Let $n(n\geq2)$ be a natural number and $a_1,a_2,...,a_n$ natural positive real numbers. Determine the least possible value of the expression $$E_n=\frac{(1+a_1)\cdot(a_1+a_2)\cdot(a_2+a_3)\cdot...\cdot(a_{n-1}+a_n)\cdot(a_n+3^{n+1})} {a_1\cdot a_2\cdot a_3\cdot...\cdot a_n}$$

A positive integer $n$ is [i]magical[/i] if $\lfloor \sqrt{\lceil \sqrt{n} \rceil} \rfloor=\lceil \sqrt{\lfloor \sqrt{n} \rfloor} \rceil$. Find the number of magical integers between $1$ and $10,000$ inclusive.

Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$

[b]p1.[/b] Let $X,Y ,Z$ be nonzero real numbers such that the quadratic function $X t^2 - Y t + Z = 0$ has the unique root $t = Y$ . Find $X$. [b]p2.[/b] Let $ABCD$ be a kite with $AB = BC = 1$ and $CD = AD =\sqrt2$. Given that $BD =\sqrt5$, find $AC$. [b]p3.[/b] Find the number of integers $n$ such that $n -2016$ divides $n^2 -2016$. An integer $a$ divides an integer $b$ if there exists a unique integer $k$ such that $ak = b$. [b]p4.[/b] The points $A(-16, 256)$ and $B(20, 400)$ lie on the parabola $y = x^2$ . There exists a point $C(a,a^2)$ on the parabola $y = x^2$ such that there exists a point $D$ on the parabola $y = -x^2$ so that $ACBD$ is a parallelogram. Find $a$. [b]p5.[/b] Figure $F_0$ is a unit square. To create figure $F_1$, divide each side of the square into equal fifths and add two new squares with sidelength $\frac15$ to each side, with one of their sides on one of the sides of the larger square. To create figure $F_{k+1}$ from $F_k$ , repeat this same process for each open side of the smallest squares created in $F_n$. Let $A_n$ be the area of $F_n$. Find $\lim_{n\to \infty} A_n$. [img]https://cdn.artofproblemsolving.com/attachments/8/9/85b764acba2a548ecc61e9ffc29aacf24b4647.png[/img] [b]p6.[/b] For a prime $p$, let $S_p$ be the set of nonnegative integers $n$ less than $p$ for which there exists a nonnegative integer $k$ such that $2016^k -n$ is divisible by $p$. Find the sum of all $p$ for which $p$ does not divide the sum of the elements of $S_p$ . [b]p7. [/b] Trapezoid $ABCD$ has $AB \parallel CD$ and $AD = AB = BC$. Unit circles $\gamma$ and $\omega$ are inscribed in the trapezoid such that circle $\gamma$ is tangent to $CD$, $AB$, and $AD$, and circle $\omega$ is tangent to $CD$, $AB$, and $BC$. If circles $\gamma$ and $\omega$ are externally tangent to each other, find $AB$. [b]p8.[/b] Let $x, y, z$ be real numbers such that $(x+y)^2+(y+z)^2+(z+x)^2 = 1$. Over all triples $(x, y, z)$, find the maximum possible value of $y -z$. [b]p9.[/b] Triangle $\vartriangle ABC$ has sidelengths $AB = 13$, $BC = 14$, and $CA = 15$. Let $P$ be a point on segment $BC$ such that $\frac{BP}{CP} = 3$, and let $I_1$ and $I_2$ be the incenters of triangles $\vartriangle ABP$ and $\vartriangle ACP$. Suppose that the circumcircle of $\vartriangle I_1PI_2$ intersects segment $AP$ for a second time at a point $X \ne P$. Find the length of segment $AX$. [b]p10.[/b] For $1 \le i \le 9$, let Ai be the answer to problem i from this section. Let $(i_1,i_2,... ,i_9)$ be a permutation of $(1, 2,... , 9)$ such that $A_{i_1} < A_{i_2} < ... < A_{i_9}$. For each $i_j$ , put the number $i_j$ in the box which is in the $j$th row from the top and the $j$th column from the left of the $9\times 9$ grid in the bonus section of the answer sheet. Then, fill in the rest of the squares with digits $1, 2,... , 9$ such that $\bullet$ each bolded $ 3\times 3$ grid contains exactly one of each digit from $ 1$ to $9$, $\bullet$ each row of the $9\times 9$ grid contains exactly one of each digit from $ 1$ to $9$, and $\bullet$ each column of the $9\times 9$ grid contains exactly one of each digit from $ 1$ to $9$. PS. You had better use hide for answers.

[u]Round 5[/u] [b]5.1[/b] A circle with a radius of $1$ is inscribed in a regular hexagon. This hexagon is inscribed in a larger circle. If the area that is outside the hexagon but inside the larger circle can be expressed as $\frac{a\pi}{b} - c\sqrt{d}$, where $a, b, c, d$ are positive integers, $a, b$ are relatively prime, and no prime perfect square divides into $d$. find the value of $a + b + c + d$. [b]5.2[/b] At a dinner party, $10$ people are to be seated at a round table. If person A cannot be seated next to person $B$ and person $C$ must be next to person $D$, how many ways can the 10 people be seated? Consider rotations of a configuration identical. [b]5.3[/b] Let $N$ be the sum of all the positive integers that are less than $2022$ and relatively prime to $1011$. Find $\frac{N}{2022}$. [u]Round 6[/u] [b]6.1[/b] The line $y = m(x - 6)$ passes through the point $ A$ $(6, 0)$, and the line $y = 8 -\frac{x}{m}$ pass through point $B$ $(0,8)$. The two lines intersect at point $C$. What is the largest possible area of triangle $ABC$? [b]6.2[/b] Let $N$ be the number of ways there are to arrange the letters of the word MATHEMATICAL such that no two As can be adjacent. Find the last $3$ digits of $\frac{N}{100}$. [b]6.3[/b] Find the number of ordered triples of integers $(a, b, c)$ such that $|a|, |b|, |c| \le 100$ and $3abc = a^3 + b^3 + c^3$. [u]Round 7[/u] [b]7.1[/b] In a given plane, let $A, B$ be points such that $AB = 6$. Let $S$ be the set of points such that for any point $C$ in $S$, the circumradius of $\vartriangle ABC$ is at most $6$. Find $a + b + c$ if the area of $S$ can be expressed as $a\pi + b\sqrt{c}$ where $a, b, c$ are positive integers, and $c$ is not divisible by the square of any prime. [b]7.2[/b] Compute $\sum_{1\le a<b<c\le 7} abc$. [b]7.3[/b] Three identical circles are centered at points $A, B$, and $C$ respectively and are drawn inside a unit circle. The circles are internally tangent to the unit circle and externally tangent to each other. A circle centered at point $D$ is externally tangent to circles $A, B$, and $C$. If a circle centered at point $E$ is externally tangent to circles $A, B$, and $D$, what is the radius of circle $E$? The radius of circle $E$ can be expressed as $\frac{a\sqrt{b}-c}{d}$ where $a, b, c$, and d are all positive integers, gcd(a, c, d) = 1, and b is not divisible by the square of any prime. What is the sum of $a + b + c + d$? [u]Round 8[/u] [b]8.[/b] Let $A$ be the number of unused Algebra problems in our problem bank. Let $B$ be the number of times the letter ’b’ appears in our problem bank. Let M be the median speed round score. Finally, let $C$ be the number of correct answers to Speed Round $1$. Estimate $$A \cdot B + M \cdot C.$$ Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2826128p24988676]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Does there exist a number $\alpha, 0 < \alpha < 1$ such that there is an infinite sequence $\{a_n\}$ of positive numbers satisfying \[ 1 + a_{n+1} \leq a_n + \frac{\alpha}{n} \cdot \alpha_n, n = 1,2, \ldots? \]

Let $a,b \in [0,1], c \in [-1,1]$ be reals chosen independently and uniformly at random. What is the probability that $p(x) = ax^2+bx+c$ has a root in $[0,1]$?

Prove for each positive real number $x, y, z$, $$\frac{x^2y}{x+2y}+\frac{y^2z}{y+2z}+\frac{z^2x}{z+2x}<\frac{(x+y+z)^2}{8}$$

Let $f$ be a function such that $f (x + y) = f (x) + f (y)$ for all $x,y \in R$ and $f (1) = 100$. Calculate $\sum_{k = 1}^{10}f (k!)$.

The vertices of a regular $n$-gon are initially marked with one of the signs $+$ or $-$ . A [i]move[/i] consists in choosing three consecutive vertices and changing the signs from the vertices , from $+$ to $-$ and from $-$ to $+$. [b]a)[/b] Prove that if $n=2015$ then for any initial configuration of signs , there exists a sequence of [i]moves[/i] such that we'll arrive at a configuration with only $+$ signs. [b]b)[/b] Prove that if $n=2016$ , then there exists an initial configuration of signs such that no matter how we make the [i]moves[/i] we'll never arrive at a configuration with only $+$ signs.

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Compute $2^2 + 0 \cdot 0 + 2^2 + 3^3$. [b]p2.[/b] How many total letters (not necessarily distinct) are there in the names Jerry, Justin, Jackie, Jason, and Jeffrey? [b]p3.[/b] What is the remainder when $20232023$ is divided by $50$? [b]p4.[/b] Let $ABCD$ be a square. The fraction of the area of $ABCD$ that is the area of the intersection of triangles $ABD$ and $ABC$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ relatively prime positive integers. Find $a + b$. [b]p5.[/b] Raymond is playing basketball. He makes a total of $15$ shots, all of which are either worth $2$ or $3$ points. Given he scored a total of $40$ points, how many $2$-point shots did he make? [b]p6.[/b] If a fair coin is flipped $4$ times, the probability that it lands on heads more often than tails is $\frac{a}{b}$ , where $a$ and $b$ relatively prime positive integers. Find $a + b$. [b]p7.[/b] What is the sum of the perfect square divisors of $640$? [b]p8.[/b] A regular hexagon and an equilateral triangle have the same perimeter. The ratio of the area between the hexagon and equilateral triangle can be expressed in the form $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p9.[/b] If a cylinder has volume $1024\pi$, radius of $r$ and height $h$, how many ordered pairs of integers $(r, h)$ are possible? [b]p10.[/b] Pump $A$ can fill up a balloon in $3$ hours, while pump $B$ can fill up a balloon in $5$ hours. Pump $A$ starts filling up a balloon at $12:00$ PM, and pump $B$ is added alongside pump $A$ at a later time. If the balloon is completely filled at $2:00$ PM, how many minutes after $12:00$ PM was Pump $B$ added? [b]p11.[/b] For some positive integer $k$, the product $81 \cdot k$ has $20$ factors. Find the smallest possible value of $k$. [b]p12.[/b] Two people wish to sit in a row of fifty chairs. How many ways can they sit in the chairs if they do not want to sit directly next to each other and they do not want to sit with exactly one empty chair between them? [b]p13.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $2$ and $M$ be the midpoint of $BC$. Let $P$ be a point in the same plane such that $2PM = BC$. The minimum value of $AP$ can be expressed as $\sqrt{a}-b$, where $a$ and $b$ are positive integers such that $a$ is not divisible by any perfect square aside from $1$. Find $a + b$. [b]p14.[/b] What are the $2022$nd to $2024$th digits after the decimal point in the decimal expansion of $\frac{1}{27}$ , expressed as a $3$ digit number in that order (i.e the $2022$nd digit is the hundreds digit, $2023$rd digit is the tens digit, and $2024$th digit is the ones digit)? [b]p15.[/b] After combining like terms, how many terms are in the expansion of $(xyz+xy+yz+xz+x+y+z)^{20}$? [b]p16.[/b] Let $ABCD$ be a trapezoid with $AB \parallel CD$ where $AB > CD$, $\angle B = 90^o$, and $BC = 12$. A line $k$ is dropped from $A$, perpendicular to line $CD$, and another line $\ell$ is dropped from $C$, perpendicular to line $AD$. $k$ and $\ell$ intersect at $X$. If $\vartriangle AXC$ is an equilateral triangle, the area of $ABCD$ can be written as $m\sqrt{n}$, where $m$ and $n$ are positive integers such that $n$ is not divisible by any perfect square aside from $1$. Find $m + n$. [b]p17.[/b] If real numbers $x$ and $y$ satisfy $2x^2 + y^2 = 8x$, maximize the expression $x^2 + y^2 + 4x$. [b]p18.[/b] Let $f(x)$ be a monic quadratic polynomial with nonzero real coefficients. Given that the minimum value of $f(x)$ is one of the roots of $f(x)$, and that $f(2022) = 1$, there are two possible values of $f(2023)$. Find the larger of these two values. [b]p19.[/b] I am thinking of a positive integer. After realizing that it is four more than a multiple of $3$, four less than a multiple of $4$, four more than a multiple of 5, and four less than a multiple of $7$, I forgot my number. What is the smallest possible value of my number? [b]p20.[/b] How many ways can Aston, Bryan, Cindy, Daniel, and Evan occupy a row of $14$ chairs such that none of them are sitting next to each other? [b]p21.[/b] Let $x$ be a positive real number. The minimum value of $\frac{1}{x^2} +\sqrt{x}$ can be expressed in the form \frac{a}{b^{(c/d)}} , where $a$, $b$, $c$, $d$ are all positive integers, $a$ and $b$ are relatively prime, $c$ and $d$ are relatively prime, and $b$ is not divisible by any perfect square. Find $a + b + c + d$. [b]p22.[/b] For all $x > 0$, the function $f(x)$ is defined as $\lfloor x \rfloor \cdot (x + \{x\})$. There are $24$ possible $x$ such that $f(x)$ is an integer between $2000$ and $2023$, inclusive. If the sum of these $24$ numbers equals $N$, then find $\lfloor N \rfloor$. Note: Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$, called the floor function. Also, $\{x\}$ is defined as $x - \lfloor x \rfloor$, called the fractional part function. [b]p23.[/b] Let $ABCD$ be a rectangle with $AD = 1$. Let $P$ be a point on diagonal $\overline{AC}$, and let $\omega$ and $\xi$ be the circumcircles of $\vartriangle APB$ and $\vartriangle CPD$, respectively. Line $\overleftrightarrow{AD}$ is extended, intersecting $\omega$ at $X$, and $\xi$ at $Y$ . If $AX = 5$ and $DY = 2$, find $[ABCD]^2$. Note: $[ABCD]$ denotes the area of the polygon $ABCD$. [b]p24.[/b] Alice writes all of the three-digit numbers on a blackboard (it’s a pretty big blackboard). Let $X_a$ be the set of three-digit numbers containing a somewhere in its representation, where a is a string of digits. (For example, $X_{12}$ would include $12$, $121$, $312$, etc.) If Bob then picks a value of $a$ at random so $0 \le a \le 999$, the expected number of elements in $X_a$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find$ m + n$. [b]p25.[/b] Let $f(x) = x^5 + 2x^4 - 2x^3 + 4x^2 + 5x + 6$ and $g(x) = x^4 - x^3 + x^2 - x + 1$. If $a$, $b$, $c$, $d$ are the roots of $g(x)$, then find $f(a) + f(b) + f(c) + f(d)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a,b$ be integers and $p$ be a prime number such that: (i) $p$ is the greatest common divisor of $a$ and $b$; (ii) $p^2$ divides $a$. Prove that the polynomial $x^{n+2}+ax^{n+1}+bx^{n}+a+b$ cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than $1$.

Do there exist two integer-valued functions $f$ and $g$ such that for every integer $x$ we have (a) $f(f(x)) = x, g(g(x)) = x, f(g(x)) > x, g(f(x)) > x$ ? (b) $f(f(x)) < x, g(g(x)) < x, f(g(x)) > x, g(f(x)) > x$ ?

Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

When Phil and Shelley stand on a scale together, the scale reads $151$ pounds. When Shelley and Ryan stand on the same scale together, the scale reads $132$ pounds. When Phil and Ryan stand on the same scale together, the scale reads $115$ pounds. Find the number of pounds Shelley weighs.

Let $P(x)$ be a polynomial of degree $n\geq 2$ with rational coefficients such that $P(x) $ has $ n$ pairwise different reel roots forming an arithmetic progression .Prove that among the roots of $P(x) $ there are two that are also the roots of some polynomial of degree $2$ with rational coefficients .

We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point). (a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$. (b) Find the largest possible size of a very set not contained in any line. (Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.) [i]Proposed by Sammy Luo[/i]

If $x,y$ are positive real numbers satisfying $x^3-xy+1=y^3$, find the minimum possible value of $y$.

Find all functions $f:R->R$ satisfying that for every $x$ (real number): $f(x)(1+|f(x)|)\geq x \geq f(x(1+|x|))$

Find all polynomials $f(x)=x^3+bx^2+cx+d$, where $b,c,d,$ are real numbers, such that $f(x^2-2)=-f(-x)f(x)$.

Prove that the equation \[x^3 - 3 \tan\frac{\pi}{12} x^2 - 3x + \tan\frac{\pi}{12}= 0\] has one root $x_1 = \tan \frac{\pi}{36}$, and find the other roots.

Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers. (a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$. (b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.

Let $P(x,y)=(x^2y^3,x^3y^5)$, $P^1=P$ and $P^{n+1}=P\circ P^n$. Also, let $p_n(x)$ be the first coordinate of $P^n(x,x)$, and $f(n)$ be the degree of $p_n(x)$. Find \[\lim_{n\to\infty}f(n)^{1/n}\]

Professor Oak is feeding his $100$ Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is $100$ kilograms. Professor Oak distributes $100$ kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of the bowl). The [i]dissatisfaction level[/i] of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $\lvert N-C\rvert$. Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute food in a way that the sum of the dissatisfaction levels over all the $100$ Pokémon is at most $D$. [i]Oleksii Masalitin, Ukraine[/i]

Let $a = (\sqrt2 +\sqrt3 +\sqrt6)(\sqrt2 +\sqrt3 -\sqrt6)(\sqrt3 +\sqrt6 -\sqrt2)(\sqrt6 +\sqrt2 -\sqrt3)$ $b = (\sqrt2 +\sqrt3 +\sqrt5)(\sqrt2 +\sqrt3 -\sqrt5)(\sqrt3 +\sqrt5 -\sqrt2)(\sqrt5 +\sqrt2 -\sqrt3)$ The difference $a - b$ belongs to the set: A. $(-\infty,-4)$ B. $[-4,0)$ C.$\{0\}$ D. $(0,4]$ E. $(4,\infty)$