Let $m$ be a positive integer and $n=m^2+1$. Determine all real numbers $x_1,x_2,\dotsc ,x_n$ satisfying $$x_i=1+\frac{2mx_i^2}{x_1^2+x_2^2+\cdots +x_n^2}\quad \text{for all }i=1,2,\dotsc ,n.$$

Let $ a, b, c, d \in \mathbb {R}, $ $ b \neq0. $ Find the functions of the $ f: \mathbb{R} \to \mathbb{R} $ such that $ f (x + d \cdot f (y)) = ax + by + c, $ for all $ x, y \in \mathbb{R}. $

Let $n\geq 2$ be an integer. For each $k$-tuple of positive integers $a_1, a_2, \ldots, a_k$ such that $a_1+a_2+\cdots +a_k=n$, consider the sums $S_i=1+2+\ldots +a_i$ for $1\leq i\leq k$. Determine, in terms of $n$, the maximum possible value of the product $S_1S_2\cdots S_k$. [i]Proposed by Misael Pelayo[/i]

If $x^x = 2012^{2012^{2013}}$ , find $x$.

Victor has four types of coins: gold, silver, bronze, and copper. All coins of the same type have the same weight, which is an integer number of grams. Victor performs two weighings: - He takes 6 gold coins, 13 silver coins, 3 bronze coins, and 7 copper coins, and the total weight on the scale is 162 grams. - He takes 15 gold coins, 5 silver coins, and 11 bronze coins, and the total weight on the scale is 110 grams. Determine the weight of each type of coin.

Quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$, both of which have real roots, are called friendly if for all $t \in [0,1]$ quadratic polynomial $tP(x)+(1-t)Q(x)$ also has real roots. a) Provide an example of quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ and which have real roots, that are not friendly. b) Prove that for any two quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ that have real roots, there is a quadratic polynomial $R(x)$ which has a leading coefficient $1$ and which is friendly to both $P$ and $Q$

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$x^2f(x)+yf(y^2)=f(x+y)f(x^2-xy+y^2)$$ for all $x,y\in\mathbb{R}$.

On a examination of $n$ questions a student answers correctly $15$ of the first $20$. Of the remaining questions he answers one third correctly. All the questions have the same credit. If the student's mark is $50\%$, how many different values of $n$ can there be? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ \text{the problem cannot be solved} $

Find all $g:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in R$, \[(4x+g(x)^2)g(y)=4g(\frac{y}{2}g(x))+4xyg(x)\]

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

[u]Round 1 [/u] [b]p1.[/b] Ben the bear has an algorithm he runs on positive integers- each second, if the integer is even, he divides it by $2$, and if the integer is odd, he adds $1$. The algorithm terminates after he reaches $1$. What is the least positive integer n such that Ben's algorithm performed on n will terminate after seven seconds? (For example, if Ben performed his algorithm on $3$, the algorithm would terminate after $3$ seconds: $3 \to 4 \to 2 \to 1$.) [b]p2.[/b] Suppose that a rectangle $R$ has length $p$ and width $q$, for prime integers $p$ and $q$. Rectangle $S$ has length $p + 1$ and width $q + 1$. The absolute difference in area between $S$ and $R$ is $21$. Find the sum of all possible values of $p$. [b]p3.[/b] Owen the origamian takes a rectangular $12 \times 16$ sheet of paper and folds it in half, along the diagonal, to form a shape. Find the area of this shape. [u]Round 2[/u] [b]p4.[/b] How many subsets of the set $\{G, O, Y, A, L, E\}$ contain the same number of consonants as vowels? (Assume that $Y$ is a consonant and not a vowel.) [b]p5.[/b] Suppose that trapezoid $ABCD$ satisfies $AB = BC = 5$, $CD = 12$, and $\angle ABC = \angle BCD = 90^o$. Let $AC$ and $BD$ intersect at $E$. The area of triangle $BEC$ can be expressed as $\frac{a}{b}$, for positive integers $a$ and $b$ with $gcd(a, b) = 1$. Find $a + b$. [b]p6.[/b] Find the largest integer $n$ for which $\frac{101^n + 103^n}{101^{n-1} + 103^{n-1}}$ is an integer. [u]Round 3[/u] [b]p7.[/b] For each positive integer n between $1$ and $1000$ (inclusive), Ben writes down a list of $n$'s factors, and then computes the median of that list. He notices that for some $n$, that median is actually a factor of $n$. Find the largest $n$ for which this is true. [b]p8.[/b] ([color=#f00]voided[/color]) Suppose triangle $ABC$ has $AB = 9$, $BC = 10$, and $CA = 17$. Let $x$ be the maximal possible area of a rectangle inscribed in $ABC$, such that two of its vertices lie on one side and the other two vertices lie on the other two sides, respectively. There exist three rectangles $R_1$, $R_2$, and $R_3$ such that each has an area of $x$. Find the area of the smallest region containing the set of points that lie in at least two of the rectangles $R_1$, $R_2$, and $R_3$. [b]p9.[/b] Let $a, b,$ and $c$ be the three smallest distinct positive values of $\theta$ satisfying $$\cos \theta + \cos 3\theta + ... + \cos 2021\theta = \sin \theta+ \sin 3 \theta+ ... + \sin 2021\theta. $$ What is $\frac{4044}{\pi}(a + b + c)$? [color=#f00]Problem 8 is voided. [/color] PS. You should use hide for answers.Rounds 4-5 have been posted [url=https://artofproblemsolving.com/community/c4h3131422p28368457]here [/url] and 6-7 [url=https://artofproblemsolving.com/community/c4h3131434p28368604]here [/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f:[0,1] \to \mathbb{R}$ such that the inequality \[(x-y)^2\leq|f(x) -f(y)|\leq|x-y|\] is satisfied for all $x,y\in [0,1]$

Let $P(x)$ denote the polynomial \[3\sum_{k=0}^{9}x^k + 2\sum_{k=10}^{1209}x^k + \sum_{k=1210}^{146409}x^k.\]Find the smallest positive integer $n$ for which there exist polynomials $f,g$ with integer coefficients satisfying $x^n - 1 = (x^{16} + 1)P(x) f(x) + 11\cdot g(x)$. [i]Victor Wang.[/i]

Find all functions $f : \mathbb R \to \mathbb R$ for which \[f(a - b) f(c - d) + f(a - d) f(b - c) \leq (a - c) f(b - d),\] for all real numbers $a, b, c$ and $d$. Note that there is only one occurrence of $f$ on the right hand side!

Determine graphically the number of roots of the equation $\sin x = \lg x$.

Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

Find the largest $c$ such that for any $\lambda \ge 1$ there is an a that satisfies the inequality $$\sin a + \sin (a\lambda ) \ge c.$$

Find the sum of the coefficients of the polynomial $P(x) = x^4- 29x^3 + ax^2 + bx + c$, given that $P(5) = 11$, $P(11) = 17$, and $P(17) = 23$.

Prove that for all the positive numbers $a,b,c$ the following inequality is valid: $$a^3+b^3+c^3+3abc>ab(a+b)+bc(b+c)+ac(a+c)$$

Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]

Let $a_1, a_2,...., a_n$ be distinct positive integers. Prove that $(a_1^5 + ...+ a_n^5) + (a_1^7 + ...+ a_n^7) \ge 2(a_1^3 + ...+ a_n^3)^2$ and find the cases of equality.

Find all real numbers $x$ for which $$\sqrt{\frac{x^3 - 8}{x}} > x - 2.$$

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 $n$.

Let $ x_1,x_2,x_3$ be the roots of: $ x^3\minus{}6x^2\plus{}ax\plus{}a\equal{}0$. Find all real numbers $ a$ for which $ (x_1\minus{}1)^3\plus{}(x_2\minus{}1)^3\plus{}(x_3\minus{}1)^3\equal{}0$. Also, for each such $ a$, determine the corresponding values of $ x_1,x_2,$ and $ x_3$.

Consider a function $f: \mathbb{R} \to \mathbb{R}$ satisfying for all $x \in \mathbb{R}$: \[ f(x+1) = \frac{1}{2} + \sqrt{f(x) - f(x)^2}. \] Prove that there exists a $b > 0$ such that $f(x + b) = f(x)$ for all $x \in \mathbb{R}$.