The triangle $ABC$ satisfies $AB=10$ and has angles $\angle{A}=75^{\circ}$, $\angle{B}=60^{\circ}$, and $\angle C = 45^{\circ}$. Let $I_A$ be the center of the excircle opposite $A$, and let $D$, $E$ be the circumcenters of triangle $BCI_A$ and $ACI_A$ respectively. If $O$ is the circumcenter of triangle $ABC$, then the area of triangle $EOD$ can be written as $\tfrac{a\sqrt{b}}{c}$ for square-free $b$ and coprime $a,c$. Find the value of $a+b+c$.

The Fibonacci sequence $\{F_n\}$ is defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$. Determine all triplets of positive integers $(k,m,n)$ such that $F_n = F_m^k$.

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] What is $2018 - 3018 + 4018$? [b]p2.[/b] What is the smallest integer greater than $100$ that is a multiple of both $6$ and $8$? [b]p3.[/b] What positive real number can be expressed as both $\frac{b}{a}$ and $a:b$ in base $10$ for nonzero digits $a$ and $b$? Express your answer as a decimal. [b]p4.[/b] A non-degenerate triangle has sides of lengths $1$, $2$, and $\sqrt{n}$, where $n$ is a positive integer. How many possible values of $n$ are there? [b]p5.[/b] When three integers are added in pairs, and the results are $20$, $18$, and $x$. If all three integers sum to $31$, what is $x$? [b]p6.[/b] A cube's volume in cubic inches is numerically equal to the sum of the lengths of all its edges, in inches. Find the surface area of the cube, in square inches. [b]p7.[/b] A $12$ hour digital clock currently displays$ 9 : 30$. Ignoring the colon, how many times in the next hour will the clock display a palindrome (a number that reads the same forwards and backwards)? [b]p8.[/b] SeaBay, an online grocery store, offers two different types of egg cartons. Small egg cartons contain $12$ eggs and cost $3$ dollars, and large egg cartons contain $18$ eggs and cost $4$ dollars. What is the maximum number of eggs that Farmer James can buy with $10$ dollars? [b]p9.[/b] What is the sum of the $3$ leftmost digits of $\underbrace{999...9}_{2018\,\,\ 9' \,\,s}\times 12$? [b]p10.[/b] Farmer James trisects the edges of a regular tetrahedron. Then, for each of the four vertices, he slices through the plane containing the three trisection points nearest to the vertex. Thus, Farmer James cuts off four smaller tetrahedra, which he throws away. How many edges does the remaining shape have? [b]p11.[/b] Farmer James is ordering takeout from Kristy's Krispy Chicken. The base cost for the dinner is $\$14.40$, the sales tax is $6.25\%$, and delivery costs $\$3.00$ (applied after tax). How much did Farmer James pay, in dollars? [b]p12.[/b] Quadrilateral $ABCD$ has $ \angle ABC = \angle BCD = \angle BDA = 90^o$. Given that $BC = 12$ and $CD = 9$, what is the area of $ABCD$? [b]p13.[/b] Farmer James has $6$ cards with the numbers $1-6$ written on them. He discards a card and makes a $5$ digit number from the rest. In how many ways can he do this so that the resulting number is divisible by $6$? [b]p14.[/b] Farmer James has a $5 \times 5$ grid of points. What is the smallest number of triangles that he may draw such that each of these $25$ points lies on the boundary of at least one triangle? [b]p15.[/b] How many ways are there to label these $15$ squares from $1$ to $15$ such that squares $1$ and $2$ are adjacent, squares $2$ and $3$ are adjacent, and so on? [img]https://cdn.artofproblemsolving.com/attachments/e/a/06dee288223a16fbc915f8b95c9e4f2e4e1c1f.png[/img] [b]p16.[/b] On Farmer James's farm, there are three henhouses located at $(4, 8)$, $(-8,-4)$, $(8,-8)$. Farmer James wants to place a feeding station within the triangle formed by these three henhouses. However, if the feeding station is too close to any one henhouse, the hens in the other henhouses will complain, so Farmer James decides the feeding station cannot be within 6 units of any of the henhouses. What is the area of the region where he could possibly place the feeding station? [b]p17.[/b] At Eggs-Eater Academy, every student attends at least one of $3$ clubs. $8$ students attend frying club, $12$ students attend scrambling club, and $20$ students attend poaching club. Additionally, $10$ students attend at least two clubs, and $3$ students attend all three clubs. How many students are there in total at Eggs-Eater Academy? [b]p18.[/b] Let $x, y, z$ be real numbers such that $8^x = 9$, $27^y = 25$, and $125^z = 128$. What is the value of $xyz$? [b]p19.[/b] Let $p$ be a prime number and $x, y$ be positive integers. Given that $9xy = p(p + 3x + 6y)$, find the maximum possible value of $p^2 + x^2 + y^2$. [b]p20.[/b] Farmer James's hens like to drop eggs. Hen Hao drops $6$ eggs uniformly at random in a unit square. Farmer James then draws the smallest possible rectangle (by area), with sides parallel to the sides of the square, that contain all $6$ eggs. What is the probability that at least one of the $6$ eggs is a vertex of this rectangle? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is $X$, in centimeters? [asy] pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R; A=(4,0); B=(7,0); C=(7,4); D=(8,4); E=(8,5); F=(10,5); G=(10,7); H=(7,7); I=(7,8); J=(5,8); K=(5,7); L=(4,7); M=(4,6); N=(0,6); O=(0,5); P=(2,5); Q=(2,3); R=(4,3); draw(A--B--C--D--E--F--G--H--I--J--K--L--M--N--O--P--Q--R--cycle); label("$X$",(3.4,1.5)); label("6",(7.6,1.5)); label("1",(7.6,3.5)); label("1",(8.4,4.6)); label("2",(9.4,4.6)); label("2",(10.4,6)); label("3",(8.4,7.4)); label("1",(7.5,7.8)); label("2",(6,8.5)); label("1",(4.7,7.8)); label("1",(4.3,7.5)); label("1",(3.5,6.5)); label("4",(1.8,6.5)); label("1",(-0.5,5.5)); label("2",(0.8,4.5)); label("2",(1.5,3.8)); label("2",(2.8,2.6)); [/asy] $\textbf{(A)}\hspace{.05in}1 \qquad \textbf{(B)}\hspace{.05in}2 \qquad \textbf{(C)}\hspace{.05in}3 \qquad \textbf{(D)}\hspace{.05in}4 \qquad \textbf{(E)}\hspace{.05in}5 $

Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality.

In $\triangle ABC$, let $D$ be the foot of the altitude from $A$ to $BC$. A circle centered at $D$ with radius $AD$ intersects lines $AB$ and $AC$ at $P$ and $Q$, respectively. Show that $\triangle AQP\sim\triangle ABC$.

Let $f : \left[ 0, 1 \right] \to \mathbb R$ be a continuous function and $g : \left[ 0, 1 \right] \to \left( 0, \infty \right)$. Prove that if $f$ is increasing, then \[\int_{0}^{t}f(x) g(x) \, dx \cdot \int_{0}^{1}g(x) \, dx \leq \int_{0}^{t}g(x) \, dx \cdot \int_{0}^{1}f(x) g(x) \, dx .\]

Prove that for positive real numbers $x$ and $y$ smaller than or equal to $1/2$, \[\frac{(x+y)^2}{xy} \geq \frac{(2-xy)^2}{(1-x)(1-y)}.\]

The following figure shows a $3^2 \times 3^2$ grid divided into $3^2$ subgrids of size $3 \times 3$. This grid has $81$ cells, $9$ in each subgrid. [asy] draw((0,0)--(9,0)--(9,9)--(0,9)--cycle, linewidth(2)); draw((0,1)--(9,1)); draw((0,2)--(9,2)); draw((0,3)--(9,3), linewidth(2)); draw((0,4)--(9,4)); draw((0,5)--(9,5)); draw((0,6)--(9,6), linewidth(2)); draw((0,7)--(9,7)); draw((0,8)--(9,8)); draw((1,0)--(1,9)); draw((2,0)--(2,9)); draw((3,0)--(3,9), linewidth(2)); draw((4,0)--(4,9)); draw((5,0)--(5,9)); draw((6,0)--(6,9), linewidth(2)); draw((7,0)--(7,9)); draw((8,0)--(8,9)); [/asy] Now consider an $n^2 \times n^2$ grid divided into $n^2$ subgrids of size $n \times n$. Find the number of ways in which you can select $n^2$ cells from this grid such that there is exactly one cell coming from each subgrid, one from each row and one from each column.

Find the smallest constant $ k$ such that $ \frac {x}{\sqrt {x \plus{} y}} \plus{} \frac {y}{\sqrt {y \plus{} z}} \plus{} \frac {z}{\sqrt {z \plus{} x}}\leq k\sqrt {x \plus{} y \plus{} z}$ for all positive $ x$, $ y$, $ z$.

Albert has a very large bag of candies and he wants to share all of it with his friends. At first, he splits the candies evenly amongst his $20$ friends and himself and he finds that there are five left over. Ante arrives, and they redistribute the candies evenly again. This time, there are three left over. If the bag contains over $500$ candies, what is the fewest number of candies the bag can contain?

There is a rectangle $ABCD$ such that $AB=12$ and $BC=7$. $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that $\frac{AE}{EB} = 1$ and $\frac{CF}{FD} = \frac{1}{2}$. Call $X$ the intersection of $AF$ and $DE$. What is the area of pentagon $BCFXE$? Proposed by Minseok Eli Park (wolfpack)

Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions: [b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0$; [b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}$; [b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.

A [i]lazy[/i] rook can only move from a square to a vertical or a horizontal neighbour. It follows a path which visits each square of an $8 \times 8$ chessboard exactly once. Prove that the number of such paths starting at a corner square is greater than the number of such paths starting at a diagonal neighbour of a corner square. [i](7 points)[/i]

A bug starts at vertex $A$ of triangle $ABC$. Six times, the bug travels to a randomly chosen adjacent vertex. For example, the bug could go from $A$, to $B$, to $C$, back to $B$, and back to $C$. What is the probability that the bug ends up at $A$ after its six moves?

Determine all $f: \mathbb{R}\rightarrow\mathbb{R}$ for which \[ 2\cdot f(x)-g(x)=f(y)-y \textrm{ and } f(x)\cdot g(x) \geq x+1. \]

Let $ABC$ be a triangle with incircle $\gamma$. The circle through $A$ and $B$ tangent to $\gamma$ touches it at $C_2$ and the common tangent at $C_2$ intersects $AB$ at $C_1$. Define the points $A_1$, $B_1$, $A_2$, $B_2$ analogously. Prove that: a) the points $A_1$, $B_1$, $C_1$ are collinear; b) the lines $AA_2$, $BB_2$, $CC_2$ are concurrent.

How many three-digit positive integers $\underline{a}$ $\underline{b}$ $\underline{c}$ are there whose nonzero digits $a$, $b$, and $c$ satisfy $$0.\overline{\underline{a}~\underline{b}~\underline{c}} = \frac{1}{3} (0.\overline{a} + 0.\overline{b} + 0.\overline{c})?$$ (The bar indicates repetition, thus $0.\overline{\underline{a}~\underline{b}~\underline{c}}$ in the infinite repeating decimal $0.\underline{a}~\underline{b}~\underline{c}~\underline{a}~\underline{b}~\underline{c}~\cdots$) $\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }13\qquad\textbf{(E) }14$

Let $ABCD$ be an isosceles trapezoid with $AD = BC = 255$ and $AB = 128$. Let $M$ be the midpoint of $CD$ and let $N$ be the foot of the perpendicular from $A$ to $CD$. If $\angle MBC = 90^o$, compute $\tan\angle NBM$.

$101$ wise men stand in a circle. Each of them either thinks that the Earth orbits Jupiter or that Jupiter orbits the Earth. Once a minute, all the wise men express their opinion at the same time. Right after that, every wise man who stands between two people with a different opinion from him changes his opinion himself. The rest do not change. Prove that at one point they will all stop changing opinions.

The repeating decimals $0.abab\overline{ab}$ and $0.abcabc\overline{abc}$ satisfy \[0.abab\overline{ab}+0.abcabc\overline{abc}=\frac{33}{37},\] where $a,b$, and $c$ are (not necessarily distinct) digits. Find the three-digit number $abc$.

In a triangle $ABC$, let $A_0$ be the point where the interior angle bisector of angle $A$ meets with side $BC$. Similarly define $B_0$ and $C_0$. Prove that $\angle B_0A_0C_0 = 90^o$ if and only if $\angle BAC = 120^o$.

Given a circle centered at $ O $ and two points $ A $ and $ B $ lying on it. $ A $ and $ B $ do not form a diameter. The point $ C $ is chosen on the circle so that the line $ AC $ divides the segment $ OB $ in half. Let lines $ AB $ and $ OC $ intersect at $ D $, and let lines $ BC $ and $ AO $ intersect at $ F $. Prove that $ AF = CD $.

For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$. [i]Proposed by Juhan Aru, Estonia[/i]

Let $G$ be a finite graph. Prove that one can partition $G$ into two graphs $A \cup B=G$ such that if we erase all edges conecting a vertex from $A$ to a vertex from $B$, each vertex of the new graph has even degree.