Given some monic polynomials $P_1, \ldots, P_n$ with real coefficients, for any real number $y$, let $S_y$ be the set of real number $x$ such that $y = P_i(x)$ for some $i = 1, 2, ..., n$. If the sets $S_{y_1}, S_{y_2}$ have the same size for any two real numbers $y_1, y_2$, show that $P_1, \ldots, P_n$ have the same degree. [i] Proposed by usjl[/i]

Let $x_1, x_2, . . . , x_{2022}$ be nonzero real numbers. Suppose that $x_k + \frac{1}{x_{k+1}} < 0$ for each $1 \leq k \leq 2022$, where $x_{2023}=x_1$. Compute the maximum possible number of integers $1 \leq n \leq 2022$ such that $x_n > 0$.

Bartek patiently performs operations on fractions. In each move, he adds its inverse to the current result, obtaining a new result. Bartek starts with the number $1$: after the first move, he receives the result 2, after the second move, the result is $\frac{5}{2}$, after the third move $\frac{29}{10}$, etc. After $300$ moves, Bartek receives the result $x$. Determine the largest integer not greater than $x$.

Let $ a_1\equal{}2, a_2\equal{}5$ and $ a_{n\plus{}2}\equal{}(2\minus{}n^2)a_{n\plus{}1}\plus{} (2\plus{}n^2)a_n$ for $ n\geq 1$. Do there exist $ p,q,r$ so that $ a_pa_q \equal{}a_r$?

Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ we have: $$xf(x+y)+yf(y-x) = f(x^2+y^2)\,.$$ [i]Authored by Nikola Velov[/i]

Suppose that $n$ is a natural number and $n$ is not divisible by $3$. Prove that $(n^{2n}+n^n+n+1)^{2n}+(n^{2n}+n^n+n+1)^n+1$ has at least $2d(n)$ distinct prime factors where $d(n)$ is the number of positive divisors of $n$. [i]proposed by Mahyar Sefidgaran[/i]

Johnny writes down quadratic equation \[ax^2 + bx + c = 0.\] with positive integer coefficients $a, b, c$. Then Pete changes one, two, or none “$+$” signs to “$-$”. Johnny wins, if both roots of the (changed) equation are integers. Otherwise (if there are no real roots or at least one of them is not an integer), Pete wins. Can Johnny choose the coefficients in such a way that he will always win?

Three friends A, B and C have a total of $120$ kroner. First, A gives as much money to B as B already has. Next, B gives as many money to C that C already has. In the end, C gives the same amount of money to A as A now has. After these transactions, A, B and C have equal amounts of money. How many money did each of the three companions have originally?

Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that: [i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$ [i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$ [i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$

An arithmetic function is a real-valued function whose domain is the set of positive integers. Define the convolution product of two arithmetic functions $ f$ and $ g$ to be the arithmetic function $ f * g$, where \[ (f * g)(n) \equal{} \sum_{ij\equal{}n} f(i) \cdot g(j),\] and $ f^{*k} \equal{} f * f * \ldots * f$ ($ k$ times) We say that two arithmetic functions $ f$ and $ g$ are dependent if there exists a nontrivial polynomial of two variables $ P(x, y) \equal{} \sum_{i,j} a_{ij} x^i y^j$ with real coefficients such that \[ P(f,g) \equal{} \sum_{i,j} a_{ij} f^{*i} * g^{*j} \equal{} 0,\] and say that they are independent if they are not dependent. Let $ p$ and $ q$ be two distinct primes and set \[ f_1(n) \equal{} \begin{cases} 1 & \text{ if } n \equal{} p, \\ 0 & \text{ otherwise}. \end{cases}\] \[ f_2(n) \equal{} \begin{cases} 1 & \text{ if } n \equal{} q, \\ 0 & \text{ otherwise}. \end{cases}\] Prove that $ f_1$ and $ f_2$ are independent.

We build a modified version of Pascal’s triangle as follows: in the first row we write a $2$ and a $3$, and in the further rows, every number is the sum of the two numbers directly above it (and rows always begin with a $2$ and end with a $3$). In the $13$th row, what is the $5$th number from the left? [img]https://cdn.artofproblemsolving.com/attachments/7/2/58e1a9f43fa7c304bfd285fc1b73bed883e9a6.png[/img]

Determine all polynomials $P(x)$ with real coeffcients such that $(x^3+3x^2+3x+2)P(x-1)=(x^3-3x^2+3x-2)P(x)$.

Prove that the sum $\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}} +\sqrt[3]{\frac{a+1}{2}-\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}}$ is independent of $a$ for $ a \ge - \frac{3}{4}$ and evaluate it.

Does there exist positive reals $a_0, a_1,\ldots ,a_{19}$, such that the polynomial $P(x)=x^{20}+a_{19}x^{19}+\ldots +a_1x+a_0$ does not have any real roots, yet all polynomials formed from swapping any two coefficients $a_i,a_j$ has at least one real root?

For $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2 \geq 3$,show that: $\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$.

For every positive $ \alpha$, natural number $ n$, and at most $ \alpha n$ points $ x_i$, construct a trigonometric polynomial $ P(x)$ of degree at most $ n$ for which \[ P(x_i) \leq 1, \; \int_0^{2 \pi} P(x)dx=0,\ \; \textrm{and}\ \; \max P(x) > cn\ ,\] where the constant $ c$ depends only on $ \alpha$. [i]G. Halasz[/i]

Let $a \geq 3$ be a real number, and $P$ a polynomial of degree $n$ and having real coefficients. Prove that at least one of the following numbers is greater than or equal to $1:$ $$|a^0- P(0)|, \ |a^1-P(1)| , \ |a^2-P(2)|, \cdots, |a^{n + 1}-P(n + 1)|.$$

Find the amount of different values given by the following expression: $\frac{n^2-2}{n^2-n+2}$ where $ n \in \{1,2,3,..,100\}$

[b]a)[/b] Let be two nonegative integers $ n\ge 1,k, $ and $ n $ real numbers $ a,b,\ldots ,c. $ Prove that $$ (1/a+1/b+\cdots 1/c)\left( a^{1+k} +b^{1+k}+\cdots c^{1+k} \right)\ge n\left(a^k+b^k+\cdots +c^k\right) . $$ [b]b)[/b] If $ 1\le d\le e\le f\le g\le h\le i\le 1000 $ are six real numbers, determine the minimum value the expression $$ d/e+f/g+h/i $$ can take.

find the number of ordered triples $(x,y,z)$ of real numbers that satisfy the system of equations: $x+y+z=7; x^2+y^2+z^2=27; xyz=5$.

Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfy the following property for all real numbers $x$ and all polynomials $P$ with real coefficients: If $P(f(x)) = 0$, then $f(P(x)) = 0$.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(2x + y + f(x + y)) + f(xy) = y f(x) \] for all real numbers $x$ and $y$.

The expression $\frac{x^2-3x+2}{x^2-5x+6}\div \frac{x^2-5x+4}{x^2-7x+12}$, when simplified is: $ \textbf{(A)}\ \frac{(x-1)(x-6)}{(x-3)(x-4)} \qquad\textbf{(B)}\ \frac{x+3}{x-3}\qquad\textbf{(C)}\ \frac{x+1}{x-1}\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2$

[b]p1.[/b] Nikhil computes the sum of the first $10$ positive integers, starting from $1$. He then divides that sum by 5. What remainder does he get? [b]p2.[/b] In class, starting at $8:00$, Ava claps her hands once every $4$ minutes, while Ella claps her hands once every $6$ minutes. What is the smallest number of minutes after $8:00$ such that both Ava and Ella clap their hands at the same time? [b]p3.[/b] A triangle has side lengths $3$, $4$, and $5$. If all of the side lengths of the triangle are doubled, how many times larger is the area? [b]p4.[/b] There are $50$ students in a room. Every student is wearing either $0$, $1$, or $2$ shoes. An even number of the students are wearing exactly $1$ shoe. Of the remaining students, exactly half of them have $2$ shoes and half of them have $0$ shoes. How many shoes are worn in total by the $50$ students? [b]p5.[/b] What is the value of $-2 + 4 - 6 + 8 - ... + 8088$? [b]p6.[/b] Suppose Lauren has $2$ cats and $2$ dogs. If she chooses $2$ of the $4$ pets uniformly at random, what is the probability that the 2 chosen pets are either both cats or both dogs? [b]p7.[/b] Let triangle $\vartriangle ABC$ be equilateral with side length $6$. Points $E$ and $F$ lie on $BC$ such that $E$ is closer to $B$ than it is to $C$ and $F$ is closer to $C$ than it is to $B$. If $BE = EF = FC$, what is the area of triangle $\vartriangle AFE$? [b]p8.[/b] The two equations $x^2 + ax - 4 = 0$ and $x^2 - 4x + a = 0$ share exactly one common solution for $x$. Compute the value of $a$. [b]p9.[/b] At Shreymart, Shreyas sells apples at a price $c$. A customer who buys $n$ apples pays $nc$ dollars, rounded to the nearest integer, where we always round up if the cost ends in $.5$. For example, if the cost of the apples is $4.2$ dollars, a customer pays $4$ dollars. Similarly, if the cost of the apples is $4.5$ dollars, a customer pays $5$ dollars. If Justin buys $7$ apples for $3$ dollars and $4$ apples for $1$ dollar, how many dollars should he pay for $20$ apples? [b]p10.[/b] In triangle $\vartriangle ABC$, the angle trisector of $\angle BAC$ closer to $\overline{AC}$ than $\overline{AB}$ intersects $\overline{BC}$ at $D$. Given that triangle $\vartriangle ABD$ is equilateral with area $1$, compute the area of triangle $\vartriangle ABC$. [b]p11.[/b] Wanda lists out all the primes less than $100$ for which the last digit of that prime equals the last digit of that prime's square. For instance, $71$ is in Wanda's list because its square, $5041$, also has $1$ as its last digit. What is the product of the last digits of all the primes in Wanda's list? [b]p12.[/b] How many ways are there to arrange the letters of $SUSBUS$ such that $SUS$ appears as a contiguous substring? For example, $SUSBUS$ and $USSUSB$ are both valid arrangements, but $SUBSSU$ is not. [b]p13.[/b] Suppose that $x$ and $y$ are integers such that $x \ge 5$, $y \ge 3$, and $\sqrt{x - 5} +\sqrt{y - 3} = \sqrt{x + y}$. Compute the maximum possible value of $xy$. [b]p14.[/b] What is the largest integer $k$ divisible by $14$ such that $x^2-100x+k = 0$ has two distinct integer roots? [b]p15.[/b] What is the sum of the first $16$ positive integers whose digits consist of only $0$s and $1$s? [b]p16.[/b] Jonathan and Ajit are flipping two unfair coins. Jonathan's coin lands on heads with probability $\frac{1}{20}$ while Ajit's coin lands on heads with probability $\frac{1}{22}$ . Each year, they flip their coins at thesame time, independently of their previous flips. Compute the probability that Jonathan's coin lands on heads strictly before Ajit's coin does. [b]p17.[/b] A point is chosen uniformly at random in square $ABCD$. What is the probability that it is closer to one of the $4$ sides than to one of the $2$ diagonals? [b]p18.[/b] Two integers are coprime if they share no common positive factors other than $1$. For example, $3$ and $5$ are coprime because their only common factor is $1$. Compute the sum of all positive integers that are coprime to $198$ and less than $198$. [b]p19.[/b] Sumith lists out the positive integer factors of $12$ in a line, writing them out in increasing order as $1$, $2$, $3$, $4$, $6$, $12$. Luke, being the mischievious person he is, writes down a permutation of those factors and lists it right under Sumith's as $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, $a_6$. Luke then calculates $$gcd(a_1, 2a_2, 3a_3, 4a_4, 6a_5, 12a_6).$$ Given that Luke's result is greater than $1$, how many possible permutations could he have written? [b]p20.[/b] Tetrahedron $ABCD$ is drawn such that $DA = DB = DC = 2$, $\angle ADB = \angle ADC = 90^o$, and $\angle BDC = 120^o$. Compute the radius of the sphere that passes through $A$, $B$, $C$, and $D$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Do there exist a infinite sequence of positive integers $(a_{n})$ ,such that for any $n\ge 1$ the relation $ a_{n+2}=\sqrt{a_{n+1}}+a_{n} $?