Consider the sequence of positive integers defined by $s_1,s_2,s_3, \dotsc $ of positive integers defined by [list] [*]$s_1=2$, and[/*] [*]for each positive integer $n$, $s_{n+1}$ is equal to $s_n$ plus the product of prime factors of $s_n$.[/*] [/list] The first terms of the sequence are $2,4,6,12,18,24$. Prove that the product of the $2019$ smallest primes is a term of the sequence.

The number $27,\,000,\,001$ has exactly four prime factors. Find their sum.

If $x$, $y$, and $z$ are integers satisfying $xyz+4(x+y+z)=2(xy+xz+yz)+7$, list all possibilities for the ordered triple $(x, y, z)$.

Let $ABCD$ be a cyclic quadrilateral of area 8. If there exists a point $O$ in the plane of the quadrilateral such that $OA+OB+OC+OD = 8$, prove that $ABCD$ is an isosceles trapezoid.

Let $ a_{n}$ is a positive number such that $ \int_{0}^{a_{n}}\frac{e^{x}\minus{}1}{1\plus{}e^{x}}\ dx \equal{}\ln n$. Find $ \lim_{n\to\infty}(a_{n}\minus{}\ln n)$.

Call a positive integer $n$ [i]good[/i] if there are $n$ integers, positive or negative, and not necessarily distinct, such that their sum and products are both equal to $n$. Show that the integers of the form $4k+1$ and $4l$ are good.

The [i]equatorial algebra[/i] is defined as the real numbers equipped with the three binary operations $\natural$, $\sharp$, $\flat$ such that for all $x, y\in \mathbb{R}$, we have \[x\mathbin\natural y = x + y,\quad x\mathbin\sharp y = \max\{x, y\},\quad x\mathbin\flat y = \min\{x, y\}.\] An [i]equatorial expression[/i] over three real variables $x$, $y$, $z$, along with the [i]complexity[/i] of such expression, is defined recursively by the following: [list] [*] $x$, $y$, and $z$ are equatorial expressions of complexity 0; [*] when $P$ and $Q$ are equatorial expressions with complexity $p$ and $q$ respectively, all of $P\mathbin\natural Q$, $P\mathbin\sharp Q$, $P\mathbin\flat Q$ are equatorial expressions with complexity $1+p+q$. [/list] Compute the number of distinct functions $f: \mathbb{R}^3\rightarrow \mathbb{R}$ that can be expressed as equatorial expressions of complexity at most 3. [i]Proposed by Yannick Yao[/i]

Let $I$ be the incenter of $ABC$ and $I_A$ the excenter of the side $BC$, let $M$ be the midpoint of $CB$ and $N$ the midpoint of arc $BC$(with the point $A$). If $T$ is the symmetric of the point $N$ by the point $A$, prove that the quadrilateral $I_AMIT$ is cyclic.

[b]p1.[/b] For positive real numbers $x, y$, the operation $\otimes$ is given by $x \otimes y =\sqrt{x^2 - y}$ and the operation $\oplus$ is given by $x \oplus y =\sqrt{x^2 + y}$. Compute $(((5\otimes 4)\oplus 3)\otimes2)\oplus 1$. [b]p2.[/b] Janabel is cutting up a pizza for a party. She knows there will either be $4$, $5$, or $6$ people at the party including herself, but can’t remember which. What is the least number of slices Janabel can cut her pizza to guarantee that everyone at the party will be able to eat an equal number of slices? [b]p3.[/b] If the numerator of a certain fraction is added to the numerator and the denominator, the result is $\frac{20}{19}$ . What is the fraction? [b]p4.[/b] Let trapezoid $ABCD$ be such that $AB \parallel CD$. Additionally, $AC = AD = 5$, $CD = 6$, and $AB = 3$. Find $BC$. [b]p5.[/b] AtMerrick’s Ice Cream Parlor, customers can order one of three flavors of ice cream and can have their ice cream in either a cup or a cone. Additionally, customers can choose any combination of the following three toppings: sprinkles, fudge, and cherries. How many ways are there to buy ice cream? [b]p6.[/b] Find the minimum possible value of the expression $|x+1|+|x-4|+|x-6|$. [b]p7.[/b] How many $3$ digit numbers have an even number of even digits? [b]p8.[/b] Given that the number $1a99b67$ is divisible by $7$, $9$, and $11$, what are $a$ and $b$? Express your answer as an ordered pair. [b]p9.[/b] Let $O$ be the center of a quarter circle with radius $1$ and arc $AB$ be the quarter of the circle’s circumference. Let $M$,$N$ be the midpoints of $AO$ and $BO$, respectively. Let $X$ be the intersection of $AN$ and $BM$. Find the area of the region enclosed by arc $AB$, $AX$,$BX$. [b]p10.[/b] Each square of a $5$-by-$1$ grid of squares is labeled with a digit between $0$ and $9$, inclusive, such that the sum of the numbers on any two adjacent squares is divisible by $3$. How many such labelings are possible if each digit can be used more than once? [b]p11.[/b] A two-digit number has the property that the difference between the number and the sum of its digits is divisible by the units digit. If the tens digit is $5$, how many different possible values of the units digit are there? [b]p12.[/b] There are $2019$ red balls and $2019$ white balls in a jar. One ball is drawn and replaced with a ball of the other color. The jar is then shaken and one ball is chosen. What is the probability that this ball is red? [b]p13.[/b] Let $ABCD$ be a square with side length $2$. Let $\ell$ denote the line perpendicular to diagonal $AC$ through point $C$, and let $E$ and $F$ be themidpoints of segments $BC$ and $CD$, respectively. Let lines $AE$ and $AF$ meet $\ell$ at points $X$ and $Y$ , respectively. Compute the area of $\vartriangle AXY$ . [b]p14.[/b] Express $\sqrt{21-6\sqrt6}+\sqrt{21+6\sqrt6}$ in simplest radical form. [b]p15.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length two. Let $D$ and $E$ be on $AB$ and $AC$ respectively such that $\angle ABE =\angle ACD = 15^o$. Find the length of $DE$. [b]p16.[/b] $2018$ ants walk on a line that is $1$ inch long. At integer time $t$ seconds, the ant with label $1 \le t \le 2018$ enters on the left side of the line and walks among the line at a speed of $\frac{1}{t}$ inches per second, until it reaches the right end and walks off. Determine the number of ants on the line when $t = 2019$ seconds. [b]p17.[/b] Determine the number of ordered tuples $(a_1,a_2,... ,a_5)$ of positive integers that satisfy $a_1 \le a_2 \le ... \le a_5 \le 5$. [b]p18.[/b] Find the sum of all positive integer values of $k$ for which the equation $$\gcd (n^2 -n -2019,n +1) = k$$ has a positive integer solution for $n$. [b]p19.[/b] Let $a_0 = 2$, $b_0 = 1$, and for $n \ge 0$, let $$a_{n+1} = 2a_n +b_n +1,$$ $$b_{n+1} = a_n +2b_n +1.$$ Find the remainder when $a_{2019}$ is divided by $100$. [b]p20.[/b] In $\vartriangle ABC$, let $AD$ be the angle bisector of $\angle BAC$ such that $D$ is on segment $BC$. Let $T$ be the intersection of ray $\overrightarrow{CB}$ and the line tangent to the circumcircle of $\vartriangle ABC$ at $A$. Given that $BD = 2$ and $TC = 10$, find the length of $AT$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For each non-zero complex number $ z,$ let $\arg(z)$ be the unique real number $ t$ such that $ \minus{}\pi < t \leq \pi$ and $ z \equal{} |z|(\cos(t) \plus{} \textrm{i} sin(t)).$ Given a real number $ c > 0$ and a complex number $ z \neq 0$ with $\arg z \neq \pi,$ define \[ B(c, z) \equal{} \{b \in \mathbb{R} \ ; \ |w \minus{} z| < b \Rightarrow |\arg(w) \minus{} \arg(z)| < c\}.\] Determine necessary and sufficient conditions, in terms of $ c$ and $ z,$ such that $ B(c, z)$ has a maximum element, and determine what this maximum element is in this case.

$ 5 $ points are situated in the plane so that any three of them form a triangle of area at most $ 1. $ Prove that there is a trapezoid of area at most $ 3 $ which contains all these points ('including' here means that the points can also be on the sides of the trapezoid).

The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$? [asy]//Choice A size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("$x$", (4,0), N); label("$y$", (0,4), W); label("$-4$", (-4,-ticks), S); label("$-1$", (-1,-ticks), S); label("$1$", (1,-ticks), S); label("$4$", (4,-ticks), S); real f(real x) { return 0.101562 x^4+0.265625 x^3+0.0546875 x^2-0.109375 x+0.125; } real g(real x) { return 0.0625 x^4+0.0520833 x^3-0.21875 x^2-0.145833 x-2.5; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("$\textbf{(A)}$", (-5,4.5)); [/asy] [asy]//Choice B size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("$x$", (4,0), N); label("$y$", (0,4), W); label("$-4$", (-4,-ticks), S); label("$-1$", (-1,-ticks), S); label("$1$", (1,-ticks), S); label("$4$", (4,-ticks), S); real f(real x) { return 0.541667 x^4+0.458333 x^3-0.510417 x^2-0.927083 x-2; } real g(real x) { return -0.791667 x^4-0.208333 x^3-0.177083 x^2-0.260417 x-1; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("$\textbf{(B)}$", (-5,4.5)); [/asy] [asy]//Choice C size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("$x$", (4,0), N); label("$y$", (0,4), W); label("$-4$", (-4,-ticks), S); label("$-1$", (-1,-ticks), S); label("$1$", (1,-ticks), S); label("$4$", (4,-ticks), S); real f(real x) { return 0.21875 x^2+0.28125 x+0.5; } real g(real x) { return -0.375 x^2-0.75 x+0.5; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("$\textbf{(C)}$", (-5,4.5)); [/asy] [asy]//Choice D size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("$x$", (4,0), N); label("$y$", (0,4), W); label("$-4$", (-4,-ticks), S); label("$-1$", (-1,-ticks), S); label("$1$", (1,-ticks), S); label("$4$", (4,-ticks), S); real f(real x) { return 0.015625 x^5-0.244792 x^3+0.416667 x+0.6875; } real g(real x) { return 0.0284722 x^6-0.340278 x^4+0.874306 x^2-1.5625; } real z=3.14; draw(graph(f,-z, z), heavygray); draw(graph(g,-z, z), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("$\textbf{(D)}$", (-5,4.5)); [/asy] [asy]//Choice E size(100);defaultpen(linewidth(0.7)+fontsize(8)); real end=4.5; draw((-end,0)--(end,0), EndArrow(5)); draw((0,-end)--(0,end), EndArrow(5)); real ticks=0.2, four=3.7, r=0.1; draw((1,ticks)--(1,-ticks)^^(-1,ticks)--(-1,-ticks)^^(four,ticks)--(four,-ticks)^^(-four,ticks)--(-four,-ticks)); label("$x$", (4,0), N); label("$y$", (0,4), W); label("$-4$", (-4,-ticks), S); label("$-1$", (-1,-ticks), S); label("$1$", (1,-ticks), S); label("$4$", (4,-ticks), S); real f(real x) { return 0.026067 x^4-0.0136612 x^3-0.157131 x^2-0.00961796 x+1.21598; } real g(real x) { return -0.166667 x^3+0.125 x^2+0.479167 x-0.375; } draw(graph(f,-four, four), heavygray); draw(graph(g,-four, four), black); clip((-end-r,-end-r)--(-end-r, end+r)--(end+r,end+r)--(end+r, -end-r)--cycle); label("$\textbf{(E)}$", (-5,4.5)); [/asy]

(1) Let $a,b$ be coprime positive integers. Prove that there exists constants $\lambda$ and $\beta$ such that for all integers $m$, $$\left| \sum\limits_{k=1}^{m-1} \left\{ \frac{ak}{m} \right\}\left\{ \frac{bk}{m} \right\} - \lambda m \right| \le \beta$$ (2) Prove that there exists $N$ such that for all $p>N$ (where $p$ is a prime number), and any positive integers $a,b,c$ positive integers satisfying $p\nmid (a+b)(b+c)(c+a)$, there are at least $\lfloor \frac{p}{12} \rfloor$ solutions $k\in \{1,\cdots,p-1\}$ such that $$ \left\{\frac{ak}{p}\right\} + \left\{\frac{bk}{p}\right\} + \left\{\frac{ck}{p}\right\} \le 1 $$

Let $p$ be a prime number and $k$ is a integer with $p|2^k-1$ for $a\in \{ 1,2,\ldots,p-1\}$ , let $m_{a}$ be the only element that satisfies $p|am_{a}-1$ and define $T_{a}= \{ x \in \{1,2,\ldots p-1\} | \{ \frac {m_{a}x}{p} - \frac {x}{ap} \} < \frac{1}{2}$and there exists integer y satisfying $p | x-y^\[k+1\] \}$ Try to proof that there exists an integer $m$ and integers $1 \le a_1 <a_2< \ldots < a_{m} \le p-1$ satisfying $$ |T_{a_1}| = |T_{a_2}| = \ldots = |T_{a_{m}}| = m $$

Let $a$ be a constant. In the $xy$ palne, the curve $C_1:y=\frac{\ln x}{x}$ touches $C_2:y=ax^2$. Find the volume of the solid generated by a rotation of the part enclosed by $C_1,\ C_2$ and the $x$ axis about the $x$ axis. [i]2011 Yokohama National Universty entrance exam/Engineering[/i]

66 players take part in the chess tournament, each player plays one game against each other, and the games take place in four cities. Prove that three players play all their games in the same city.

Let $O{}$ be the center of the circumscribed sphere of the tetrahedron $ABCD$. Let $L,M,N$ respectively be the midpoints of the segments $BC,CA,AB$. It is known that $AB+BC=AD+CD$, $BC+CA=BD+AD$, $CA+AB=CD+BD$. Prove that $\angle LOM=\angle MON=\angle NOL$. Find their value.

Let $f(x)=\frac 12e^{2x}+2e^x+x$. Answer the following questions. (1) For a real number $t$, set $g(x)=tx-f(x).$ When $x$ moves in the range of all real numbers, find the range of $t$ for which $g(x)$ has maximum value, then for the range of $t$, find the maximum value of $g(x)$ and the value of $x$ which gives the maximum value. (2) Denote by $m(t)$ the maximum value found in $(1)$. Let $a$ be a constant, consider a function of $t$, $h(t)=at-m(t)$. When $t$ moves in the range of $t$ found in $(1)$, find the maximum value of $h(t)$.

Determine the smallest real number $a$ such that there is a square of side $a$ such that contains $5$ unit circles inside it without common interior points in pairs.

An infinite arithmetic progression is given. The products of the pairs of its members are considered. Prove that two of these numbers differ by no more than 1. [i]Proposed by A. Kuznetsov[/i]

Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\frac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \cup S$, and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S$. See the figure (not drawn to scale). [asy] //kaaaaaaaaaante314 size(8cm); import olympiad; label(scale(.8)*"$y$", (0,60), N); label(scale(.8)*"$x$", (60,0), E); filldraw((0,0)--(55,0)--(55,55)--(0,55)--cycle, yellow+orange+white+white); label(scale(1.3)*"$R$", (55/2,55/2)); filldraw((0,0)--(0,28)--(-28,28)--(-28,0)--cycle, green+white+white); label(scale(1.3)*"$S$",(-14,14)); filldraw((-10,0)--(15,0)--(15,25)--(-10,25)--cycle, red+white+white); label(scale(1.3)*"$T$",(3.5,25/2)); draw((0,-10)--(0,60),EndArrow(TeXHead)); draw((-34,0)--(60,0),EndArrow(TeXHead));[/asy] The fraction of lattice points in $S$ that are in $S \cap T$ is 27 times the fraction of lattice points in $R$ that are in $R \cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$? $\textbf{(A) }336\qquad\textbf{(B) }337\qquad\textbf{(C) }338\qquad\textbf{(D) }339\qquad\textbf{(E) }340$

Let $p \geq 5$ be a prime and let \begin{align*} (p-1)^p +1 = \prod _{i=1}^n q_i^{\beta_i} \end{align*} where $q_i$ are primes. Prove, \begin{align*} \sum_{i=1}^n q_i \beta_i >p^2 \end{align*}

Find a right triangle that can be cut into $365$ equal triangles.

Let $ABCD$ be a parallelogram. Let $X$ and $Y$ be arbitrary points on sides $BC$ and $CD$, respectively. Segments $BY$ and $DX$ intersect at $P$. Prove that the line going through the midpoints of segments $BD$ and $XY$ is either parallel to or coincides with line $AP$.

What should the angle at the vertex of an isosceles triangle be so that it is possible to construct a triangle with sides equal to the height, base, and one of the other sides of the isosceles triangle?