[b]p1.[/b] A shape made by joining four identical regular hexagons side-to-side is called a hexo. Two hexos are considered the same if one can be rotated / reflected to match the other. Find the number of different hexos. [b]p2.[/b] The sequence $1, 2, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6,... $ consists of numbers written in increasing order, where every even number $2n$ is written once, and every odd number $2n + 1$ is written $2n + 1$ times. What is the $2019^{th}$ term of this sequence? [b]p3.[/b] On planet EMCCarth, months can only have lengths of $35$, $36$, or $42$ days, and there is at least one month of each length. Victor knows that an EMCCarth year has $n$ days, but realizes that he cannot figure out how many months there are in an EMCCarth year. What is the least possible value of $n$? [b]p4.[/b] In triangle $ABC$, $AB = 5$ and $AC = 9$. If a circle centered at $A$ passing through $B$ intersects $BC$ again at $D$ and $CD = 7$, what is $BC$? [b]p5.[/b] How many nonempty subsets $S$ of the set $\{1, 2, 3,..., 11, 12\}$ are there such that the greatest common factor of all elements in $S$ is greater than $1$? [b]p6.[/b] Jasmine rolls a fair $6$-sided die, with faces labeled from $1$ to $6$, and a fair $20$-sided die, with faces labeled from $1$ to $20$. What is the probability that the product of these two rolls, added to the sum of these two rolls, is a multiple of $3$? [b]p7.[/b] Let $\{a_n\}$ be a sequence such that $a_n$ is either $2a_{n-1}$ or $a_{n-1} - 1$. Given that $a_1 = 1$ and $a_{12} = 120$, how many possible sequences $a_1$, $a_2$, $...$, $a_{12}$ are there? [b]p8.[/b] A tetrahedron has two opposite edges of length $2$ and the remaining edges have length $10$. What is the volume of this tetrahedron? [b]p9.[/b] In the garden of EMCCden, there is a tree planted at every lattice point $-10 \le x, y \le 10$ except the origin. We say that a tree is visible to an observer if the line between the tree and the observer does not intersect any other tree (assume that all trees have negligible thickness). What fraction of all the trees in the garden of EMCCden are visible to an observer standing at the origin? [b]p10.[/b] Point $P$ lies inside regular pentagon $\zeta$, which lies entirely within regular hexagon $\eta$. A point $Q$ on the boundary of pentagon $\zeta$ is called projective if there exists a point $R$ on the boundary of hexagon $\eta$ such that $P$, $Q$, $R$ are collinear and $2019 \cdot \overline{PQ} = \overline{QR}$. Given that no two sides of $\zeta$ and $\eta$ are parallel, what is the maximum possible number of projective points on $\zeta$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

4. Let $a \geq b \geq c \geq d>0$. Show that \[ \frac{b^3}{a} + \frac{c^3}{b} + \frac{d^3}{c} + \frac{a^3}{d} + 3 \left( ab+bc+cd+da \right) \geq 4 {\left( a^2 + b^2 + c^2 +d^2 \right)}. \] Other problems (in Korean) are also available at https://www.facebook.com/KoreanMathOlympiad

The complex numbers $\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$ are the four distinct roots of the equation $x^4+2x^3+2=0$. Determine the unordered set \[\{\alpha_1\alpha_2+\alpha_3\alpha_4,\alpha_1\alpha_3+\alpha_2\alpha_4,\alpha_1\alpha_4+\alpha_2\alpha_3\}.\]

A triangle $ABC$ is to be constructed given a side $a$ (oppisite angle $A$). angle $B$, and $h_c$, the altitude from $C$. If $N$ is the number of noncongruent solutions, then $N$ $\textbf{(A)}\ \text{is} \; 1\qquad \textbf{(B)}\ \text{is} \; 2\qquad \textbf{(C)}\ \text{must be zero}\qquad \textbf{(D)}\ \text{must be infinite}\qquad \textbf{(E)}\ \text{must be zero or infinite}$

Given is an equilateral triangle $ABC$ with circumcenter $O$. Let $D$ be a point on to minor arc $BC$ of its circumcircle such that $DB>DC$. The perpendicular bisector of $OD$ meets the circumcircle at $E, F$, with $E$ lying on the minor arc $BC$. The lines $BE$ and $CF$ meet at $P$. Prove that $PD \perp BC$.

Find all positive integers such that they have $6$ divisors (without $1$ and the number itself) and the sum of the divisors is $14133$.

Let $ S$ be the set of points $ (a,b)$ with $ 0\le a,b\le1$ such that the equation \[x^4 \plus{} ax^3 \minus{} bx^2 \plus{} ax \plus{} 1 \equal{} 0\] has at least one real root. Determine the area of the graph of $ S$.

Prove that for every real number $x$ and positive integer $n$ $$|\cos x|+|\cos2x|+|\cos2^2x|+\ldots+|\cos2^nx|\ge\frac n{2\sqrt2}.$$

Let $P(x)$ be a quadratic polynomial with real coefficients. Suppose that $P(1) = 20, P(-1) = 22,$ and $P(P(0)) = 400.$ Compute the largest possible value of $P(10).$

For which positive integers $n\geq4$ does there exist a convex $n$-gon with side lengths $1, 2, \dots, n$ (in some order) and with all of its sides tangent to the same circle?

Let $n>1$ be a positive integer. Show that the number of residues modulo $n^2$ of the elements of the set $\{ x^n + y^n : x,y \in \mathbb{N} \}$ is at most $\frac{n(n+1)}{2}$. [I]Proposed by N. Safaei (Iran)[/i]

Compute the limit: \[ \lim_{n\to\infty} \frac{1}{n^2}\sum\limits_{1\leq i <j\leq n}\sin \frac{i+j}{n}\].

A mouse is playing a game of mouse hopscotch. In mouse hopscotch there is a straight line of $11$ squares, and starting on the first square the mouse must reach the last square by jumping forward $1$, $2$, or $3$ squares at a time (so in particular the mouse’s first jump can be to the second, third, or fourth square). The mouse cannot jump past the last square. Compute the number of ways there are to complete mouse hopscotch.

[b]4.[/b] Denoting by $a(n)$ the greatest prime factor of the positive integer $n$, show that $S= \sum_{n=1}^{\infty } \frac{1}{na(n)}$ is convergente. [b](N. 13)[/b]

A convex quadrilateral $ABCD$ is inscribed in a circle $\omega$. The rays $AD$ and $BC$ meet in point $K$ and the rays $AB$ and $DC$ meet in $L$. Prove that the circumcircle of triangle $AKL$ is tangent to $\omega$ if and only if so is the circumcircle of triangle $CKL$.

With $76$ tiles, of which some are white, other blue and the remaining red, they form a rectangle of $4 \times 19$. Show that there is a rectangle, inside the largest, that has its vertices of the same color.

The sequence $ (a_n)$ is given by $ a_1\equal{}1,a_2\equal{}0$ and: $ a_{2k\plus{}1}\equal{}a_k\plus{}a_{k\plus{}1}, a_{2k\plus{}2}\equal{}2a_{k\plus{}1}$ for $ k \in \mathbb{N}.$ Find $ a_m$ for $ m\equal{}2^{19}\plus{}91.$

Given a sequence $(a_n)$ of real numbers such that the set $\{a_n\}$ is finite. If for every $k>1$ subsequence $(a_{kn})$ is periodic, is it true that the sequence $(a_n)$ must be periodic?

Find the functions $ f(x),\ g(x)$ satisfying the following equations. (1) $ f'(x) \equal{} 2f(x) \plus{} 10,\ f(0) \equal{} 0$ (2) $ \int_0^x u^3g(u)du \equal{} x^4 \plus{} g(x)$

Alice and Bob take turns alternatively on a $2020\times2020$ board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees? [i]Proposed by Seyed Reza Hosseini[/i]

Find the number of ways to order the integers $1,2,3,4,5,6,7$ from left to right so that each integer has all its divisors besides itself appearing to the left of it.

Let $T=\text{TNFTPP}$. Triangle $ABC$ has $AB=6T-3$ and $AC=7T+1$. Point $D$ is on $BC$ so that $AD$ bisects angle $BAC$. The circle through $A$, $B$, and $D$ has center $O_1$ and intersects line $AC$ again at $B'$, and likewise the circle through $A$, $C$, and $D$ has center $O_2$ and intersects line $AB$ again at $C'$. If the four points $B'$, $C'$, $O_1$, and $O_2$ lie on a circle, find the length of $BC$.

Determine the ratio of the sides of the rectangle circumscribed around a corner of five cells (see figure). (M. Evdokimov) [img]https://cdn.artofproblemsolving.com/attachments/f/f/9c3e345f33cabbbd83f65d7240aac29a163b19.png[/img]

$\textbf{Bake Sale}$ Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown. $\circ$ Art's cookies are trapezoids: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label("5 in", (2.5,0), S); label("3 in", (5,1.5), E); label("3 in", (3.5,3), N);[/asy] $\circ$ Roger's cookies are rectangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label("4 in", (2,0), S); label("2 in", (4,1), E);[/asy] $\circ$ Paul's cookies are parallelograms: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label("3 in", (1.5,0), S); label("2 in", (2.5,1), W);[/asy] $\circ$ Trisha's cookies are triangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label("3 in", (1.5,0), S); label("4 in", (3,2), E);[/asy] Each friend uses the same amount of dough, and Art makes exactly 12 cookies. How many cookies will be in one batch of Trisha's cookies? $ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 24$