For how many integers $n$ between $ 1$ and $2021$ does the infinite nested expression $$\sqrt{n + \sqrt{n +\sqrt{n + \sqrt{...}}}}$$ give a rational number?

Show that if $a,b$ are positive real numbers such that $a^{2000}+b^{2000}=a^{1998}+b^{1998}$ then $a^2+b^2 \le 2$.

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 $x_n > 0$.

Let $\mathbb Q_{\geq 0}$ be the non-negative rational numbers, $f: \mathbb Q_{\geq 0} \to \mathbb Q_{\geq 0}$ such that $f(z+1) = f(z)+1$, $f(1/z) = f(z)$ for $z\neq 0$, and $f(0) = 0.$ Define a sequence $P_n$ of non-negative integers recursively via $$P_0 = 0,\quad P_1 = 1,\quad P_n = 2 P_{n-1}+P_{n-2}$$ for every $n \geq 2$. Find $f\left(\frac{P_{20}}{P_{24}}\right).$ [i]Proposed by Robert Trosten[/i]

For any positive integer $ n$, prove that there exists a polynomial $ P$ of degree $ n$ such that all coeffients of this polynomial $ P$ are integers, and such that the numbers $ P\left(0\right)$, $ P\left(1\right)$, $ P\left(2\right)$, ..., $ P\left(n\right)$ are pairwisely distinct powers of $ 2$.

Let be three complex numbers $ a,b,c $ such that $ |a|=|b|=|c|=1=a^2+b^2+c^2. $ Calculate $ \left| a^{2019} +b^{2019} +c^{2019} \right| . $ [i]Costică Ambrinoc[/i]

If positive reals $a,b,c,d$ satisfy $abcd = 1.$ Prove the following inequality $$1<\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{d}{cd+d+1}+\frac{a}{da+a+1}<2.$$

Define \[ S = \tan^{-1}(2020) + \sum_{j = 0}^{2020} \tan^{-1}(j^2 - j + 1). \] Then $S$ can be written as $\frac{m \pi}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Find all possible values of $x-\lfloor x\rfloor$ if $\sin \alpha = 3/5$ and $x=5^{2003}\sin {(2004\alpha)}$.

A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?

The equations are given $$ \begin{array}{c} x^2 + p_1x + q_1 = 0\\ x^2 + p_2x + q_2 = 0\\ x^2 + p_3x + q_3 = 0 \end{array}$$ each two of which have a common root, but all three have no common root. Prove that: 1) $2 (p_1p_2 + p_2p_3 + p_3p_1) - (p_1^2 + p_2^2 + p_3^2) = 4 (q_1 + q_2+ q_3)$ 2) he roots of these equations are rational when the numbers $p_1$, $p_2$ and $p_3$ are rational}.

Let $n$ be a given positive integer. Solve the system \[x_1 + x_2^2 + x_3^3 + \cdots + x_n^n = n,\] \[x_1 + 2x_2 + 3x_3 + \cdots + nx_n = \frac{n(n+1)}{2}\] in the set of nonnegative real numbers.

[u]Round 1[/u] [b]p1.[/b] Let $n$ be a two-digit positive integer. What is the maximum possible sum of the prime factors of $n^2 - 25$ ? [b]p2.[/b] Angela has ten numbers $a_1, a_2, a_3, ... , a_{10}$. She wants them to be a permutation of the numbers $\{1, 2, 3, ... , 10\}$ such that for each $1 \le i \le 5$, $a_i \le 2i$, and for each $6 \le i \le 10$, $a_i \le - 10$. How many ways can Angela choose $a_1$ through $a_{10}$? [b]p3.[/b] Find the number of three-by-three grids such that $\bullet$ the sum of the entries in each row, column, and diagonal passing through the center square is the same, and $\bullet$ the entries in the nine squares are the integers between $1$ and $9$ inclusive, each integer appearing in exactly one square. [u]Round 2 [/u] [b]p4.[/b] Suppose that $P(x)$ is a quadratic polynomial such that the sum and product of its two roots are equal to each other. There is a real number $a$ that $P(1)$ can never be equal to. Find $a$. [b]p5.[/b] Find the number of ordered pairs $(x, y)$ of positive integers such that $\frac{1}{x} +\frac{1}{y} =\frac{1}{k}$ and k is a factor of $60$. [b]p6.[/b] Let $ABC$ be a triangle with $AB = 5$, $AC = 4$, and $BC = 3$. With $B = B_0$ and $C = C_0$, define the infinite sequences of points $\{B_i\}$ and $\{C_i\}$ as follows: for all $i \ge 1$, let $B_i$ be the foot of the perpendicular from $C_{i-1}$ to $AB$, and let $C_i$ be the foot of the perpendicular from $B_i$ to $AC$. Find $C_0C_1(AC_0 + AC_1 + AC_2 + AC_3 + ...)$. [u]Round 3 [/u] [b]p7.[/b] If $\ell_1, \ell_2, ... ,\ell_{10}$ are distinct lines in the plane and $p_1, ... , p_{100}$ are distinct points in the plane, then what is the maximum possible number of ordered pairs $(\ell_i, p_j )$ such that $p_j$ lies on $\ell_i$? [b]p8.[/b] Before Andres goes to school each day, he has to put on a shirt, a jacket, pants, socks, and shoes. He can put these clothes on in any order obeying the following restrictions: socks come before shoes, and the shirt comes before the jacket. How many distinct orders are there for Andres to put his clothes on? [b]p9. [/b]There are ten towns, numbered $1$ through $10$, and each pair of towns is connected by a road. Define a backwards move to be taking a road from some town $a$ to another town $b$ such that $a > b$, and define a forwards move to be taking a road from some town $a$ to another town $b$ such that $a < b$. How many distinct paths can Ali take from town $1$ to town $10$ under the conditions that $\bullet$ she takes exactly one backwards move and the rest of her moves are forward moves, and $\bullet$ the only time she visits town $10$ is at the very end? One possible path is $1 \to 3 \to 8 \to 6 \to 7 \to 8 \to 10$. [u]Round 4[/u] [b]p10.[/b] How many prime numbers $p$ less than $100$ have the properties that $p^5 - 1$ is divisible by $6$ and $p^6 - 1$ is divisible by $5$? [b]p11.[/b] Call a four-digit integer $\overline{d_1d_2d_3d_4}$ [i]primed [/i] if 1) $d_1$, $d_2$, $d_3$, and $d_4$ are all prime numbers, and 2) the two-digit numbers $\overline{d_1d_2}$ and $\overline{d_3d_4}$ are both prime numbers. Find the sum of all primed integers. [b]p12.[/b] Suppose that $ABC$ is an isosceles triangle with $AB = AC$, and suppose that $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, with $\overline{DE} \parallel \overline{BC}$. Let $r$ be the length of the inradius of triangle $ADE$. Suppose that it is possible to construct two circles of radius $r$, each tangent to one another and internally tangent to three sides of the trapezoid $BDEC$. If $\frac{BC}{r} = a + \sqrt{b}$ forpositive integers $a$ and $b$ with $b$ squarefree, then find $a + b$. PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2800986p24675177]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p(x) = x^2 +ax +b$ be a quadratic polynomial with $a,b \in \mathbb{Z}$. Given any integer $n$ , show that there is an integer $M$ such that $p(n) p(n+1) = p(M)$.

Solve for real values of parameter $a$, the inequality : $$\sqrt{a+x}+ \sqrt{a-x}>a , \ \ x\in\mathbb{R}$$

We say that a strictly increasing sequence of positive integers $n_1, n_2,\ldots$ is [i]non-decelerating[/i] if $n_{k+1}-n_k\le n_{k+2}-n_{k+1}$ holds for all positive integers $k$. We say that a strictly increasing sequence $n_1, n_2, \ldots$ is [i]convergence-inducing[/i], if the following statement is true for all real sequences $a_1, a_2, \ldots$: if subsequence $a_{m+n_1}, a_{m+n_2}, \ldots$ is convergent and tends to $0$ for all positive integers $m$, then sequence $a_1, a_2, \ldots$ is also convergent and tends to $0$. Prove that a non-decelerating sequence $n_1, n_2,\ldots$ is convergence-inducing if and only if sequence $n_2-n_1$, $n_3-n_2$, $\ldots$ is bounded from above. [i]Proposed by András Imolay[/i]

[b]p1.[/b] A boy has as many sisters as brothers. How ever, his sister has twice as many brothers as sisters. How many boys and girls are there in the family? [b]p2.[/b] Solve each of the following problems. (1) Find a pair of numbers with a sum of $11$ and a product of $24$. (2) Find a pair of numbers with a sum of $40$ and a product of $400$. (3) Find three consecutive numbers with a sum of $333$. (4) Find two consecutive numbers with a product of $182$. [b]p3.[/b] $2008$ integers are written on a piece of paper. It is known that the sum of any $100$ numbers is positive. Show that the sum of all numbers is positive. [b]p4.[/b] Let $p$ and $q$ be prime numbers greater than $3$. Prove that $p^2 - q^2$ is divisible by $24$. [b]p5.[/b] Four villages $A,B,C$, and $D$ are connected by trails as shown on the map. [img]https://cdn.artofproblemsolving.com/attachments/4/9/33ecc416792dacba65930caa61adbae09b8296.png[/img] On each path $A \to B \to C$ and $B \to C \to D$ there are $10$ hills, on the path $A \to B \to D$ there are $22$ hills, on the path $A \to D \to B$ there are $45$ hills. A group of tourists starts from $A$ and wants to reach $D$. They choose the path with the minimal number of hills. What is the best path for them? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $\omega=e^{\frac{2\pi i}{2017}}$ and $\zeta = e^{\frac{2\pi i}{2019}}$. Let $S=\{(a,b)\in\mathbb{Z}\,|\,0\leq a \leq 2016, 0 \leq b \leq 2018, (a,b)\neq (0,0)\}$. Compute $$\prod_{(a,b)\in S}(\omega^a-\zeta^b).$$

From the digits $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ all possible four-digit numbers with different digits are formed. Find the sum of these numbers.

Suppose that $x$ and $y$ are nonzero real numbers such that \[\frac{3x+y}{x-3y}= -2.\] What is the value of \[\frac{x+3y}{3x-y}?\] $\textbf{(A) } {-3} \qquad \textbf{(B) } {-1} \qquad \textbf{(C) } 1 \qquad \textbf{(D) }2 \qquad \textbf{(E) } 3$

Prove that for each $x,y,z \in R$ it holds that $$\frac{x^2-y^2}{2x^2+1} +\frac{y^2-z^2}{2y^2+1}+\frac{z^2-x^2}{2z^2+1}\le 0$$

Find all function $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that \[ f(x \plus{} y)(f(x) \minus{} y) \equal{} xf(x) \minus{} yf(y) \] for all $ x,y \in \mathbb{R}$.

If $\mid ax^2+bx+c \mid \leq 1$ for all $x \in [-1,1]$ prove that: $a)$ $\mid c \mid \leq 1$ $b)$ $\mid a+c \mid \leq 1$ $c)$ $a^2+b^2+c^2 \leq 5$

Function $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies the following condition. (Condition) For each positive real number $x$, there exists a positive real number $y$ such that $(x + f(y))(y + f(x)) \leq 4$, and the number of $y$ is finite. Prove $f(x) > f(y)$ for any positive real numbers $x < y$. ($\mathbb{R^+}$ is a set for all positive real numbers.)

Find all values of the parameter $a$ for which the inequality \[\sqrt{x-x^2-a}+\sqrt{6a-2x-x^2}\leq \sqrt{10a-2x-4x^2}\] has a unique solution.