Let $ABC$ be a right triangle with $\angle A=90^o$ and $AB<AC$. Let $AH,AD,AM$ be altitude, angle bisector and median respectively. Prove that $\frac{BD}{CD}<\frac{HD}{MD}.$

A Mathlon is a competition where there are $M$ athletic events. $A, B$ and $C$ were the only participants of a Mathlon. In each event, $p_1$ points were given to the first place, $p_2$ points to the second place and $p_3$ points to third place, with $p_1> p_2> p_3> 0$ where $p_1$, $p_2$ and $p_3$ are integer numbers. The final result was $22$ points for $A$, $9$ for $B$, and $9$ for $C$. $B$ won the $100$ meter dash. Determine $M$ and who was the second in high jump.

Given $ \triangle ABC$ with point $ D$ inside. Let $ A_0\equal{}AD\cap BC$, $ B_0\equal{}BD\cap AC$, $ C_0 \equal{}CD\cap AB$ and $ A_1$, $ B_1$, $ C_1$, $ A_2$, $ B_2$, $ C_2$ are midpoints of $ BC$, $ AC$, $ AB$, $ AD$, $ BD$, $ CD$ respectively. Two lines parallel to $ A_1A_2$ and $ C_1C_2$ and passes through point $ B_0$ intersects $ B_1B_2$ in points $ A_3$ and $ C_3$respectively. Prove that $ \frac{A_3B_1}{A_3B_2}\equal{}\frac{C_3B_1}{C_3B_2}$.

John found all real numbers $p$ such that in the polynomial $g(x) = (x -1)^2(p + 2x)^2$ , the quadratic term has coefficient $2021$. What is the sum of all of these values $p$?

In an isosceles triangle $ABC$ ($AB = BC$), point $O$ is the center of the circumcircle. Point $M$ lies on the segment $BO$, point $M' $ is symmetric to $M$ wrt the midpoint of $AB$. Point K is the intersection point of of $M'O$ and $AB$. Point $L$ lies on side BC such that $\angle CLO = \angle BLM$. Prove that points $O, K,B,L$ lie on the same circle

A circle inscribed in the square $ABCD$, with side $10$ cm, intersects sides $BC$ and $AD$ at points $M$ and $N$ respectively. The point $I$ is the intersection of $AM$ with the circle different from $M$, and $P$ is the orthogonal projection of $I$ into $MN$. Find the value of segment $PI$.

Find the number of $(a_1,a_2, ... ,a_{2014})$ permutations of the $(1,2, . . . ,2014)$ such that, for all $1\leq i<j\leq2014$, $i+a_i \leq j+a_j$.

Let $k$ be a positive integer. At the European Chess Cup every pair of players played a game in which somebody won (there were no draws). For any $k$ players there was a player against whom they all lost, and the number of players was the least possible for such $k$. Is it possible that at the Closing Ceremony all the participants were seated at the round table in such a way that every participant was seated next to both a person he won against and a person he lost against. [i]Proposed by Matija Bucić.[/i]

Given a convex quadrilateral $ABCD$, let $P$ and $Q$ be points on $BC$ and $CD$ respectively such that $\angle BAP = \angle DAQ$. Prove that the triangles $ABP$ and $ADQ$ have the same area if the line connecting their orthocenters is perpendicular to $AC$.

Evaluate $ \int_0^{n\pi} e^x\sin ^ 4 x\ dx\ (n\equal{}1,\ 2,\ \cdots).$

Let $x_0$, $x_1$, $\ldots$, $x_{1368}$ be complex numbers. For an integer $m$, let $d(m)$, $r(m)$ be the unique integers satisfying $0\leq r(m) < 37$ and $m = 37d(m) + r(m)$. Define the $1369\times 1369$ matrix $A = \{a_{i,j}\}_{0\leq i, j\leq 1368}$ as follows: \[ a_{i,j} = \begin{cases} x_{37d(j)+d(i)} & r(i) = r(j),\ i\neq j\\ -x_{37r(i)+r(j)} & d(i) = d(j),\ i \neq j \\ x_{38d(i)} - x_{38r(i)} & i = j \\ 0 & \text{otherwise} \end{cases}. \]We say $A$ is $r$-\emph{murine} if there exists a $1369\times 1369$ matrix $M$ such that $r$ columns of $MA-I_{1369}$ are filled with zeroes, where $I_{1369}$ is the identity $1369\times 1369$ matrix. Let $\operatorname{rk}(A)$ be the maximum $r$ such that $A$ is $r$-murine. Let $S$ be the set of possible values of $\operatorname{rk}(A)$ as $\{x_i\}$ varies. Compute the sum of the $15$ smallest elements of $S$. [i]Proposed by Brandon Wang[/i]

Find the amplitude of $y = 4 \sin (x) + 3 \cos (x)$.

Determine whether there exist $100$ disks $D_2,D_3,\ldots ,D_{101}$ in the plane such that the following conditions hold for all pairs $(a,b)$ of indices satisfying $2\le a< b\le 101$: [list] [*] If $a|b$ then $D_a$ is contained in $D_b$. [*] If $\gcd (a,b)=1$ then $D_a$ and $D_b$ are disjoint. [/list] (A disk $D(O,r)$ is a set of points in the plane whose distance to a given point $O$ is at most a given positive real number $r$.)

A quadrilateral is given, in which the lengths of some two sides are equal to $1$ and $4$. Also, the diagonal of length $2$ divides it into two isosceles triangles. Find the perimeter of this quadrilateral.

Find the sum of all integers $m$ with $1 \le m \le 300$ such that for any integer $n$ with $n \ge 2$, if $2013m$ divides $n^n-1$ then $2013m$ also divides $n-1$. [i]Proposed by Evan Chen[/i]

A scout troop buys $ 1000$ candy bars at a price of five for $ \$2$. They sell all the candy bars at a price of two for $ \$1$. What was their profit, in dollars? $ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 200 \qquad \textbf{(C)}\ 300 \qquad \textbf{(D)}\ 400 \qquad \textbf{(E)}\ 500$

$12$ students need to form five study groups. They will form three study groups with $2$ students each and two study groups with $3$ students each. In how many ways can these groups be formed?

In a triangle $ABC$ , let $P$ be a point on its circumscribed circle (on the arc $AC$ that does not contain $B$). Let $H,H_1,H_2$ and $H_3$ be the orthocenters of triangles $ABC, BCP, ACP$ and $ABP$, respectively. Let $L = PB \cap AC$ and $J = HH_2 \cap H_1H_3$. If $M$ and $N$ are the midpoints of $JH$ and $LP$, respectively, prove that $MN$ and $JL$ intersect at their midpoint.

Let $ K $ be the group of Klein. Prove that: [b]a)[/b] There is an unique division ring (up to isomorphism), $ D, $ such that $ (D,+)\cong K. $ [b]b)[/b] There are no division rings $ A $ such that $ (A\setminus\{ 0\} ,+)\cong K. $

Two finite sequences $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ are just rearranged sequence $1, 1/2, ... , 1/n$ with $$a_1+b_1\ge a_2+b_2\ge...\ge a_n+b_n.$$ Prove that $a_m+a_n\ge 4/m$ for every $m$ ($1\le m\le n$) .

Find all positive integers $n$ with the property that the set \[\{n,n+1,n+2,n+3,n+4,n+5\}\] can be partitioned into two sets such that the product of the numbers in one set equals the product of the numbers in the other set.

Let $a, b, c, n$ be four integers, where n$\ge 2$, and let $p$ be a prime dividing both $a^2+ab+b^2$ and $a^n+b^n+c^n$, but not $a+b+c$. for instance, $a \equiv b \equiv -1 (mod \,\, 3), c \equiv 1 (mod \,\, 3), n$ a positive even integer, and $p = 3$ or $a = 4, b = 7, c = -13, n = 5$, and $p = 31$ satisfy these conditions. Show that $n$ and $p - 1$ are not coprime.

On the grid board shown, a token is placed on each white space. [img]https://cdn.artofproblemsolving.com/attachments/3/2/0060b2436edb0ce25160d2f94f379defef237c.png[/img] A move consists of choosing four squares on the board that form a "$T$" in any of the shapes shown below, and add a token to each of these four squares. [img]https://cdn.artofproblemsolving.com/attachments/8/c/3890aed5289ec9ea2d147f8000a0422c233029.png[/img] Is it possible, after carrying out several moves, to get the $25$ squares to have the same amount of chips?

An acute triangle $ABC$ is given. Let $D$ be the foot of altitude dropped for $A$. Tangents from $D$ to circles with diameters $AB$ and $AC$ intersects with the said circles at $K$ and $L$, in respective. Point $S$ in the plane is given so that $\angle ABC + \angle ABS = \angle ACB + \angle ACS = 180^\circ$. Prove that $A, K, L$ and $S$ lie on a circle.

We randomly choose 5 distinct positive integers less than or equal to 90. What is the floor of 10 times the expected value of the fourth largest number?