Find all pairs $ (x,y)$ of real numbers which satisfy $ x^3\minus{}y^3\equal{}4(x\minus{}y)$ and $ x^3\plus{}y^3\equal{}2(x\plus{}y)$.

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions: $(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$ $(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.

Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that \[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]

Let $(x, y)$ be an intersection of the equations $y = 4x^2 - 28x + 41$ and $x^2 + 25y^2 - 7x + 100y +\frac{349}{4}= 0$. Find the sum of all possible values of $x$.

Let $f$ and $g$ be two polynomials with integer coefficients such that the leading coefficients of both the polynomials are positive. Suppose $\deg(f)$ is odd and the sets $\{f(a)\mid a\in \mathbb{Z}\}$ and $\{g(a)\mid a\in \mathbb{Z}\}$ are the same. Prove that there exists an integer $k$ such that $g(x)=f(x+k)$.

Assume that complex numbers $z_1,z_2,...,z_n$ satisfy $|z_i-z_j| \le 1$ for any $1 \le i <j \le n$. Let $$S= \sum_{1 \le i <j \le n} |z_i-z_j|^2.$$ (1) If $n = 6063$, find the maximum value of $S$. (2) If $n= 2021$, find the maximum value of $S$.

It is known about the function $f : R \to R$ that $\bullet$ $f(x) > f(y)$ for all real $x > y$ $\bullet$ $f(x) > x$ for all real $x$ $\bullet$ $f(2x - f (x)) = x$ for all real $x$. Prove that $f(x) = x + f(0)$ for all real numbers $x$.

Let $a, b,$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove the following inequality $$a \sqrt[3]{\frac{b}{a}} + b \sqrt[3]{\frac{c}{b}} + c \sqrt[3]{\frac{a}{c}} \le ab + bc + ca + \frac{2}{3}.$$ Proposed by [i]Anastasija Trajanova, Macedonia[/i]

Let $f(X)$ be a monic irreducible polynomial over $\mathbb{Z}$; therefore, by Gauss's Lemma, $f$ is also irreducible over $\mathbb{Q}$ (you may assume this). Moreover, assume $f(X) \mid f\left(X^2+n\right)$ where $n$ is an integer such that $n \notin\{-1,0,1\}$. Show that $n^2 \nmid f(0)$.

In this question we re-define the operations addition and multiplication as follows: $a + b$ is defined as the minimum of $a$ and $b$, while $a * b$ is defined to be the sum of $a$ and $b$. For example, $3+4 = 3$, $3*4 = 7$, and $$3*4^2+5*4+7 = \min(\text{3 plus 4 plus 4}, \text{5 plus 4}, 7) = \min(11, 9, 7) = 7.$$ Let $a, b, c$ be real numbers. Characterize, in terms of $a, b, c$, what the graph of $y = ax^2+bx+c$ looks like.

Let $ {{a} _ {1}}, {{a} _ {2}}, \ldots, {{a} _ {n}} $ be permutation of numbers $ 1,2, \ldots, n $, where $ n \geq 2 $. Find the maximum value of the sum $$ S (n) = | {{a} _ {1}} - {{a} _ {2}} | + | {{a} _ {2}} - {{a} _ {3}} | + \cdots + | {{a} _ {n-1}} - {{a} _ {n}} |. $$

(Eisentein's Criterion) Let $f(x)=a_{n}x^{n} +\cdots +a_{1}x+a_{0}$ be a nonconstant polynomial with integer coefficients. If there is a prime $p$ such that $p$ divides each of $a_{0}$, $a_{1}$, $\cdots$,$a_{n-1}$ but $p$ does not divide $a_{n}$ and $p^2$ does not divide $a_{0}$, then $f(x)$ is irreducible in $\mathbb{Q}[x]$.

Let $f(x)=\frac{\sqrt{2+\sqrt2}x+\sqrt{2-\sqrt2}}{-\sqrt{2-\sqrt2}x+\sqrt{2+\sqrt2}}$. Find $\underbrace{f(f(\cdots f}_{1987\text{ times}}(x)\cdots))$.

Find all functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $f (0)=2022$ and $f (x+g(y)) =xf(y)+(2023-y)f(x)+g(x)$ for all $x, y \in \mathbb{R}$.

Alan is assigning values to lattice points on the 3d coordinate plane. First, Alan computes the roots of the cubic $20x^3-22x^2+2x+1$ and finds that they are $\alpha$, $\beta$, and $\gamma$. He finds out that each of these roots satisfy $|\alpha|,|\beta|,|\gamma|\leq 1$ On each point $(x,y,z)$ where $x,y,$ and $z$ are all nonnegative integers, Alan writes down $\alpha^x\beta^y\gamma^z$. What is the value of the sum of all numbers he writes down? [i]Proposed by Alan Abraham[/i]

Let $n$ be a positive integer. Prove that $$\frac{a_1}{b_1}+ \frac{a_2}{b_2}+ ...+\frac{a_n}{b_n} \ge n $$ where $(b_1, b_2, ..., b_n)$ is any permutation of the positive real numbers $a_1, a_2, ..., a_n$.

Show that there do not exist non-zero real numbers $a,b,c$ such that the following statements hold simultaneously: $\bullet$ the equation $ax^2 + bx + c = 0$ has two distinct roots $x_1,x_2$; $\bullet$ the equation $bx^2 + cx + a = 0$ has two distinct roots $x_2,x_3$; $\bullet$ the equation $cx^2 + ax + b = 0$ has two distinct roots $x_3,x_1$. (Note that $x_1,x_2,x_3$ may be real or complex numbers.)

Consider a sequence of 100 positive integers. Each member of the sequence, starting from the second one, is derived by either multiplying the previous number by 2 or dividing it by 16. Is it possible for the sum of these 100 numbers to be equal to $2^{2023}$? [i]Proposed by Nika Glunchadze, Georgia[/i]

Let $A$ and $B$ be $2\times 2$ matrices with integer entries such that $A, A+B, A+2B, A+3B,$ and $A+4B$ are all invertible matrices whose inverses have integer entries. Show that $A+5B$ is invertible and that its inverse has integer entries.

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Somya has a football game $4$ days from today. If the day before yesterday was Wednesday, what day of the week is the game? [b]p2.[/b] Sammy writes the following equation: $$\frac{2 + 2}{8 + 8}=\frac{x}{8}.$$ What is the value of $x$ in Sammy's equation? [b]p3.[/b] On $\pi$ day, Peter buys $7$ pies. The pies costed $\$3$, $\$1$, $\$4$, $\$1$, $\$5$, $\$9$, and $\$2$. What was the median price of Peter's $7$ pies in dollars? [b]p4.[/b] Antonio draws a line on the coordinate plane. If the line passes through the points ($1, 3$) and ($-1,-1$), what is slope of the line? [b]p5.[/b] Professor Varun has $25$ students in his science class. He divides his students into the maximum possible number of groups of $4$, but $x$ students are left over. What is $x$? [b]p6.[/b] Evaluate the following: $$4 \times 5 \div 6 \times 3 \div \frac47$$ [b]p7.[/b] Jonny, a geometry expert, draws many rectangles with perimeter $16$. What is the area of the largest possible rectangle he can draw? [b]p8.[/b] David always drives at $60$ miles per hour. Today, he begins his trip to MIT by driving $60$ miles. He stops to take a $20$ minute lunch break and then drives for another $30$ miles to reach the campus. What is the total time in minutes he spends getting to MIT? [b]p9.[/b] Richard has $5$ hats: blue, green, orange, red, and purple. Richard also has 5 shirts of the same colors: blue, green, orange, red, and purple. If Richard needs a shirt and a hat of different colors, how many outts can he wear? [b]p10.[/b] Poonam has $9$ numbers in her bag: $1, 2, 3, 4, 5, 6, 7, 8, 9$. Eric runs by with the number $36$. How many of Poonam's numbers evenly divide Eric's number? [b]p11.[/b] Serena drives at $45$ miles per hour. If her car runs at $6$ miles per gallon, and each gallon of gas costs $2$ dollars, how many dollars does she spend on gas for a $135$ mile trip? [b]p12.[/b] Grace is thinking of two integers. Emmie observes that the sum of the two numbers is $56$ but the difference of the two numbers is $30$. What is the sum of the squares of Grace's two numbers? [b]p13.[/b] Chang stands at the point ($3,-3$). Fang stands at ($-3, 3$). Wang stands in-between Chang and Fang; Wang is twice as close to Fang as to Chang. What is the ordered pair that Wang stands at? [b]p14.[/b] Nithin has a right triangle. The longest side has length $37$ inches. If one of the shorter sides has length $12$ inches, what is the perimeter of the triangle in inches? [b]p15.[/b] Dora has $2$ red socks, $2$ blue socks, $2$ green socks, $2$ purple socks, $3$ black socks, and $4$ gray socks. After a long snowstorm, her family loses electricity. She picks socks one-by-one from the drawer in the dark. How many socks does she have to pick to guarantee a pair of socks that are the same color? [b]p16.[/b] Justin selects a random positive $2$-digit integer. What is the probability that the sum of the two digits of Justin's number equals $11$? [b]p17.[/b] Eddie correctly computes $1! + 2! + .. + 9! + 10!$. What is the remainder when Eddie's sum is divided by $80$? [b]p18.[/b] $\vartriangle PQR$ is drawn such that the distance from $P$ to $\overline{QR}$ is $3$, the distance from $Q$ to $\overline{PR}$ is $4$, and the distance from $R$ to $\overline{PQ}$ is $5$. The angle bisector of $\angle PQR$ and the angle bisector of $\angle PRQ$ intersect at $I$. What is the distance from $I$ to $\overline{PR}$? [b]p19.[/b] Maxwell graphs the quadrilateral $|x - 2| + |y + 2| = 6$. What is the area of the quadrilateral? [b]p20.[/b] Uncle Gowri hits a speed bump on his way to the hospital. At the hospital, patients who get a rare disease are given the option to choose treatment $A$ or treatment $B$. Treatment $A$ will cure the disease $\frac34$ of the time, but since the treatment is more expensive, only $\frac{8}{25}$ of the patients will choose this treatment. Treatment $B$ will only cure the disease $\frac{1}{2}$ of the time, but since it is much more aordable, $\frac{17}{25}$ of the patients will end up selecting this treatment. Given that a patient was cured, what is the probability that the patient selected treatment $A$? [b]p21.[/b] In convex quadrilateral $ABCD$, $AC = 28$ and $BD = 15$. Let $P, Q, R, S$ be the midpoints of $AB$, $BC$, $CD$ and $AD$ respectively. Compute $PR^2 + QS^2$. [b]p22.[/b] Charlotte writes the polynomial $p(x) = x^{24} - 6x + 5$. Let its roots be $r_1$, $r_2$, $...$, $r_{24}$. Compute $r^{24}_1 +r^{24}_2 + r^{24}_3 + ... + r^{24}_24$. [b]p23.[/b] In rectangle $ABCD$, $AB = 6$ and $BC = 4$. Let $E$ be a point on $CD$, and let $F$ be the point on $AB$ which lies on the bisector of $\angle BED$. If $FD^2 + EF^2 = 52$, what is the length of $BE$? [b]p24.[/b] In $\vartriangle ABC$, the measure of $\angle A$ is $60^o$ and the measure of $\angle B$ is $45^o$. Let $O$ be the center of the circle that circumscribes $\vartriangle ABC$. Let $I$ be the center of the circle that is inscribed in $\vartriangle ABC$. Finally, let $H$ be the intersection of the $3$ altitudes of the triangle. What is the angle measure of $\angle OIH$ in degrees? [b]p25.[/b] Kaitlyn fully expands the polynomial $(x^2 + x + 1)^{2018}$. How many of the coecients are not divisible by $3$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the function $ h$, defined for all positive real numbers, such that: $$10x -6h(x) = 4h \left(\frac{2020}{x}\right) $$ for all $x > 0$. Find $h(x)$ and the value of $h(4)$.

The polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0,P_{1}(x)=x$ and \[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2\] For every natural number $n\geq 1$, find all real numbers $x$ satisfying the equation $P_{n}(x)=0$.

Given three positive real numbers $a,b,c$ such that following holds $a^2=b^2+bc$, $b^2=c^2+ac$ Prove that $\frac{1}{c}=\frac{1}{a}+\frac{1}{b}$.

Given that $P(x)$ is the least degree polynomial with rational coefficients such that \[P(\sqrt{2} + \sqrt{3}) = \sqrt{2},\] find $P(10)$.

For some positive numbers $a$, $b$, $c$ and $d$, we know that $$ \frac{1}{a^3 + 1}+ \frac{1}{b^3 + 1}+ \frac{1}{c^3 + 1} + \frac{1}{d^3 + 1} = 2. $$ Prove that $$ \frac{1 - a}{a^2 - a + 1} + \frac{1-b}{b^2 - b + 1} + \frac{1-c}{c^2 - c + 1} +\frac{1-d}{d^2 - d + 1} \geq 0. $$