cdot| BC | \cdot| AK| = | AC | \cdot| AK |\cdot |CL |.$ Prove that if $P\ne S$, then the intersection of $KN$ and $ML$ lies on the perpendicular bisector of the $PS$. (Jan Mazák)

[b]p1.[/b] What is $2 \cdot 24 + 20 \cdot 24 + 202 \cdot 4 + 2024$? [b]p2.[/b] Jerry has $300$ legos. Tie can either make cars, which require $17$ legos, or bikes, which require $13$ legos. Assuming he uses all of his legos, how many ordered pairs $(a, b)$ are there such that he makes $a$ cars and $b$ bikes? [b]p3.[/b] Patrick has $7$ unique textbooks: $2$ Geometry books, $3$ Precalculus books and $2$ Algebra II books. How many ways can he arrange his books on a bookshelf such that all the books of the same subjects are adjacent to each other? [b]p4.[/b] After a hurricane, a $32$ meter tall flagpole at the Act on-Boxborough Regional High School snapped and fell over. Given that the snapped part remains in contact with the original pole, and the top of the polo falls $24$ meters away from the bottom of the pole, at which height did the polo snap? (Assume the flagpole is perpendicular to the ground.) [b]p5.[/b] Jimmy is selling lemonade. Iio has $200$ cups of lemonade, and he will sell them all by the end of the day. Being the ethically dubious individual he is, Jimmy intends to dilute a few of the cups of lemonade with water to conserve resources. Jimmy sells each cup for $\$4$. It costs him $\$ 1$ to make a diluted cup of lemonade, and it costs him $\$2.75$ to make a cup of normal lemonade. What is the minimum number of diluted cups Jimmy must sell to make a profit of over $\$400$? [b]p6.[/b] Jeffrey has a bag filled with five fair dice: one with $4$ sides, one with $6$ sides, one with $8$ sides, one with $12$ sides, and one with $20$ sides. The dice are numbered from $1$ to the number of sides on the die. Now, Marco will randomly pick a die from .Jeffrey's bag and roll it. The probability that Marco rolls a $7$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$. [b]p7.[/b] What is the remainder when the sum of the first $2024$ odd numbers is divided by $6072$? [b]p8.[/b] A rhombus $ABCD$ with $\angle A = 60^o$ and $AB = 600$ cm is drawn on a piece of paper. Three ants start moving from point $A$ to the three other points on the rhombus. One ant walks from $A$ to $B$ at a leisurely speed of $10$ cm/s. The second ant runs from $A$ to $C$ at a slightly quicker pace of $6\sqrt3$ cm/s, arriving to $C$ $x$ seconds after the first ant. The third ant travels from $A$ to $B$ to $D$ at a constant speed, arriving at $D$ $x$ seconds after the second ant. The speed of the last ant can be written as $\frac{m}{n}$ cm/s, where $m$ and $n$ are relatively prime positive integers. Find $mn$. [b]p9.[/b] This year, the Apple family has harvested so many apples that they cannot sell them all! Applejack decides to make $40$ glasses of apple cider to give to her friends. If Twilight and Fluttershy each want $1$ or $2$ glasses; Pinkie Pic wants cither $2$, $14$, or $15$ glasses; Rarity wants an amount of glasses that is a power of three; and Rainbow Dash wants any odd number of glasses, then how many ways can Applejack give her apple cider to her friends? Note: $1$ is considered to be a power of $3$. [b]p10.[/b] Let $g_x$ be a geometric sequence with first term $27$ and successive ratio $2n$ (so $g_{x+1}/g_x = 2n$). Then, define a function $f$ as $f(x) = \log_n(g_x)$, where $n$ is the base of the logarithm. It is known that the sum of the first seven terms of $f(x)$ is $42$. Find $g_2$, the second term of the geometric sequence. Note: The logarithm base $b$ of $x$, denoted $\log_b(x)$ is equal to the value $y$ such that $b^y = x$. In other words, if $\log_b(x) = y$, then $b^y = x$. [b]p11.[/b] Let $\varepsilon$ be an ellipse centered around the origin, such that its minor axis is perpendicular to the $x$-axis. The length of the ellipse's major and minor axes is $8$ and $6$, respectively. Then, let $ABCD$ be a rectangle centered around the origin, such that $AB$ is parallel to the $x$-axis. The lengths of $AB$ and $BC$ are $8$ and $3\sqrt2$, respectively. The area outside the ellipse but inside the rectangle can be expressed as $a\sqrt{b}-c-d\pi$, for positive integers $a$, $b$, $c$, $d$ where $b$ is not divisible by a perfect square of any prime. Find $a + b + c + d$. [img]https://cdn.artofproblemsolving.com/attachments/e/c/9d943966763ee7830d037ef98c21139cf6f529.png[/img] [b]p12.[/b] Let $N = 2^7 \cdot 3^7 \cdot 5^5$. Find the number of ways to express $N$ as the product of squares and cubes, all of which are integers greater than $1$. [b]p13.[/b] Jerry and Eric are playing a $10$-card game where Jerry is deemed the ’’landlord" and Eric is deemed the ' peasant'’. To deal the cards, the landlord keeps one card to himself. Then, the rest of the $9$ cards are dealt out, such that each card has a $1/2$ chance to go to each player. Once all $10$ cards are dealt out, the landlord compares the number of cards he owns with his peasant. The probability that the landlord wins is the fraction of cards he has. (For example, if Jerry has $5$ cards and Eric has $2$ cards, Jerry has a$ 5/7$ ths chance of winning.) The probability that Jerry wins the game can be written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime. Find $p + q$. [b]p14.[/b] Define $P(x) = 20x^4 + 24x^3 + 10x^2 + 21x+ 7$ to have roots $a$, $b$, $c$, and $d$. If $Q(x)$ has roots $\frac{1}{a-2}$,$\frac{1}{b-2}$,$ \frac{1}{c-2}$, $\frac{1}{d-2}$ and integer coefficients with a greatest common divisor of $1$, then find $Q(2)$. [b]p15.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 14$, $BC = 13$, and $AC = 15$. The incircle of $\vartriangle ABC$ is drawn with center $I$, tangent to $\overline{AB}$ at $X$. The line $\overleftrightarrow{IX}$ intersects the incircle again at $Y$ and intersects $\overline{AC}$ at $Z$. The area of $\vartriangle AYZ$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle. Let $E$ be a point on the segment $BC$ such that $BE = 2EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AE$ in $Q$. Determine $BQ:QF$.

If the length of a rectangle is increased by $20\% $ and its width is increased by $50\% $, then the area is increased by $\text{(A)}\ 10\% \qquad \text{(B)}\ 30\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 80\% \qquad \text{(E)}\ 100\% $

In the given figure, $ABCD$ is a parallelogram. We know that $\angle D = 60^\circ$, $AD = 2$ and $AB = \sqrt3 + 1$. Point $M$ is the midpoint of $AD$. Segment $CK$ is the angle bisector of $C$. Find the angle $CKB$. [i]Proposed by Mahdi Etesamifard[/i]

$D$ is midpoint of $AC$ for $\triangle ABC$. Bisectors of $\angle ACB,\angle ABD$ are perpendicular. Find max value for $\angle BAC$ [i](S. Berlov)[/i]

Let $P$ be a set of $n$ points and $S$ a set of $l$ segments. It is known that: $(i)$ No four points of $P$ are coplanar. $(ii)$ Any segment from $S$ has its endpoints at $P$. $(iii)$ There is a point, say $g$, in $P$ that is the endpoint of a maximal number of segments from $S$ and that is not a vertex of a tetrahedron having all its edges in $S$. Prove that $l \leq \frac{n^2}{3}$

Let $ M $ be a point in the interior of a triangle $ ABC, $ let $ D $ be the intersection of $ AM $ with $ BC, $ let $ E $ be the intersection of $ M $ with AC, let $ F $ be the intersection of $ CM $ with $ AB. $ Knowing that the expression $$ \frac{MA}{MD}\cdot \frac{MB}{ME}\cdot \frac{MC}{MF} $$ is minimized, describe the point $ M. $

In triangle $ABC$, $\angle C = 30^{\circ}$, $O$ and $I$ are the circumcenter and incenter respectively, Points $D \in AC$ and $E \in BC$, such that $AD = BE = AB$. Prove that $OI = DE$ and $OI \bot DE$.

Title is self explanatory. Pick two points on the unit sphere. What is the expected distance between them?

Let $\triangle$$ABC$ a triangle with circumcircle $\Gamma$. Suppose there exist points $R$ and $S$ on sides $AB$ and $AC$, respectively, such that $BR=RS=SC$. A tangent line through $A$ to $\Gamma$ meet the line $RS$ at $P$. Let $I$ the incenter of triangle $\triangle$$ARS$. Prove that $PA=PI$

Let $ABC$ be an isosceles triangle ($AB=AC$). The altitude $AH$ and the perpendiculare bisector $(e)$ of side $AB$ intersect at point $M$ . The perpendicular on line $(e)$ passing through $M$ intersects $BC$ at point $D$. If the circumscribed circle of the triangle $BMD$ intersects line $(e)$ at point $S$ , the prove that: a) $BS // AM$ . b) quadrilateral $AMBS$ is rhombus.

* On a plane two points $A$ and $B$ are on the same side of a line. Find point $M$ on the line such that $MA +MB$ is equal to a given length.

Let $ABC$ be an acute-angled triangle, $O$ its circumcenter and $H$ its orthocenter. The orthogonal projection of the vertex $A$ to the line $BC$ lies on the perpendicular bisector of the segment $AC$. Compute $\frac{CH}{BO}$ . (J. Willemson)

The bisectors of the angles $A$ and $B$ of the triangle $ABC$ meet the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Assuming that $AE+BD=AB$, determine the angle $C$.