For all positive real numbers $x$ and $y$ let \[f(x,y)=\min\left( x,\frac{y}{x^2+y^2}\right) \] Show that there exist $x_0$ and $y_0$ such that $f(x, y)\le f(x_0, y_0)$ for all positive $x$ and $y$, and find $f(x_0,y_0)$.

Find the maximum value of: $E(a,b)=\frac{a+b}{(4a^2+3)(4b^2+3)}$ For $a,b$ real numbers.

Prove that for any polynomial $P$ with integer coefficients and any natural number $k$ there exists a natural number $n$ such that $P(1) + P(2) + ...+ P(n)$ is divisible by $k$.

Edward’s formula for the stock market predicts correctly that the price of HMMT is directly proportional to a secret quantity $x$ and inversely proportional to $y$, the number of hours he slept the night before. If the price of HMMT is $\$12$ when $x = 8$ and $y = 4$, how many dollars does it cost when $x = 4$ and $y = 8$?

Let $a,b,c\ge0$ and $a+b+c=3.$ Prove that $(3a-bc)(3b-ac)(3c-ab)\le8.$ (O. Rudenko)

Let \( P_1(x) \) and \( P_2(x) \) be monic quadratic trinomials, and let \( A_1 \) and \( A_2 \) be the vertices of the parabolas \( y = P_1(x) \) and \( y = P_2(x) \), respectively. Let \( m(g(x)) \) denote the minimum value of the function \( g(x) \). It is known that the differences \( m(P_1(P_2(x))) - m(P_1(x)) \) and \( m(P_2(P_1(x))) - m(P_2(x)) \) are equal positive numbers. Find the angle between the line \( A_1A_2 \) and the $x$-axis. \\

An infinite increasing sequence of positive integers $n_j (j = 1, 2, \ldots )$ has the property that for a certain $c$, \[\frac{1}{N}\sum_{n_j\le N} n_j \le c,\] for every $N >0$. Prove that there exist finitely many sequences $m^{(i)}_j (i = 1, 2,\ldots, k)$ such that \[\{n_1, n_2, \ldots \} =\bigcup_{i=1}^k\{m^{(i)}_1 ,m^{(i)}_2 ,\ldots\}\] and \[m^{(i)}_{j+1} > 2m^{(i)}_j (1 \le i \le k, j = 1, 2,\ldots).\]

What is the sum of the positive solutions to $2x^2 -\lfloor x \rfloor = 5$, where $\lfloor x \rfloor$ is the largest integer less than or equal to $x$?

The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.

[b]p1.[/b] There exist real numbers $B$, $M$, and $T$ such that $B + M + T = 23$ and $B - M - T = 20$. Compute $M + T$. [b]p2.[/b] Kaity has a rectangular garden that measures $10$ yards by $12$ yards. Austin’s triangular garden has side lengths $6$ yards, $8$ yards, and $10$ yards. Compute the ratio of the area of Kaity’s garden to the area of Austin’s garden. [b]p3.[/b] Nikhil’s mom and brother both have ages under $100$ years that are perfect squares. His mom is $33$ years older than his brother. Compute the sum of their ages. [b]p4.[/b] Madison wants to arrange $3$ identical blue books and $2$ identical pink books on a shelf so that each book is next to at least one book of the other color. In how many ways can Madison arrange the books? [b]p5.[/b] Two friends, Anna and Bruno, are biking together at the same initial speed from school to the mall, which is $6$ miles away. Suddenly, $1$ mile in, Anna realizes that she forgot her calculator at school. If she bikes $4$ miles per hour faster than her initial speed, she could head back to school and still reach the mall at the same time as Bruno, assuming Bruno continues biking towards the mall at their initial speed. In miles per hour, what is Anna and Bruno’s initial speed, before Anna has changed her speed? (Assume that the rate at which Anna and Bruno bike is constant.) [b]p6.[/b] Let a number be “almost-perfect” if the sum of its digits is $28$. Compute the sum of the third smallest and third largest almost-perfect $4$-digit positive integers. [b]p7.[/b] Regular hexagon $ABCDEF$ is contained in rectangle $PQRS$ such that line $\overline{AB}$ lies on line $\overline{PQ}$, point $C$ lies on line $\overline{QR}$, line $\overline{DE}$ lies on line $\overline{RS}$, and point $F$ lies on line $\overline{SP}$. Given that $PQ = 4$, compute the perimeter of $AQCDSF$. [img]https://cdn.artofproblemsolving.com/attachments/6/7/5db3d5806eaefa00d7fc90fb786a41c0466a90.png[/img] [b]p8.[/b] Compute the number of ordered pairs $(m, n)$, where $m$ and $n$ are relatively prime positive integers and $mn = 2520$. (Note that positive integers $x$ and $y$ are relatively prime if they share no common divisors other than $1$. For example, this means that $1$ is relatively prime to every positive integer.) [b]p9.[/b] A geometric sequence with more than two terms has first term $x$, last term $2023$, and common ratio $y$, where $x$ and $y$ are both positive integers greater than $1$. An arithmetic sequence with a finite number of terms has first term $x$ and common difference $y$. Also, of all arithmetic sequences with first term $x$, common difference $y$, and no terms exceeding $2023$, this sequence is the longest. What is the last term of the arithmetic sequence? [b]p10.[/b] Andrew is playing a game where he must choose three slips, uniformly at random and without replacement, from a jar that has nine slips labeled $1$ through $9$. He wins if the sum of the three chosen numbers is divisible by $3$ and one of the numbers is $1$. What is the probability Andrew wins? [b]p11.[/b] Circle $O$ is inscribed in square $ABCD$. Let $E$ be the point where $O$ meets line segment $\overline{AB}$. Line segments $\overline{EC}$ and $\overline{ED}$ intersect $O$ at points $P$ and $Q$, respectively. Compute the ratio of the area of triangle $\vartriangle EPQ$ to the area of triangle $\vartriangle ECD$. [b]p12.[/b] Define a recursive sequence by $a_1 = \frac12$ and $a_2 = 1$, and $$a_n =\frac{1 + a_{n-1}}{a_{n-2}}$$ for n ≥ 3. The product $a_1a_2a_3 ... a_{2023}$ can be expressed in the form $a^b \cdot c^d \cdot e^f$ , where $a$, $b$, $c$, $d$, $e$, and $f$ are positive (not necessarily distinct) integers, and a, c, and e are prime. Compute $a + b + c + d + e + f$. [b]p13.[/b] An increasing sequence of $3$-digit positive integers satisfies the following properties: $\bullet$ Each number is a multiple of $2$, $3$, or $5$. $\bullet$ Adjacent numbers differ by only one digit and are relatively prime. (Note that positive integers x and y are relatively prime if they share no common divisors other than $1$.) What is the maximum possible length of the sequence? [b]p14.[/b] Circles $O_A$ and $O_B$ with centers $A$ and $B$, respectively, have radii $3$ and $8$, respectively, and are internally tangent to each other at point $P$. Point $C$ is on circle $O_A$ such that line $\overline{BC}$ is tangent to circle $OA$. Extend line $\overline{PC}$ to intersect circle $O_B$ at point $D \ne P$. Compute $CD$. [b]p15.[/b] Compute the product of all real solutions $x$ to the equation $x^2 + 20x - 23 = 2 \sqrt{x^2 + 20x + 1}$. [b]p16.[/b] Compute the number of divisors of $729, 000, 000$ that are perfect powers. (A perfect power is an integer that can be written in the form $a^b$, where $a$ and $b$ are positive integers and $b > 1$.) [b]p17.[/b] The arithmetic mean of two positive integers $x$ and $y$, each less than $100$, is $4$ more than their geometric mean. Given $x > y$, compute the sum of all possible values for $x + y$. (Note that the geometric mean of $x$ and $y$ is defined to be $\sqrt{xy}$.) [b]p18.[/b] Ankit and Richard are playing a game. Ankit repeatedly writes the digits $2$, $0$, $2$, $3$, in that order, from left to right on a board until Richard tells him to stop. Richard wins if the resulting number, interpreted as a base-$10$ integer, is divisible by as many positive integers less than or equal to $12$ as possible. For example, if Richard stops Ankit after $7$ digits have been written, the number would be $2023202$, which is divisible by $1$ and $2$. Richard wants to win the game as early as possible. Assuming Ankit must write at least one digit, after how many digits should Richard stop Ankit? [b]p19.[/b] Eight chairs are set around a circular table. Among these chairs, two are red, two are blue, two are green, and two are yellow. Chairs that are the same color are identical. If rotations and reflections of arrangements of chairs are considered distinct, how many arrangements of chairs satisfy the property that each pair of adjacent chairs are different colors? [b]p20.[/b] Four congruent spheres are placed inside a right-circular cone such that they are all tangent to the base and the lateral face of the cone, and each sphere is tangent to exactly two other spheres. If the radius of the cone is $1$ and the height of the cone is $2\sqrt2$, what is the radius of one of the spheres? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On the plane, a Cartesian coordinate system is chosen. Given points $ A_1,A_2,A_3,A_4$ on the parabola $ y \equal{} x^2$, and points $ B_1,B_2,B_3,B_4$ on the parabola $ y \equal{} 2009x^2$. Points $ A_1,A_2,A_3,A_4$ are concyclic, and points $ A_i$ and $ B_i$ have equal abscissas for each $ i \equal{} 1,2,3,4$. Prove that points $ B_1,B_2,B_3,B_4$ are also concyclic.

Given an integer $N \geq 4$, determine the largest value the sum $$\sum_{i=1}^{\left \lfloor{\frac{k}{2}}\right \rfloor+1}\left( \left \lfloor{\frac{n_i}{2}}\right \rfloor+1\right)$$ may achieve, where $k, n_1, \ldots, n_k$ run through the integers subject to $k \geq 3$, $n_1 \geq \ldots\geq n_k\geq 1$ and $n_1 + \ldots + n_k = N$.

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $ x,y$: $ x f(x)\minus{}yf(y)\equal{}(x\minus{}y)f(x\plus{}y)$.

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

(a) Show that for every $n\in\mathbb{N}$ there is exactly one $x\in\mathbb{R}^+$ so that $x^n+x^{n+1}=1$. Call this $x_n$. (b) Find $\lim\limits_{n\rightarrow+\infty}x_n$.

Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$. [i]Carl Lian and Brian Hamrick.[/i]

Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$ [i]Proposed by Pierre Bornsztein, France[/i]

Let $P(x,\, y)=2x^{2}-6xy+5y^{2}$. Let us say an integer number $a$ is a value of $P$ if there exist integer numbers $b$, $c$ such that $P(b,\, c)=a$. a) Find all values of $P$ lying between 1 and 100. b) Show that if $r$ and $s$ are values of $P$, then so is $rs$.

Prove that for every $x$ such that $\sin x \ne 0$, there is such natural $n$, which $$ | \sin nx| \ge \frac{\sqrt3}{2}.$$

Compute the sum of all real numbers $x$ such that \[2x^6-3x^5+3x^4+x^3-3x^2+3x-1=0.\]

Let $n$ be a positive integer and let $a_1, \ldots, a_{n-1} $ be arbitrary real numbers. Define the sequences $u_0, \ldots, u_n $ and $v_0, \ldots, v_n $ inductively by $u_0 = u_1 = v_0 = v_1 = 1$, and $u_{k+1} = u_k + a_k u_{k-1}$, $v_{k+1} = v_k + a_{n-k} v_{k-1}$ for $k=1, \ldots, n-1.$ Prove that $u_n = v_n.$

Find all the real number sequences $\{b_n\}_{n \geq 1}$ and $\{c_n\}_{n \geq 1}$ that satisfy the following conditions: (i) For any positive integer $n$, $b_n \leq c_n$; (ii) For any positive integer $n$, $b_{n+1}$ and $c_{n+1}$ is the two roots of the equation $x^2+b_nx+c_n=0$.

The unknown real numbers $x_1,x_2,\dots,x_n$ satisfy $x_1 < x_2 < \cdots < x_n,$ where $n \geq 3$. The numbers $s$, $t$ and $d_1,d_2,\dots,d_{n - 2}$ are given, such that \[ \begin{aligned} s & = \sum\limits_{i = 1}^nx_i, \\ t & = \sum\limits_{i = 1}^nx_i^2,\\ d_i & = x_{i + 2} - x_i,\ \ i = 1,2,\dots,n - 2. \end{aligned} \] For which $n$ is this information always sufficient to determine $x_1,x_2,\dots,x_n$ uniquely?

For the system of equations $x^2 + x^2y^2 + x^2y^4 = 525$ and $x + xy + xy^2 = 35$, the sum of the real $y$ values that satisfy the equations is $ \mathrm{(A) \ } 2 \qquad \mathrm{(B) \ } \frac{5}{2} \qquad \mathrm {(C) \ } 5 \qquad \mathrm{(D) \ } 20 \qquad \mathrm{(E) \ } \frac{55}{2}$