Let $A_1A_2 \dots A_{4000}$ be a regular $4000$-gon. Let $X$ be the foot of the altitude from $A_{1986}$ onto diagonal $A_{1000}A_{3000}$, and let $Y$ be the foot of the altitude from $A_{2014}$ onto $A_{2000}A_{4000}$. If $XY = 1$, what is the area of square $A_{500}A_{1500}A_{2500}A_{3500}$? [i]Proposed by Evan Chen[/i]

A convex pentagon $ABCDE$ is given with $AB=BC$, $CD=DE$ and $\angle A=\angle C=\angle E>90^{\circ}$. Prove that the pentagon is circumscribed [I]Proposed by F. Baharev[/i]

If a triangle has side lengths $a,b,c$ where $a\leq 2 \leq b \leq 3$, what is the largest possible value of its area? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ \text{None of above} $

The circular base of a hemisphere of radius $2$ rests on the base of a square pyramid of height $6$. The hemisphere is tangent to the other four faces of the pyramid. What is the edge-length of the base of the pyramid? $ \textbf{(A)}\ 3\sqrt{2} \qquad \textbf{(B)}\ \frac{13}{3} \qquad \textbf{(C)}\ 4\sqrt{2} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ \frac{13}{2} $

$AH$ is the altitude of triangle $ABC$ and $H^\prime$ is the reflection of $H$ trough the midpoint of $BC$. If the tangent lines to the circumcircle of $ABC$ at $B$ and $C$, intersect each other at $X$ and the perpendicular line to $XH^\prime$ at $H^\prime$, intersects $AB$ and $AC$ at $Y$ and $Z$ respectively, prove that $\angle ZXC=\angle YXB$.

Point $D$ on the side $BC$ of the acute triangle $ABC$ is chosen so that $AD = AC$. Let $P$ and $Q$ be the feet of the perpendiculars from $C$ and $D$ on the side $AB$, respectively. Suppose that $AP^2 + 3BP^2 = AQ^2 + 3BQ^2$. Determine the measure of angle $\angle ABC$.

[u]Round 6[/u] 16. If you roll four fair $6$-sided dice, what is the probability that at least three of them will show the same value. 17. In a tetrahedron with volume $1$, four congruent speres are placed each tangent to three walls and three other spheres. What is the radii of each of the spheres. 18. let $f(x)$ be twice the number of letters in $x$. What is the sum of the unique $x,y$ such that $x \ne y$ and $f(x)=y$ and $f(y)=x$. [u]Round 7[/u] [b]p19.[/b] How many $4$ digit numbers with distinct digits $ABCD$ with $A$ not equal to $0$ are divisible by $11$? [b]p20.[/b] How many ($2$-dimensional) faces does a $2014$-dimensional hypercube have? [b]p21.[/b] How many subsets of the numbers $1,2,3,4...2^{2014}$ have a sum of $2014$ mod $2^{2014}$? [u]Round 8[/u] [b]p22.[/b] Two diagonals of a dodecagon measure $1$ unit and $2$ units. What is the area of this dodecagon? [b]p23.[/b] Square $ABCD$ has point $X$ on AB and $Y$ on $BC$ such that angle $ADX = 15$ degrees and angle $CDY = 30$ degrees. what is the degree measure of angle $DXY$? [b]p24.[/b] A $4\times 4$ grid has the numbers $1$ through $16$ placed in it, $1$ per cell, such that no adjacent boxes have cells adding to a number divisible by $3$. In how many ways is this possible? [u]Round 9[/u] [b]p25.[/b] Let $B$ and $C$ be the answers to $26$ and $27$ respectively.If $S(x)$ is the sum of the digits in $x$, what is the unique integer $A$ such that $S(A), S(B), S(C) \subset A,B,C$. [b]p26.[/b] Let $A$ and $C$ be the answers to $25$ and $27$ respectively. What is the third angle of a triangle with two of its angles equal to $A$ and $C$ degrees. [b]p27.[/b] Let $A$ and $B$ be the answers to $25$ and $26$ respectively. How many ways are there to put $A$ people in a line, with exactly $B$ places where a girl and a boy are next to each other. [u]Round 10[/u] [b]p28.[/b] What is the sum of all the squares of the digits to answers to problems on the individual, team, and theme rounds of this years LMT? If the correct answer is $N$ and you submit $M$, you will recieve $\lfloor 15 - 10  \times \log (M - N) \rfloor $. [b]p29.[/b] How many primes have all distinct digits, like $2$ or $109$ for example, but not $101$. If the correct answer is $N$ and you submit $M$, you will recieve $\left\lfloor 15 \min \left( \frac{M}{N} , \frac{N}{M} \right)\right\rfloor $. [b]p30.[/b] For this problem, you can use any $10$ mathematical symbols that you want, to try to achieve the highest possible finite number. (So "Twenty-one", " $\frac{12}{100} +843$" and "$\sum^{10}_{i=0} i^2 +1$" are all valid submissions.) If your team has the $N$th highest number, you will recieve $\max (16 - N, 0)$. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3156859p28695035]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A non-square rectangle is cut into $N$ rectangles of various shapes and sizes. Prove that one can always cut each of these rectangles into two rectangles so that one can construct a square and rectangle, each figure consisting of $N$ pieces. [i](6 points)[/i]

The figure shows an arc $\ell$ on the unit circle and two regions $A$ and $B$. Prove that the area of $A$ plus the area of $B$ equals the length of $\ell$. [img]https://1.bp.blogspot.com/-SYoSrFowZ30/XzRz0ygiOVI/AAAAAAAAMUs/0FCduUoxKGwq0gSR-b3dtb3SvDjZ89x_ACLcBGAsYHQ/s0/2017%2BMohr%2Bp3.png[/img]

Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$, $M_3$, $M_5$, and $M_7$ be the midpoints of sides $\overline{A_1 A_2}$, $\overline{A_3 A_4}$, $\overline{A_5 A_6}$, and $\overline{A_7 A_8}$, respectively. For $i = 1, 3, 5, 7$, ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \perp R_3$, $R_3 \perp R_5$, $R_5 \perp R_7$, and $R_7 \perp R_1$. Pairs of rays $R_1$ and $R_3$, $R_3$ and $R_5$, $R_5$ and $R_7$, and $R_7$ and $R_1$ meet at $B_1$, $B_3$, $B_5$, $B_7$ respectively. If $B_1 B_3 = A_1 A_2$, then $\cos 2 \angle A_3 M_3 B_1$ can be written in the form $m - \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.

A triangle has inradius $r$ and circumradius $R$. Its longest altitude has length $H$. Show that if the triangle does not have an obtuse angle, then $H \ge r+R$. When does equality hold?

Given is a triangle $ABC$ with barycenter $G$ and circumcenter $O$.The perpendicular bisectors of $GA,GB,GC$ intersect at $A_1,B_1,C_1$.Show that $O$ is the barycenter of $\triangle{A_1B_1C_1}$.

Let $O$ be the centre of a regular $n$-gon whose vertices are labelled $A_1$,$...$, $A_n$. Let $a_1>a_2>...>a_n>0$. Prove that the vector $$a_1\overrightarrow{OA_1}+a_2\overrightarrow{OA_2}+...+a_n\overrightarrow{OA_n}$$ is not equal to the zero vector. (D. Fomin, Alexey Kirichenko)

Given a triangle $ABC$ such that $AB<AC$ . On sides $AB$ and $BC$, points $P$ and $Q$ are marked respectively such that the lines $AQ$ and $CP$ are perpendicular, and the circle inscribed in the triangle $ABC$ touches the length $PQ$. The line $CP$ intersects the circle circumscribed around the triangle $ABC$ at the points $C$ and $T$. If the lines $CA,PQ$ and $BT$ intersect at one point, prove that the angle $\angle CAB$ is right.

Let $ CL$ be a bisector of triangle $ ABC$. Points $ A_1$ and $ B_1$ are the reflections of $ A$ and $ B$ in $ CL$, points $ A_2$ and $ B_2$ are the reflections of $ A$ and $ B$ in $ L$. Let $ O_1$ and $ O_2$ be the circumcenters of triangles $ AB_1B_2$ and $ BA_1A_2$ respectively. Prove that angles $ O_1CA$ and $ O_2CB$ are equal.

$AB$ is the diameter of circle $O$. A random point $P$ is selected on $O$ so that $AP = 4$ and $BP = 3$. Points $C$ and $D$ are drawn on circle $O$ so that $OC$ bisects $AP$ and $OD$ bisects $BP$. What is the degree measure of $\angle COD$?

Let $ABCD$ be a quadrilateral such that $\angle A=\angle C=90^{\circ}$. If $A, D$ and the midpoints of $BA, BC$ are concyclic, show that the midpoints of $AD, DC$ and $B, C$ are concyclic.

It is known that $\triangle ABC$ is an acute triangle. Let $C'$ be the foott of the perpendicular from $C$ to $AB$, and $D$, $E$ two distinct points on $CC'$. The feet of the perpendiculars from $D$ to $AC$ and $BC$ are $F$ and $G$, respectively. Show that if $DGEF$ is a parallelogram then $ABC$ is isosceles.

[u]Round 5[/u] [b]5.1.[/b] Quadrilateral $ABCD$ is such that $\angle ABC = \angle ADC = 90^o$ , $\angle BAD = 150^o$ , $AD = 3$, and $AB = \sqrt3$. The area of $ABCD$ can be expressed as $p\sqrt{q}$ for positive integers $p, q$ where $q$ is not divisible by the square of any prime. Find $p + q$. [b]5.2.[/b] Neetin wants to gamble, so his friend Akshay describes a game to him. The game will consist of three dice: a $100$-sided one with the numbers $1$ to $100$, a tetrahedral one with the numbers $1$ to $4$, and a normal $6$-sided die. If Neetin rolls numbers with a product that is divisible by $21$, he wins. Otherwise, he pays Akshay $100$ dollars. The number of dollars that Akshay must pay Neetin for a win in order to make this game fair is $a/b$ for relatively prime positive integers $a, b$. Find $a + b$. (Fair means the expected net gain is $0$. ) [b]5.3.[/b] What is the sum of the fourth powers of the roots of the polynomial $P(x) = x^2 + 2x + 3$? [u]Round 6[/u] [b]6.1.[/b] Consider the set $S = \{1, 2, 3, 4,..., 25\}$. How many ordered $n$-tuples $S_1 = (a_1, a_2, a_3,..., a_n)$ of pairwise distinct ai exist such that $a_i \in S$ and $i^2 | a_i$ for all $1 \le i \le n$? [b]6.2.[/b] How many ways are there to place $2$ identical rooks and $ 1$ queen on a $ 4 \times 4$ chessboard such that no piece attacks another piece? (A queen can move diagonally, vertically or horizontally and a rook can move vertically or horizontally) [b]6.3.[/b] Let $L$ be an ordered list $\ell_1$, $\ell_2$, $...$, $\ell_{36}$ of consecutive positive integers who all have the sum of their digits not divisible by $11$. It is given that $\ell_1$ is the least element of $L$. Find the least possible value of $\ell_1$. [u]Round 7[/u] [b]7.1.[/b] Spencer, Candice, and Heather love to play cards, but they especially love the highest cards in the deck - the face cards (jacks, queens, and kings). They also each have a unique favorite suit: Spencer’s favorite suit is spades, Candice’s favorite suit is clubs, and Heather’s favorite suit is hearts. A dealer pulls out the $9$ face cards from every suit except the diamonds and wants to deal them out to the $3$ friends. How many ways can he do this so that none of the $3$ friends will see a single card that is part of their favorite suit? [b]7.2.[/b] Suppose a sequence of integers satisfies the recurrence $a_{n+3} = 7a_{n+2} - 14a_{n+1} + 8a_n$. If $a_0 = 4$, $a_1 = 9$, and $a_2 = 25$, find $a_{16}$. Your answer will be in the form $2^a + 2^b + c$, where $2^a < a_{16} < 2^{a+1}$ and $b$ is as large as possible. Find $a + b + c$. [b]7.3.[/b] Parallel lines $\ell_1$ and $\ell_2$ are $1$ unit apart. Unit square $WXYZ$ lies in the same plane with vertex $W$ on $\ell_1$. Line $\ell_2$ intersects segments $YX$ and $YZ$ at points $U$ and $O$, respectively. Given $UO =\frac{9}{10}$, the inradius of $\vartriangle YOU$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$. [u]Round 8[/u] [b]8.[/b] Let $A$ be the number of contestants who participated in at least one of the three rounds of the 2020 ABMC April contest. Let $B$ be the number of times the letter b appears in the Accuracy Round. Let $M$ be the number of people who submitted both the speed and accuracy rounds before 2:00 PM EST. Further, let $C$ be the number of times the letter c appears in the Speed Round. Estimate $$A \cdot B + M \cdot C.$$Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2766239p24226402]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The bases of a trapezoid have lengths 10 and 21, and the legs have lengths $\sqrt{34}$ and $3 \sqrt{5}$. What is the area of the trapezoid?

[b]p1.[/b] $ABCD$ is a convex quadrilateral such that $AB = 20$, $BC = 24$, $CD = 7$, $DA = 15$, and $\angle DAB$ is a right angle. What is the area of $ABCD$? [b]p2.[/b] A triangular number is one that can be written in the form $1 + 2 +...·+n$ for some positive number $n$. $ 1$ is clearly both triangular and square. What is the next largest number that is both triangular and square? [b]p3.[/b] Find the last (i.e. rightmost) three digits of $9^{2008}$. [b]p4.[/b] When expressing numbers in a base $b \ge 11$, you use letters to represent digits greater than $9$. For example, $A$ represents $10$ and $B$ represents $11$, so that the number $110$ in base $10$ is $A0$ in base $11$. What is the smallest positive integer that has four digits when written in base $10$, has at least one letter in its base $12$ representation, and no letters in its base $16$ representation? [b]p5.[/b] A fly starts from the point $(0, 16)$, then flies straight to the point $(8, 0)$, then straight to the point $(0, -4)$, then straight to the point $(-2, 0)$, and so on, spiraling to the origin, each time intersecting the coordinate axes at a point half as far from the origin as its previous intercept. If the fly flies at a constant speed of $2$ units per second, how many seconds will it take the fly to reach the origin? [b]p6.[/b] A line segment is divided into two unequal lengths so that the ratio of the length of the short part to the length of the long part is the same as the ratio of the length of the long part to the length of the whole line segment. Let $D$ be this ratio. Compute $$D^{-1} + D^{[D^{-1}+D^{(D^{-1}+D^2)}]}.$$ [b]p7.[/b] Let $f(x) = 4x + 2$. Find the ordered pair of integers $(P, Q)$ such that their greatest common divisor is $1, P$ is positive, and for any two real numbers $a$ and $b$, the sentence: “$P a + Qb \ge 0$” is true if and only if the following sentence is true: “For all real numbers x, if $|f(x) - 6| < b$, then $|x - 1| < a$.” [b]p8.[/b] Call a rectangle “simple” if all four of its vertices have integers as both of their coordinates and has one vertex at the origin. How many simple rectangles are there whose area is less than or equal to $6$? [b]p9.[/b] A square is divided into eight congruent triangles by the diagonals and the perpendicular bisectors of its sides. How many ways are there to color the triangles red and blue if two ways that are reflections or rotations of each other are considered the same? [b]p10.[/b] In chess, a knight can move by jumping to any square whose center is $\sqrt5$ units away from the center of the square that it is currently on. For example, a knight on the square marked by the horse in the diagram below can move to any of the squares marked with an “X” and to no other squares. How many ways can a knight on the square marked by the horse in the diagram move to the square with a circle in exactly four moves? [img]https://cdn.artofproblemsolving.com/attachments/d/9/2ef9939642362182af12089f95836d4e294725.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The incircle of a right triangle $ABC$ touches the hypotenuse $AB$ at a point $D$. Show that the area of $\triangle ABC$ equals $AD\cdot DB$.

1-Let $ \triangle ABC$ be a triangle and $ (O)$ its circumcircle. $ D$ is the midpoint of arc $ BC$ which doesn't contain $ A$. We draw a circle $ W$ that is tangent internally to $ (O)$ at $ D$ and tangent to $ BC$.We draw the tangent $ AT$ from $ A$ to circle $ W$.$ P$ is taken on $ AB$ such that $ AP \equal{} AT$.$ P$ and $ T$ are at the same side wrt $ A$.PROVE $ \angle APD \equal{} 90^\circ$.

Triangle $ABC$ is given. Consider ellipse $ \Omega _1$, passes through $C$ with focuses in $A$ and $B$. Similarly define ellipses $ \Omega _2 , \Omega _3$ with focuses $B,C$ and $C,A$ respectively. Prove, that if all ellipses have common point $D$ then $A,B,C,D$ lies on the circle. Ellipse with focuses $X,Y$, passes through $Z$- locus of point $T$, such that $XT+YT=XZ+YZ$

Let $ABC$ be an acute scalene triangle with orthocenter $H$. Let $M$ be the midpoint of $BC$. $N$ is the point on line $AM$ such that $(BMN)$ is tangent to $AB$. Finally, let $H'$ be the reflection of $H$ in $B$. Prove that $\angle ANH'=90^{\circ}$. [i]Proposed by Malay Mahajan and Siddharth Choppara[/i]