Let $ABC$ be a triangle with incenter $I$, and $A$-excenter $\Gamma$. Let $A_1,B_1,C_1$ be the points of tangency of $\Gamma$ with $BC,AC$ and $AB$, respectively. Suppose $IA_1, IB_1$ and $IC_1$ intersect $\Gamma$ for the second time at points $A_2,B_2,C_2$, respectively. $M$ is the midpoint of segment $AA_1$. If the intersection of $A_1B_1$ and $A_2B_2$ is $X$, and the intersection of $A_1C_1$ and $A_2C_2$ is $Y$, prove that $MX=MY$.

A square-shaped quilt is divided into $16 = 4 \times 4$ equal squares. We say that the quilt is [i]UMD certified[/i] if each of these $16$ squares is colored red, yellow, or black, so that (i) all three colors are used at least once and (ii) the quilt looks the same when it is rotated $90, 180,$ or $270$ degrees about its center. How many distinct UMD certified quilts are there? \[\rm a. ~33\qquad \mathrm b. ~36 \qquad \mathrm c. ~45\qquad\mathrm d. ~54\qquad\mathrm e. ~81\]

Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech? $\textbf{(A)}\ 2250 \qquad \textbf{(B)}\ 3000 \qquad \textbf{(C)}\ 4200 \qquad \textbf{(D)}\ 4350 \qquad \textbf{(E)}\ 5650$

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [b]L.10[/b] Given the following system of equations where $x, y, z$ are nonzero, find $x^2 + y^2 + z^2$. $$x + 2y = xy$$ $$3y + z = yz$$ $$3x + 2z = xz$$ [u]Set 4[/u] [b]L.16 / D.23[/b] Anson, Billiam, and Connor are looking at a $3D$ figure. The figure is made of unit cubes and is sitting on the ground. No cubes are floating; in other words, each unit cube must either have another unit cube or the ground directly under it. Anson looks from the left side and says, “I see a $5 \times 5$ square.” Billiam looks from the front and says the same thing. Connor looks from the top and says the same thing. Find the absolute difference between the minimum and maximum volume of the figure. [b]L.17[/b] The repeating decimal $0.\overline{MBMT}$ is equal to $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, and $M, B, T$ are distinct digits. Find the minimum value of $q$. [b]L.18[/b] Annie, Bob, and Claire have a bag containing the numbers $1, 2, 3, . . . , 9$. Annie randomly chooses three numbers without replacement, then Bob chooses, then Claire gets the remaining three numbers. Find the probability that everyone is holding an arithmetic sequence. (Order does not matter, so $123$, $213$, and $321$ all count as arithmetic sequences.) [b]L.19[/b] Consider a set $S$ of positive integers. Define the operation $f(S)$ to be the smallest integer $n > 1$ such that the base $2^k$ representation of $n$ consists only of ones and zeros for all $k \in S$. Find the size of the largest set $S$ such that $f(S) < 2^{2019}$. [b]L.20 / D.25[/b] Find the largest solution to the equation $$2019(x^{2019x^{2019}-2019^2+2019})^{2019} = 2019^{x^{2019}+1}.$$ [u]Set 5[/u] [b]L.21[/b] Steven is concerned about his artistic abilities. To make himself feel better, he creates a $100 \times 100$ square grid and randomly paints each square either white or black, each with probability $\frac12$. Then, he divides the white squares into connected components, groups of white squares that are connected to each other, possibly using corners. (For example, there are three connected components in the following diagram.) What is the expected number of connected components with 1 square, to the nearest integer? [img]https://cdn.artofproblemsolving.com/attachments/e/d/c76e81cd44c3e1e818f6cf89877e56da2fc42f.png[/img] [b]L.22[/b] Let x be chosen uniformly at random from $[0, 1]$. Let n be the smallest positive integer such that $3^n x$ is at most $\frac14$ away from an integer. What is the expected value of $n$? [b]L.23[/b] Let $A$ and $B$ be two points in the plane with $AB = 1$. Let $\ell$ be a variable line through $A$. Let $\ell'$ be a line through $B$ perpendicular to $\ell$. Let X be on $\ell$ and $Y$ be on $\ell'$ with $AX = BY = 1$. Find the length of the locus of the midpoint of $XY$ . [b]L.24[/b] Each of the numbers $a_i$, where $1 \le i \le n$, is either $-1$ or $1$. Also, $$a_1a_2a_3a_4+a_2a_3a_4a_5+...+a_{n-3}a_{n-2}a_{n-1}a_n+a_{n-2}a_{n-1}a_na_1+a_{n-1}a_na_1a_2+a_na_1a_2a_3 = 0.$$ Find the number of possible values for $n$ between $4$ and $100$, inclusive. [b]L.25[/b] Let $S$ be the set of positive integers less than $3^{2019}$ that have only zeros and ones in their base $3$ representation. Find the sum of the squares of the elements of $S$. Express your answer in the form $a^b(c^d - 1)(e^f - 1)$, where $a, b, c, d, e, f$ are positive integers and $a, c, e$ are not perfect powers. 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 D.16-30/ L10-15 [url=https://artofproblemsolving.com/community/c3h2790818p24541688]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Ben picks a positive number $n$ less than $2017$ uniformly at random. Then Rex, starting with the number $ 1$, repeatedly multiplies his number by $n$ and then finds the remainder when dividing by $2017$. Rex does this until he gets back to the number $ 1$. What is the probability that, during this process, Rex reaches every positive number less than $2017$ before returning back to $ 1$?

Consider the following array: \[ 3, 5\\3, 8, 5\\3, 11, 13, 5\\3, 14, 24, 18, 5\\3, 17, 38, 42, 23, 5\\ \ldots \] Find the 5-th number on the $ n$-th row with $ n>5$.

Find all functions $f:\mathbb N\to\mathbb N$ such that \begin{align*} x+f(y)&\mid f(y+f(x))\\ f(x)-2017&\mid x-2017\end{align*}

Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.

Let $H$ be the orthocenter of an acute-angled triangle $ABC$. Show that the triangles $ABH,BCH$ and $CAH$ have the same perimeter if and only if the triangle $ABC$ is equilateral.

Let $a_1,a_2,...$ be an infinite sequence of integers that satisfies $a_{n+2}=a_{n+1}+a_n$ for all $n \ge 1$. There exists a positive integer $k$ such that $a_k=a_{k+2018}$. Prove that there exists a term of the sequence which is equal to zero.

The letters $P$, $Q$, and $R$ are entered in a $20\times 20$ grid according to the pattern shown below. How many $P$s, $Q$s, and $R$s will appear in the completed table? [asy] usepackage("mathdots"); size(5cm); draw((0,0)--(6,0),linewidth(1.5)+mediumgray); draw((0,1)--(6,1),linewidth(1.5)+mediumgray); draw((0,2)--(6,2),linewidth(1.5)+mediumgray); draw((0,3)--(6,3),linewidth(1.5)+mediumgray); draw((0,4)--(6,4),linewidth(1.5)+mediumgray); draw((0,5)--(6,5),linewidth(1.5)+mediumgray); draw((0,0)--(0,6),linewidth(1.5)+mediumgray); draw((1,0)--(1,6),linewidth(1.5)+mediumgray); draw((2,0)--(2,6),linewidth(1.5)+mediumgray); draw((3,0)--(3,6),linewidth(1.5)+mediumgray); draw((4,0)--(4,6),linewidth(1.5)+mediumgray); draw((5,0)--(5,6),linewidth(1.5)+mediumgray); label(scale(.9)*"\textsf{P}", (.5,.5)); label(scale(.9)*"\textsf{Q}", (.5,1.5)); label(scale(.9)*"\textsf{R}", (.5,2.5)); label(scale(.9)*"\textsf{P}", (.5,3.5)); label(scale(.9)*"\textsf{Q}", (.5,4.5)); label("$\vdots$", (.5,5.6)); label(scale(.9)*"\textsf{Q}", (1.5,.5)); label(scale(.9)*"\textsf{R}", (1.5,1.5)); label(scale(.9)*"\textsf{P}", (1.5,2.5)); label(scale(.9)*"\textsf{Q}", (1.5,3.5)); label(scale(.9)*"\textsf{R}", (1.5,4.5)); label("$\vdots$", (1.5,5.6)); label(scale(.9)*"\textsf{R}", (2.5,.5)); label(scale(.9)*"\textsf{P}", (2.5,1.5)); label(scale(.9)*"\textsf{Q}", (2.5,2.5)); label(scale(.9)*"\textsf{R}", (2.5,3.5)); label(scale(.9)*"\textsf{P}", (2.5,4.5)); label("$\vdots$", (2.5,5.6)); label(scale(.9)*"\textsf{P}", (3.5,.5)); label(scale(.9)*"\textsf{Q}", (3.5,1.5)); label(scale(.9)*"\textsf{R}", (3.5,2.5)); label(scale(.9)*"\textsf{P}", (3.5,3.5)); label(scale(.9)*"\textsf{Q}", (3.5,4.5)); label("$\vdots$", (3.5,5.6)); label(scale(.9)*"\textsf{Q}", (4.5,.5)); label(scale(.9)*"\textsf{R}", (4.5,1.5)); label(scale(.9)*"\textsf{P}", (4.5,2.5)); label(scale(.9)*"\textsf{Q}", (4.5,3.5)); label(scale(.9)*"\textsf{R}", (4.5,4.5)); label("$\vdots$", (4.5,5.6)); label(scale(.9)*"$\dots$", (5.5,.5)); label(scale(.9)*"$\dots$", (5.5,1.5)); label(scale(.9)*"$\dots$", (5.5,2.5)); label(scale(.9)*"$\dots$", (5.5,3.5)); label(scale(.9)*"$\dots$", (5.5,4.5)); label(scale(.9)*"$\iddots$", (5.5,5.6)); [/asy] $\textbf{(A)}~132~\text{Ps}, 134~\text{Qs}, 134~\text{Rs}\qquad\textbf{(B)}~133~\text{Ps}, 133~\text{Qs}, 134~\text{Rs}\qquad\textbf{(C)}~133~\text{Ps}, 134~\text{Qs}, 133~\text{Rs}$\\ $\textbf{(D)}~134~\text{Ps}, 132~\text{Qs}, 134~\text{Rs}\qquad\textbf{(E)}~134~\text{Ps}, 133~\text{Qs}, 133~\text{Rs}\qquad$

Find all functions $f:\mathbb{N} \to \mathbb{N}$ so that for any distinct positive integers $x,y,z$ the value of $x+y+z$ is a perfect square if and only if $f(x)+f(y)+f(z)$ is a perfect square.

At the end of the school year it became clear that for any arbitrarily chosen group of no less than 5 students, 80% of the marks “F” received by this group were given to no more than 20% of the students in the group. Prove that at least 3/4 of all “F” marks were given to the same student.

The segments $BF$ and $CN$ are the altitudes in the acute-angled triangle $ABC$. The line $OI$, which connects the centers of the circumscribed and inscribed circles of triangle $ABC$, is parallel to the line $FN$. Find the length of the altitude $AK$ in the triangle $ABC$ if the radii of its circumscribed and inscribed circles are $R$ and $r$, respectively. (Grigory Filippovsky)

Show that for every prime $p$ and integer $n$, there is an irreducible polynomial of degree $n$ in $\mathbb{Z}_p[x]$ and use that to show there is a field of size $p^n$.

The first row of this table is filled with the numbers $1$ through $10$, in that order. The second row is filled with the numbers from $1$ to $10$, in any order. In each box of the third row the sum of the two numbers written above is written. Is there a way to fill in the second row so that the ones digits of the numbers in the third row are all different? [img]https://cdn.artofproblemsolving.com/attachments/8/5/41117d105cc880bf452fa46132c20f2167aa5b.png[/img]