Let $ n$ be positive integer. For $ n \equal{} 1,\ 2,\ 3,\ \cdots n$, let denote $ S_k$ be the area of $ \triangle{AOB_k}$ such that $ \angle{AOB_k} \equal{} \frac {k}{2n}\pi ,\ OA \equal{} 1,\ OB_k \equal{} k$. Find the limit $ \lim_{n\to\infty}\frac {1}{n^2}\sum_{k \equal{} 1}^n S_k$.

If the inradius of a triangle is half of its circumradius, prove that the triangle is equilateral.

Given two circles $\Gamma_1$ and $\Gamma_2$ which intersect at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ at points $C$ and $D$, respectively. Let $M$ be the midpoint of arc $BC$ in $\Gamma_1$ ,which does not contains $A$, and $N$ is the midpoint of the arc $BD$ in $\Gamma_2$, which does not contain $A$. If $K$ is the midpoint of $CD$, prove that $\angle MKN = 90^o.$

Let $ABCDEF$ be a convex hexagon with $AB = BC, CD = DE$ and $EF = FA$. Prove that the lines through $C,E,A$ perpendicular to $BD,DF,FB$ are concurrent.

Pentagon $ABCDE$ is given with the following conditions: (a) $\angle CBD + \angle DAE = \angle BAD = 45^o$, $\angle BCD + \angle DEA = 300^o$ (b) $\frac{BA}{DA} =\frac{ 2\sqrt2}{3}$ , $CD =\frac{ 7\sqrt5}{3} $, and $DE = \frac{15\sqrt2}{4}$ (c) $AD^2 \cdot BC = AB \cdot AE \cdot BD$ Compute $BD$.

Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different)

Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$? $\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad$

Point $P$ is located inside traingle $ABC$ so that angles $PAB, PBC,$ and $PCA$ are all congruent. The sides of the triangle have lengths $AB=13, BC=14,$ and $CA=15,$ and the tangent of angle $PAB$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Let $ABCM$ be a quadrilateral and $D$ be an interior point such that $ABCD$ is a parallelogram. It is known that $\angle AMB =\angle CMD$. Prove that $\angle MAD =\angle MCD$.

In acute-angled triangle $ABC, I$ is the center of the inscribed circle and holds $| AC | + | AI | = | BC |$. Prove that $\angle BAC = 2 \angle ABC$.

Let $n$ be an integer greater than 1. Suppose $2n$ points are given in the plane, no three of which are collinear. Suppose $n$ of the given $2n$ points are colored blue and the other $n$ colored red. A line in the plane is called a [i]balancing line[/i] if it passes through one blue and one red point and, for each side of the line, the number of blue points on that side is equal to the number of red points on the same side. Prove that there exist at least two balancing lines.

Let $ABC$ be a triangle and $I$ its incentre. Let $\varrho_1$ and $\varrho_2$ be the inradii of triangles $IAB$ and $IAC$ respectively. (a) Show that there exists a function $f: ( 0, \pi ) \mapsto \mathbb{R}$ such that \[ \frac{ \varrho_1}{ \varrho_2} = \frac{f(C)}{f(B)} \] where $B = \angle ABC$ and $C = \angle BCA$ (b) Prove that \[ 2 ( \sqrt{2} -1 ) < \frac{ \varrho_1} { \varrho_2} < \frac{ 1 + \sqrt{2}}{2} \]

We have to divide a cube onto $k$ non-overlapping tetrahedrons. For what smallest $k$ is it possible?

Let $ABC$ be a triangle with $ \angle ACB = 90^o + \frac12 \angle ABC$ . The point $M$ is the midpoint of the side $BC$ . A circle with center at vertex $A$ intersects the line $BC$ at points $M$ and $D$. Prove that $MD = AB$.

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [u]Set 1[/u] [b]D.1 / L.1[/b] Find the units digit of $3^{1^{3^{3^7}}}$. [b]D.2[/b] Find the positive solution to the equation $x^3 - x^2 = x - 1$. [b]D.3[/b] Points $A$ and $B$ lie on a unit circle centered at O and are distance $1$ apart. What is the degree measure of $\angle AOB$? [b]D.4[/b] A number is a perfect square if it is equal to an integer multiplied by itself. How many perfect squares are there between $1$ and $2019$, inclusive? [b]D.5[/b] Ted has four children of ages $10$, $12$, $15$, and $17$. In fifteen years, the sum of the ages of his children will be twice Ted’s age in fifteen years. How old is Ted now? [u]Set 2[/u] [b]D.6[/b] Mr. Schwartz is on the show Wipeout, and is standing on the first of $5$ balls, all in a row. To reach the finish, he has to jump onto each of the balls and collect the prize on the final ball. The probability that he makes a jump from a ball to the next is $1/2$, and if he doesn’t make the jump he will wipe out and no longer be able to finish. Find the probability that he will finish. [b]D.7 / L. 5[/b] Kevin has written $5$ MBMT questions. The shortest question is $5$ words long, and every other question has exactly twice as many words as a different question. Given that no two questions have the same number of words, how many words long is the longest question? [b]D.8 / L. 3[/b] Square $ABCD$ with side length $1$ is rolled into a cylinder by attaching side $AD$ to side $BC$. What is the volume of that cylinder? [b]D.9 / L.4[/b] Haydn is selling pies to Grace. He has $4$ pumpkin pies, $3$ apple pies, and $1$ blueberry pie. If Grace wants $3$ pies, how many different pie orders can she have? [b]D.10[/b] Daniel has enough dough to make $8$ $12$-inch pizzas and $12$ $8$-inch pizzas. However, he only wants to make $10$-inch pizzas. At most how many $10$-inch pizzas can he make? [u]Set 3[/u] [b]D.11 / L.2[/b] A standard deck of cards contains $13$ cards of each suit (clubs, diamonds, hearts, and spades). After drawing $51$ cards from a randomly ordered deck, what is the probability that you have drawn an odd number of clubs? [b]D.12 / L. 7[/b] Let $s(n)$ be the sum of the digits of $n$. Let $g(n)$ be the number of times s must be applied to n until it has only $1$ digit. Find the smallest n greater than $2019$ such that $g(n) \ne g(n + 1)$. [b]D.13 / L. 8[/b] In the Montgomery Blair Meterology Tournament, individuals are ranked (without ties) in ten categories. Their overall score is their average rank, and the person with the lowest overall score wins. Alice, one of the $2019$ contestants, is secretly told that her score is $S$. Based on this information, she deduces that she has won first place, without tying with anyone. What is the maximum possible value of $S$? [b]D.14 / L. 9[/b] Let $A$ and $B$ be opposite vertices on a cube with side length $1$, and let $X$ be a point on that cube. Given that the distance along the surface of the cube from $A$ to $X$ is $1$, find the maximum possible distance along the surface of the cube from $B$ to $X$. [b]D.15[/b] A function $f$ with $f(2) > 0$ satisfies the identity $f(ab) = f(a) + f(b)$ for all $a, b > 0$. Compute $\frac{f(2^{2019})}{f(23)}$. PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here [/url] and L10,16-30 [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In triangle $ABC$, $\angle{BAC}=15^{\circ}$. The circumcenter $O$ of triangle $ABC$ lies in its interior. Find $\angle{OBC}$. [asy] size(3cm,0); dot((0,0)); draw(Circle((0,0),1)); draw(dir(70)--dir(220)); draw(dir(220)--dir(310)); draw(dir(310)--dir(70)); draw((0,0)--dir(220)); label("$A$",dir(70),NE); label("$B$",dir(220),SW); label("$C$",dir(310),SE); label("$O$",(0,0),NE); [/asy] $\text{(A) }30^{\circ}\qquad\text{(B) }75^{\circ}\qquad\text{(C) }45^{\circ}\qquad\text{(D) }60^{\circ}\qquad\text{(E) }15^{\circ}$

An equilateral triangle of side $1$ is covered by five congruent equilateral triangles of side $s < 1$ with sides parallel to those of the larger triangle. Show that some four of these smaller triangles also cover the large triangle.

In a certain geometry we operate with two types of elements, points and lines, related to each other by the following axioms: [b]I.[/b] Given two points $A$ and $B$, there is a unique line $(AB)$ that passes through both. [b]II. [/b]There are at least two points on a line. There are three points not situated on a straight line. [b]III.[/b] When a point $B$ is located between $A$ and $C$, then $B$ is also between $C$ and $A$. ($A, B, C$ are three different points on a line.) [b]IV.[/b] Given two points $A$ and $C$, there exists at least one point $B$ on the line $(AC)$ of the form that C is between $A$ and $B$. [b]V.[/b] Among three points located on the same straight line, one at most is between the other two. [b]VI.[/b] If $A, B, C$ are three points not lying on the same line and a is a line that does not contain any of the three, when the line passes through a point on segment [AB] , then it goes through one of the $[BC]$ , or it goes through one of the [AC] . (We designate by [AB] the set of points that lie between $A$ and $B$.) From the previous axioms, prove the following propositions: Theorem 1. Between points A and C there is at least one point $B$. Theorem 2. Among three points located on a line, one is always between the two others.

Let $A_1A_2 . . . A_n$ be a cyclic convex polygon whose circumcenter is strictly in its interior. Let $B_1, B_2, ..., B_n$ be arbitrary points on the sides $A_1A_2, A_2A_3, ..., A_nA_1$, respectively, other than the vertices. Prove that $\frac{B_1B_2}{A_1A_3}+ \frac{B_2B_3}{A_2A_4}+...+\frac{B_nB_1}{A_nA_2}>1$.

The incircle $\odot (I)$ of $\triangle ABC$ touch $AC$ and $AB$ at $E$ and $F$ respectively. Let $H$ be the foot of the altitude from $A$, if $R \equiv IC \cap AH, \ \ Q \equiv BI \cap AH$ prove that the midpoint of $AH$ lies on the radical axis between $\odot (REC)$ and $\odot (QFB)$ I hope that this is not repost :)

Three (not necessarily distinct) points in the plane which have integer coordinates between $ 1$ and $2020$, inclusive, are chosen uniformly at random. The probability that the area of the triangle with these three vertices is an integer is $a/b$ in lowest terms. If the three points are collinear, the area of the degenerate triangle is $0$. Find $a + b$.

Parallelogram ${ABCD}$ is given with ${AC>BD}$, and ${O}$ intersection point of ${AC}$ and ${BD}$. Circle with center at ${O}$and radius ${OA}$ intersects extensions of ${AD}$and ${AB}$at points ${G}$ and ${L}$, respectively. Let ${Z}$ be intersection point of lines ${BD}$and ${GL}$. Prove that $\angle ZCA={{90}^{{}^\circ }}$.

In triangle $ABC$, the altitude $AH$ passes through midpoint of the median $BM$. Prove that in the triangle $BMC$ also one of the altitudes passes through the midpoint of one of the medians.

In a convex quadrilateral $ABCD$, ${\angle}ABD=65^\circ$,${\angle}CBD=35^\circ$, ${\angle}ADC=130^\circ$ and $BC=AB$.Find the angles of $ABCD$.

Let $ABCD$ be a trapezoid with bases $AB$ and $CD$, inscribed in a circle of center $O$. Let $P$ be the intersection of the lines $BC$ and $AD$. A circle through $O$ and $P$ intersects the segments $BC$ and $AD$ at interior points $F$ and $G$, respectively. Show that $BF=DG$.