Let $\omega$ be the circumcircle of isosceles triangle $ABC$ ($AB=AC$). Points $P$ and $Q$ lie on $\omega$ and $BC$ respectively such that $AP=AQ$ .$AP$ and $BC$ intersect at $R$. Prove that the tangents from $B$ and $C$ to the incircle of $\triangle AQR$ (different from $BC$) are concurrent on $\omega$. [i]Proposed by Ali Zamani, Hooman Fattahi[/i]

In rectangle $ABCD$, the points $M,N,P, Q$ lie on $AB$, $BC$, $CD$, $DA$ respectively such that the area of triangles $AQM$, $BMN$, $CNP$, $DPQ$ are equal. Prove that the quadrilateral $MNPQ$ is parallelogram. by Mahdi Etesami Fard

On the sides $ BC, CA $ and $ AB $ (extremities excluded) of the triangle $ ABC, $ consider the arbitrary points $ P,Q,R $ and the circumcenters $ O_1,O_2,O_3 $ of $ AQR,BRP,CPQ. $ Show that $ O_1O_2O_3\sim ABC. $

In a acute-angled triangle $ABC$, a point $D$ lies on the segment $BC$. Let $O_1,O_2$ denote the circumcentres of triangles $ABD$ and $ACD$ respectively. Prove that the line joining the circumcentre of triangle $ABC$ and the orthocentre of triangle $O_1O_2D$ is parallel to $BC$.

The point $M$ is the middle of the side $BC$ of the acute-angled triangle $ABC$ and the points $E$ and $F$ are respectively perpendicular foot of $M$ to the sides $AC$ and $AB$. The points $X$ and $Y$ lie on the plane such that $\triangle XEC\sim\triangle CEY$ and $\triangle BYF\sim\triangle XBF$(The vertices of triangles with this order are corresponded in the similarities) and the points $E$ and $F$ [u]don't[/u][neither] lie on the line $XY$. Prove that $XY\perp AM$.

A unit cube $C$ is rotated around one of its diagonals for the angle $\pi /3$ to form a cube $C'$. Find the volume of the intersection of $C$ and $C'$.

Call a scalene triangle K [i]disguisable[/i] if there exists a triangle K′ similar to K with two shorter sides precisely as long as the two longer sides of K, respectively. Call a disguisable triangle [i]integral[/i] if the lengths of all its sides are integers. (a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter. (b) Let K be an arbitrary integral disguisable triangle for which no smaller integral disguisable triangle similar to it exists. Prove that at least two side lengths of K are perfect squares.

Let $ABCD$ be a cyclic quadrilateral such that $AB \parallel CD$. Let $O$ be the circumcenter of $ABCD$ and $L$ be the point on $AD$ such that $OL$ is perpendicular to $AD$. Prove that \[ OB\cdot(AB+CD) = OL\cdot(AC + BD).\] [i]Proposed by Rijul Saini[/i]

Let $ABC$ be a triangle such that $AC>AB.$ A circle tangent to the sides $AB$ and $AC$ at $D$ and $E$ respectively, intersects the circumcircle of $ABC$ at $K$ and $L$. Let $X$ and $Y$ be points on the sides $AB$ and $AC$ respectively, satisfying \[ \frac{AX}{AB}=\frac{CE}{BD+CE} \quad \text{and} \quad \frac{AY}{AC}=\frac{BD}{BD+CE} \] Show that the lines $XY, BC$ and $KL$ are concurrent.

Let $ABCD$ be a quadrilateral. The circumcircle of the triangle $ABC$ intersects the sides $CD$ and $DA$ in the points $P$ and $Q$ respectively, while the circumcircle of $CDA$ intersects the sides $AB$ and $BC$ in the points $R$ and $S$. The lines $BP$ and $BQ$ intersect the line $RS$ in the points $M$ and $N$ respectively. Prove that the points $M, N, P$ and $Q$ lie on the same circle.

Prove that if the sides $ a $, $ b $, $ c $ of a triangle satisfy the inequality $$a < b < c$$then the angle bisectors $ d_a $, $ d_b $, $ d_c $ of opposite angles satisfy the inequality $$ d_a > d_b > d_c.$$

[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].

Let $\angle XOY = \frac{\pi}{2}$; $P$ is a point inside $\angle XOY$ and we have $OP = 1; \angle XOP = \frac{\pi}{6}.$ A line passes $P$ intersects the Rays $OX$ and $OY$ at $M$ and $N$. Find the maximum value of $OM + ON - MN.$

Let $(O)$ be the circumcircle of triangle $ABC$, and $P$ a point on the arc $BC$ not containing $A$. $(Q)$ is the $A$-mixtilinear circle of triangle $ABC$, and $(K), (L)$ are the $P$-mixtilinear circles of triangle $PAB, PAC$ respectively. Prove that there is a line tangent to all the three circles $(Q), (K)$ and $(L)$. Nguyen Van Linh, a student at Hanoi Foreign Trade University Cabinet

Let $D$ be the foot of the internal bisector of the angle $\angle A$ of the triangle $ABC$. The straight line which joins the incenters of the triangles $ABD$ and $ACD$ cut $AB$ and $AC$ at $M$ and $N$, respectively. Show that $BN$ and $CM$ meet on the bisector $AD$.

A tetrahedron $ABCD$ is given such that $AD = BC = a; AC = BD = b; AB\cdot CD = c^2$. Let $f(P) = AP + BP + CP + DP$, where $P$ is an arbitrary point in space. Compute the least value of $f(P).$

Let $ABCD$ be a square and let $\Gamma$ be the circumcircle of $ABCD$. $M$ is a point of $\Gamma$ belonging to the arc $CD$ which doesn't contain $A$. $P$ and $R$ are respectively the intersection points of $(AM)$ with $[BD]$ and $[CD]$, $Q$ and $S$ are respectively the intersection points of $(BM)$ with $[AC]$ and $[DC]$. Prove that $(PS)$ and $(QR)$ are perpendicular.

Let $n\ge3$ be an integer. A circle is divided into $2n$ arcs by $2n$ points. Each arc has one of three possible lengths, and no two adjacent arcs have the same lengths. The $2n$ points are colored alternately red and blue. Prove that the $n$-gon with red vertices and the $n$-gon with blue vertices have the same perimeter and the same area.

In the diagram below, point $A$ lies on the circle centered at $O$. $AB$ is tangent to circle $O$ with $\overline{AB} = 6$. Point $C$ is $\frac{2\pi}{3}$ radians away from point $A$ on the circle, with $BC$ intersecting circle $O$ at point $D$. The length of $BD$ is $3$. Compute the radius of the circle. [img]https://cdn.artofproblemsolving.com/attachments/7/8/baa528c776eb50455f31ae50a3ec28efc291e8.png[/img]

Let $h$ be a semicircle with diameter $AB$. The two circles $k_1$ and $k_2$, $k_1 \ne k_2$, touch the segment $AB$ at the points $C$ and $D$, respectively, and the semicircle $h$ fom the inside at the points $E$ and $F$, respectively. Prove that the four points $C$, $D$, $E$ and $F$ lie on a circle. [i](Walther Janous)[/i]

Three circles $\omega_1$, $\omega_2$, $\omega_3$ of radius $r$ pass through the point$ S$ and internally touch the circle $\omega$ of radius $R$ ($R > r$) at points $T_1$, $T_2$, $T_3$ respectively. Prove that the line $T_1T_2$ passes through the second (different from $S$) intersection point of the circles $\omega_1$ and $\omega_2$.

The vertices $X, Y , Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC, CA, AB$ of an acute-angled triangle $ABC.$ Prove that the incenter of triangle $ABC$ lies inside triangle $XYZ.$ [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Compute $\frac{2024}{2 + 0 \times 2 - 4}.$ [b]p2.[/b] Find the smallest integer that can be written as the product of three distinct positive odd integers. [b]p3.[/b] Bryan’s physics test score is a two-digit number. When Bryan reverses its digits and adds the tens digit of his test score, he once again obtains his test score. Determine Bryan’s physics test score. [b]p4.[/b] Grant took four classes today. He spent $70$ minutes in math class. Had his math class been $40$ minutes instead, he would have spent $15\%$ less total time in class today. Find how many minutes he spent in his other classes combined. [b]p5.[/b] Albert’s favorite number is a nonnegative integer. The square of Albert’s favorite number has $9$ digits. Find the number of digits in Albert’s favorite number. [b]p6.[/b] Two semicircular arcs are drawn in a rectangle, splitting it into four regions as shown below. Given the areas of two of the regions, find the area of the entire rectangle. [img]https://cdn.artofproblemsolving.com/attachments/1/a/22109b346c7bdadeaf901d62155de4c506b33c.png[/img] [b]p7.[/b] Daria is buying a tomato and a banana. She has a $20\%$-off coupon which she may use on one of the two items. If she uses it on the tomato, she will spend $\$1.21$ total, and if she uses it on the banana, she will spend $\$1.31$ total. In cents, find the absolute difference between the price of a tomato and the price of a banana. [b]p8.[/b] Celine takes an $8\times 8$ checkerboard of alternating black and white unit squares and cuts it along a line, creating two rectangles with integer side lengths, each of which contains at least $9$ black squares. Find the number of ways Celine can do this. (Rotations and reflections of the cut are considered distinct.) [b]p9.[/b] Each of the nine panes of glass in the circular window shown below has an area of $\pi$, eight of which are congruent. Find the perimeter of one of the non-circular panes. [img]https://cdn.artofproblemsolving.com/attachments/b/c/0d3644dde33b68f186ba1ff0602e08ce7996f5.png[/img] [b]p10.[/b] In Alan’s favorite book, pages are numbered with consecutive integers starting with $1$. The average of the page numbers in Chapter Five is $95$ and the average of the page numbers in Chapter Six is $114$. Find the number of pages in Chapters Five and Six combined. [b]p11.[/b] Find the number of ordered pairs $(a, b)$ of positive integers such that $a + b = 2024$ and $$\frac{a}{b}>\frac{1000}{1025}.$$ [b]p12.[/b] A square is split into three smaller rectangles $A$, $B$, and $C$. The area of $A$ is $80$, $B$ is a square, and the area of $C$ is $30$. Compute the area of $B$. [img]https://cdn.artofproblemsolving.com/attachments/d/5/43109b964eacaddefd410ddb8bf4e4354a068b.png[/img] [b]p13.[/b] A knight on a chessboard moves two spaces horizontally and one space vertically, or two spaces vertically and one space horizontally. Two knights attack each other if each knight can move onto the other knight’s square. Find the number of ways to place a white knight and a black knight on an $8 \times 8$ chessboard so that the two knights attack each other. One such possible configuration is shown below. [img]https://cdn.artofproblemsolving.com/attachments/2/2/b4a83fbbab7e54dda81ac5805728d268b6db9f.png[/img] [b]p14.[/b] Find the sum of all positive integers $N$ for which the median of the positive divisors of $N$ is $9$. [b]p15.[/b] Let $x$, $y$, and $z$ be nonzero real numbers such that $$\begin{cases} 20x + 24y = yz \\ 20y + 24x = xz \end{cases}$$ Find the sum of all possible values of $z$. [b]p16.[/b] Ava glues together $9$ standard six-sided dice in a $3 \times 3$ grid so that any two touching faces have the same number of dots. Find the number of dots visible on the surface of the resulting shape. (On a standard six-sided die, opposite faces sum to $7$.) [img]https://cdn.artofproblemsolving.com/attachments/5/5/bc71dac9b8ae52a4456154000afde2c89fd83a.png[/img] [b]p17.[/b] Harini has a regular octahedron of volume $1$. She cuts off its $6$ vertices, turning the triangular faces into regular hexagons. Find the volume of the resulting solid. [b]p18.[/b] Each second, Oron types either $O$ or $P$ with equal probability, forming a growing sequence of letters. Find the probability he types out $POP$ before $OOP$. [b]p19.[/b] For an integer $n \ge 10$, define $f(n)$ to be the number formed after removing the first digit from $n$ (and removing any leading zeros) and define $g(n)$ to be the number formed after removing the last digit from $n$. Find the sum of the solutions to the equation $f(n) + g(n) = 2024$. [b]p20.[/b] In convex trapezoid $ABCD$ with $\overline{AB} \parallel \overline{CD}$ and $AD = BC$, let $M$ be the midpoint of $\overline{BC}$. If $\angle AMB = 24^o$ and $\angle CMD = 66^o$, find $\angle ABC$, in degrees. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a triangle $ABC$, points $A_{1}$ and $A_{2}$ are chosen in the prolongations beyond $A$ of segments $AB$ and $AC$, such that $AA_{1}=AA_{2}=BC$. Define analogously points $B_{1}$, $B_{2}$, $C_{1}$, $C_{2}$. If $[ABC]$ denotes the area of triangle $ABC$, show that $[A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}] \geq 13 [ABC]$.

The bisectors of the angles $ CAB, ABC, BCA $ of the triangle $ ABC $ intersect the circle circumcribed around this triangle at points $ K, L, M $, respectively. Prove that $$ AK+BL+CM > AB+BC+CA.$$