Let $ABCD$ be a convex quadrilateral, and let $K$ be a point on side$ AB$ such that $\angle KDA = \angle BCD$. Let $L$ be a point on the diagonal $AC$ such that $KL \parallel BC$. Prove that $\angle KDB = \angle LDC$.

If $0 < x < \frac{\pi}{2}$ and $\frac{\sin x}{1 + \cos x} = \frac{1}{3}$, what is $\frac{\sin 2x}{1 + \cos 2x}$?

Jay has a $24\times 24$ grid of lights, all of which are initially off. Each of the $48$ rows and columns has a switch that toggles all the lights in that row and column, respectively, i.e. it switches lights that are on to off and lights that are off to on. Jay toggles each of the $48$ rows and columns exactly once, such that after each toggle he waits for one minute before the next toggle. Each light uses no energy while off and 1 kiloJoule of energy per minute while on. To express his creativity, Jay chooses to toggle the rows and columns in a random order. Compute the expected value of the total amount of energy in kiloJoules which has been expended by all the lights after all $48$ toggles. [i]Proposed by James Lin

Given a triangle $ABC$ and an arbitrary point $D$.The lines passing through $D$ and perpendicular to segments $DA, DB, DC$ meet lines $BC, AC, AB$ at points $A_1, B_1, C_1$ respectively. Prove that the midpoints of segments $AA_1, BB_1, CC_1$ are collinear.

Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that \[ x f(x \plus{} xy) \equal{} x f(x) \plus{} f \left( x^2 \right) f(y) \quad \forall x,y \in \mathbb{R}.\]

Let $ABC$ a triangle and let $\omega$ be its circumcircle. Let $E{}$ be the midpoint of the minor arc $BC$ of $\omega$, and $M{}$ the midpoint of $BC$. Let $V$ be the other point of intersection of $AM$ with $\omega$, $F{}$ the point of intersection of $AE$ with $BC$, $X{}$ the other point of intersection of the circumcircle of $FEM$ with $\omega$, $X'$ the reflection of $V{}$ with respect to $M{}$, $A'{}$ the foot of the perpendicular from $A{}$ to $BC$ and $S{}$ the other point of intersection of $XA'$ with $\omega$. If $Z \in \omega$ with $Z\neq X$ is such that $AX = AZ$, then prove that $S, X'$ and $Z{}$ are collinear.

Let $ABCD$ be an isosceles trapezoid with base $BC$ and $AD$. Suppose $\angle BDC=10^{\circ}$ and $\angle BDA=70^{\circ}$. Show that $AD^2=BC(AD+AB)$.

Consider the addition problem: \begin{tabular}{ccccc} &C&A&S&H\\ +&&&M&E\\ \hline O&S&I&D&E \end{tabular} where each letter represents a base-ten digit, and $C,M,O \ne 0.$ (Distinct letters are allowed to represent the same digit.) How many ways are there to assign values to the letters so that the addition problem is true?

A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square? [asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2)); draw(A--B--C--D--E--F--G--H--cycle); draw(A--D); draw(B--G); draw(C--F); draw(E--H); [/asy] $ \textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad\textbf{(E)}\ 2 - \sqrt{2}$

Find all positive integers $n$ with $n \ge 2$ such that the polynomial \[ P(a_1, a_2, ..., a_n) = a_1^n+a_2^n + ... + a_n^n - n a_1 a_2 ... a_n \] in the $n$ variables $a_1$, $a_2$, $\dots$, $a_n$ is irreducible over the real numbers, i.e. it cannot be factored as the product of two nonconstant polynomials with real coefficients. [i]Proposed by Yang Liu[/i]

The sequence $(u_n)$ is defined by $u_1 = 1, u_2 = 2$ and $u_{n+1} = 3u_n - u_{n-1}$ for $n \ge 2$. Set $v_n =\sum_{k=1}^n \text{arccot }u_k$. Compute $\lim_{n\to\infty} v_n$.

Which triplet of numbers has a sum NOT equal to 1? $ \text{(A)}\ (1/2,1/3,1/6)\qquad\text{(B)}\ (2,-2,1)\qquad\text{(C)}\ (0.1,0.3,0.6)\qquad\text{(D)}\ (1.1,-2.1,1.0)\qquad\text{(E)}\ (-3/2,-5/2,5) $

For which of the following values of real number $t$, the equation $x^4-tx+\dfrac 1t = 0$ has no root on the interval $[1,2]$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $a_1,a_2,...$ be a strictly increasing sequence of positive real numbers such that $a_1 = 1$,$a_2 = 4$, and that for every positive integer $k$, the subsequence $a_{4k-3}$,$a_{4k-2}$,$a_{4k-1}$,$a_{4k}$ is geometric and the subsequence $a_{4k-1}$,$a_{4k}$,$a_{4k+1}$,$a_{4k+2}$ is arithmetic. For each positive integer $k$, let rk be the common ratio of the geometric sequence $a_{4k-3}$,$a_{4k-2}$,$a_{4k-1}$,$a_{4k}$. Compute $$\sum_{k=1}^{\infty} (r_k -1)(r_{k+1} -1)$$

The population of a small town is $480$. The graph indicates the number of females and males in the town, but the vertical scale-values are omitted. How many males live in the town? [asy] draw((0,13)--(0,0)--(20,0)); draw((3,0)--(3,10)--(8,10)--(8,0)); draw((3,5)--(8,5)); draw((11,0)--(11,5)--(16,5)--(16,0)); label("$\textbf{POPULATION}$",(10,11),N); label("$\textbf{F}$",(5.5,0),S); label("$\textbf{M}$",(13.5,0),S); [/asy] $\text{(A)}\ 120 \qquad \text{(B)}\ 160 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 240 \qquad \text{(E)}\ 360$

Let $p=4k+1$ be a prime. Show that \[\sum^{k}_{i=1}\left \lfloor \sqrt{ ip }\right \rfloor = \frac{p^{2}-1}{12}.\]

Let $N$ be the number of polynomials $P(x_1, x_2, \dots, x_{2016})$ of degree at most $2015$ with coefficients in the set $\{0, 1, 2 \}$ such that $P(a_1,a_2,\cdots ,a_{2016}) \equiv 1 \pmod{3}$ for all $(a_1,a_2,\cdots ,a_{2016}) \in \{0, 1\}^{2016}.$ Compute the remainder when $v_3(N)$ is divided by $2011$, where $v_3(N)$ denotes the largest integer $k$ such that $3^k | N.$ [i]Proposed by Yang Liu[/i]

Does there exist a natural number $n$ such that $n!$ ends with exactly $2021$ zeros?

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Given that $a + 19b = 3$ and $a + 1019b = 5$, what is $a + 2019b$? [b]p2.[/b] How many multiples of $3$ are there between $2019$ and $2119$, inclusive? [b]p3.[/b] What is the maximum number of quadrilaterals a $12$-sided regular polygon can be quadrangulated into? Here quadrangulate means to cut the polygon along lines from vertex to vertex, none of which intersect inside the polygon, to form pieces which all have exactly $4$ sides. [b]p4.[/b] What is the value of $|2\pi - 7| + |2\pi - 6|$, rounded to the nearest hundredth? [b]p5.[/b] In the town of EMCCxeter, there is a $30\%$ chance that it will snow on Saturday, and independently, a $40\%$ chance that it will snow on Sunday. What is the probability that it snows exactly once that weekend, as a percentage? [b]p6.[/b] Define $n!$ to be the product of all integers between $1$ and $n$ inclusive. Compute $\frac{2019!}{2017!} \times \frac{2016!}{2018!}$ . [b]p7.[/b] There are $2019$ people standing in a row, and they are given positions $1$, $2$, $3$, $...$, $2019$ from left to right. Next, everyone in an odd position simultaneously leaves the row, and the remaining people are assigned new positions from $1$ to $1009$, again from left to right. This process is then repeated until one person remains. What was this person's original position? [b]p8.[/b] The product $1234\times 4321$ contains exactly one digit not in the set $\{1, 2, 3, 4\}$. What is this digit? [b]p9.[/b] A quadrilateral with positive area has four integer side lengths, with shortest side $1$ and longest side $9$. How many possible perimeters can this quadrilateral have? [b]p10.[/b] Define $s(n)$ to be the sum of the digits of $n$ when expressed in base $10$, and let $\gamma (n)$ be the sum of $s(d)$ over all natural number divisors $d$ of $n$. For instance, $n = 11$ has two divisors, $1$ and $11$, so $\gamma (11) = s(1) + s(11) = 1 + (1 + 1) = 3$. Find the value of $\gamma (2019)$. [b]p11.[/b] How many five-digit positive integers are divisible by $9$ and have $3$ as the tens digit? [b]p12.[/b] Adam owns a large rectangular block of cheese, that has a square base of side length $15$ inches, and a height of $4$ inches. He wants to remove a cylindrical cheese chunk of height $4$, by making a circular hole that goes through the top and bottom faces, but he wants the surface area of the leftover cheese block to be the same as before. What should the diameter of his hole be, in inches? [i]Αddendum on 1/26/19: the hole must have non-zero diameter. [/i] [b]p13.[/b] Find the smallest prime that does not divide $20! + 19! + 2019!$. [b]p14.[/b] Convex pentagon $ABCDE$ has angles $\angle ABC = \angle BCD = \angle DEA = \angle EAB$ and angle $\angle CDE = 60^o$. Given that $BC = 3$, $CD = 4$, and $DE = 5$, find $EA$. [i]Addendum on 1/26/19: ABCDE is specified to be convex. [/i] [b]p15.[/b] Sophia has $3$ pairs of red socks, $4$ pairs of blue socks, and $5$ pairs of green socks. She picks out two individual socks at random: what is the probability she gets a pair with matching color? [b]p16.[/b] How many real roots does the function $f(x) = 2019^x - 2019x - 2019$ have? [b]p17.[/b] A $30-60-90$ triangle is placed on a coordinate plane with its short leg of length $6$ along the $x$-axis, and its long leg along the $y$-axis. It is then rotated $90$ degrees counterclockwise, so that the short leg now lies along the $y$-axis and long leg along the $x$-axis. What is the total area swept out by the triangle during this rotation? [b]p18.[/b] Find the number of ways to color the unit cells of a $2\times 4$ grid in four colors such that all four colors are used and every cell shares an edge with another cell of the same color. [b]p19.[/b] Triangle $\vartriangle ABC$ has centroid $G$, and $X, Y,Z$ are the centroids of triangles $\vartriangle BCG$, $\vartriangle ACG$, and $\vartriangle ABG$, respectively. Furthermore, for some points $D,E, F$, vertices $A,B,C$ are themselves the centroids of triangles $\vartriangle DBC$, $\vartriangle ECA$, and $\vartriangle FAB$, respectively. If the area of $\vartriangle XY Z = 7$, what is the area of $\vartriangle DEF$? [b]p20.[/b] Fhomas orders three $2$-gallon jugs of milk from EMCCBay for his breakfast omelette. However, every jug is actually shipped with a random amount of milk (not necessarily an integer), uniformly distributed between $0$ and $2$ gallons. If Fhomas needs $2$ gallons of milk for his breakfast omelette, what is the probability he will receive enough milk? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $(x + 1) + (x + 2) + ... + (x + 20) = 174 + 176 + 178 + ... + 192$, then what is the value of $x$? $\mathrm{(A) \ } 80 \qquad \mathrm{(B) \ } 81 \qquad \mathrm {(C) \ } 82 \qquad \mathrm{(D) \ } 83 \qquad \mathrm{(E) \ } 84$

Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$. Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.

There is a circle which is 1 meter in circumference and a point marked on it. Two cockroaches start running in the same direction from the marked point with different speeds. Whenever the fast one would catch up with the slow one, the slow one would instantly turn around and start running in tho other direction with the same speed. Whenever they would meet face-to-face, the fast one would instantly turn around and start running in tho other direction with the same speed. How far from the marked point could their 100th meeting be?

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

Let $\triangle ABC$ be a acute triangle. Let $H$ the foot of the C-altitude in $AB$ such that $AH=3BH$, let $M$ and $N$ the midpoints of $AB$ and $AC$ and let $P$ be a point such that $NP=NC$ and $CP=CB$ and $B$, $P$ are located on different sides of the line $AC$. Prove that $\measuredangle APM=\measuredangle PBA$.

Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.