Find all positive integers $n$ such that there exist non-constant polynomials with integer coefficients $f_1(x),...,f_n(x)$ (not necessarily distinct) and $g(x)$ such that $$1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x)$$

Show that $\cos \frac{\pi}{7}$ is irrational.

How many real roots does the polynomial $x^5 + x^4 - x^3 - x^2 - 2x - 2$ have? $ \textbf{a)}\ 1 \qquad\textbf{b)}\ 2 \qquad\textbf{c)}\ 3 \qquad\textbf{d)}\ 4 \qquad\textbf{e)}\ \text{None of above} $

find all $f : \mathbb{C} \to \mathbb{C}$ st: $$f(f(x)+yf(y))=x+|y|^2$$ for all $x,y \in \mathbb{C}$

Let $n$ be a positive integer and let $a_1$, $a_2$,..., $a_n$ be different real numbers, placed one after the other in any order. We say we have a [i]local minimum[/i] in one of the numbers if this is less than both of their neighbors. Which is the average number of local minima over all possible ways of ordering the numbers each other?

Determine all the integers $n > 1$ such that $$\sum_{i=1}^{n}x_i^2 \ge x_n \sum_{i=1}^{n-1}x_i$$ for all real numbers $x_1, x_2, ... , x_n$.

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that any real numbers $x{}$ and $y{}$ satisfy \[f(xf(x)+f(y))=f(f(x^2))+y.\]

Determine all integers $x$ satisfying \[ \left[\frac{x}{2}\right] \left[\frac{x}{3}\right] \left[\frac{x}{4}\right] = x^2. \] ($[y]$ is the largest integer which is not larger than $y.$)

[u]Set 7[/u] [b]p19.[/b] The polynomial $x^4 + ax^3 + bx^2 - 32x$, where$ a$ and $b$ are real numbers, has roots that form a square in the complex plane. Compute the area of this square. [b]p20.[/b] Tetrahedron $ABCD$ has equilateral triangle base $ABC$ and apex $D$ such that the altitude from $D$ to $ABC$ intersects the midpoint of $\overline{BC}$. Let $M$ be the midpoint of $\overline{AC}$. If the measure of $\angle DBA$ is $67^o$, find the measure of $\angle MDC$ in degrees. [b]p21.[/b] Last year’s high school graduates started high school in year $n- 4 = 2017$, a prime year. They graduated high school and started college in year $n = 2021$, a product of two consecutive primes. They will graduate college in year $n + 4 = 2025$, a square number. Find the sum of all $n < 2021$ for which these three properties hold. That is, find the sum of those $n < 2021$ such that $n -4$ is prime, n is a product of two consecutive primes, and $n + 4$ is a square. PS. You should use hide for answers.

Prove that for every natural number $a_1>1$ there exists an increasing sequence of natural numbers $a_n$ such that $a^2_1+a^2_2+\cdots+a^2_k$ is divisible by $a_1+a_2+\cdots+a_k$ for all $k \geq 1$. [i]A. Golovanov[/i]

Find all pairs of real numbers $(x,y)$ which satisfy the system $$\begin{cases} x-y = 7 \\ \sqrt[3]{x^2}+\sqrt[3]{xy}+\sqrt[3]{y^2} = 7\end{cases}$$

Let $n\ge 2$ be a natural number. Let $a_1\le a_2\le a_3\le \cdots \le a_n$ be real numbers such that $a_1+a_2+\cdots +a_n>0$ and $n(a_1^2+a_2^2+\cdots +a_n^2)=2(a_1+a_2+\cdots +a_n)^2.$ If $m=\lfloor n/2\rfloor+1$, the smallest integer larger than $n/2$, then show that $a_m>0.$

In an electronic game of questions and answers, for each correct answer the player adds $5$ points on the screen, for each incorrect answer $2$ points are subtracted and when the player does not answer, no score is added or subtracted. Each game has $30$ questions. Francisco played $5$ games and in all of them he obtained the same number of points, greater than zero, but the number of correct answers, errors and unanswered questions in each game was different. Give all the possible scores that Francisco could obtain.

Let $a,b,c$ be three positive real numbers such that $abc=1$. Show that: \[ \displaystyle \frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+ \frac{c}{(c+1)(a+1)} \geq \frac{3}{4}. \] When is there equality?

Let $\lfloor x\rfloor$ denote the greatest integer function and $\{x\}=x-\lfloor x\rfloor$ denote the fractional part of $x$. Let $1\leq x_1<\ldots<x_{100}$ be the $100$ smallest values of $x\geq 1$ such that $\sqrt{\lfloor x\rfloor\lfloor x^3\rfloor}+\sqrt{\{x\}\{x^3\}}=x^2.$ Compute \[\sum_{k=1}^{50}\dfrac{1}{x_{2k}^2-x_{2k-1}^2}.\]

Consider a function \( n \mapsto f(n) \) that satisfies the following conditions: \( f(n) \) is an integer for each \( n \). \( f(0) = 1 \). \( f(n+1) > f(n) + f(n-1) + \cdots + f(0) \) for each \( n = 0, 1, 2, \dots \). Determine the smallest possible value of \( f(2023) \).

Find all real polynomials $p(x) $ such that $p(x-1)p(x+1)= p(x^2-1)$.

A polynomial $P(x)$ of degree $n$ has $n$ different real roots. What is the largest number of its coefficients that can be zero?

[b](a)[/b] Prove that for all positive integers $m,n$ we have \[\sum_{k=1}^n k(k+1)(k+2)\cdots (k+m-1)=\frac{n(n+1)(n+2) \cdots (n+m)}{m+1}\] [b](b)[/b] Let $P(x)$ be a polynomial with rational coefficients and degree $m.$ If $n$ tends to infinity, then prove that \[\frac{\sum_{k=1}^n P(k)}{n^{m+1}}\] Has a limit.

[u]Round 1[/u] [b]p1.[/b] Three two-digit numbers are written on a board. One starts with $5$, another with $6$, and the last one with $7$. Annie added the first and the second numbers; Benny added the second and the third numbers; Denny added the third and the first numbers. Could it be that one of these sums is equal to $148$, and the two other sums are three-digit numbers that both start with $12$? [b]p2.[/b] Three rocks, three seashells, and one pearl are placed in identical boxes on a circular plate in the order shown. The lids of the boxes are then closed, and the plate is secretly rotated. You can open one box at a time. What is the smallest number of boxes you need to open to know where the pearl is, no matter how the plate was rotated? [img]https://cdn.artofproblemsolving.com/attachments/0/2/6bb3a2a27f417a84ab9a64100b90b8768f7978.png[/img] [b]p3.[/b] Two detectives, Holmes and Watson, are hunting the thief Raffles in a library, which has the floorplan exactly as shown in the diagram. Holmes and Watson start from the center room marked $D$. Show that no matter where Raffles is or how he moves, Holmes and Watson can find him. Holmes and Watson do not need to stay together. A detective sees Raffles only if they are in the same room. A detective cannot stand in a doorway to see two rooms at the same time. [img]https://cdn.artofproblemsolving.com/attachments/c/1/6812f615e60a36aea922f145a1ffc470d0f1bc.png[/img] [b]p4.[/b] A museum has a $4\times 4$ grid of rooms. Every two rooms that share a wall are connected by a door. Each room contains some paintings. The total number of paintings along any path of $7$ rooms from the lower left to the upper right room is always the same. Furthermore, the total number of paintings along any path of $7$ rooms from the lower right to the upper left room is always the same. The guide states that the museum has exactly $500$ paintings. Show that the guide is mistaken. [img]https://cdn.artofproblemsolving.com/attachments/4/6/bf0185e142cd3f653d4a9c0882d818c55c64e4.png[/img] [b]p5.[/b] The numbers $1–14$ are placed around a circle in some order. You can swap two neighbors if they differ by more than $1$. Is it always possible to rearrange the numbers using swaps so they are ordered clockwise from $1$ to $14$? [u]Round 2[/u] [b]p6.[/b] A triangulation of a regular polygon is a way of drawing line segments between its vertices so that no two segments cross, and the interior of the polygon is divided into triangles. A flip move erases a line segment between two triangles, creating a quadrilateral, and replaces it with the opposite diagonal through that quadrilateral. This results in a new triangulation. [img]https://cdn.artofproblemsolving.com/attachments/a/a/657a7cf2382bab4d03046075c6e128374c72d4.png[/img] Given any two triangulations of a polygon, is it always possible to find a sequence of flip moves that transforms the first one into the second one? [img]https://cdn.artofproblemsolving.com/attachments/0/9/d09a3be9a01610ffc85010d2ac2f5b93fab46a.png[/img] [b]p7.[/b] Is it possible to place the numbers from $1$ to $121$ in an $11\times 11$ table so that numbers that differ by $1$ are in horizontally or vertically adjacent cells and all the perfect squares $(1, 4, 9,..., 121)$ are in one column? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all real numbers $x, y, z$ that satisfy the following system $$\sqrt{x^3 - y} = z - 1$$ $$\sqrt{y^3 - z} = x - 1$$ $$\sqrt{z^3 - x} = y - 1$$

Consider the set $S$ of $100$ numbers: $1; \frac{1}{2}; \frac{1}{3}; ... ; \frac{1}{100}$. Any two numbers, $a$ and $b$, are eliminated in $S$, and the number $a+b+ab$ is added. Now, there are $99$ numbers on $S$. After doing this operation $99$ times, there's only $1$ number on $S$. What values can this number take?

Positive integers $a$ and $b$ satisfy the condition \[\log_2(\log_{2^a}(\log_{2^b}(2^{1000})))=0.\] Find the sum of all possible values of $a+b$.

Find all pairs of positive integers $(p; q) $such that both the equations $x^2- px + q = 0 $ and $ x^2 -qx + p = 0 $ have integral solutions.

Let $X$ be a set with $|X| = n$ , and let $X_1 , X_2 ,... X_n$ be the $n$subsets eith $|X_j| \geq 2$, for $1 \leq j \leq n$. Suppose for each $2$ element subset $Y$ of $X$, there is a unique $j$ in the set $1,2,3....,n$ such that $Y \subset X_j$ . Prove that $X_j \cap X_k \not= \Phi$ for all $1 \leq j < k \leq n$