We call a polynomial $P$ [i]square-friendly[/i] if it is monic, has integer coefficients, and there is a polynomial $Q$ for which $P(n^2)=P(n)Q(n)$ for all integers $n$. We say $P$ is [i]minimally square-friendly[/i] if it is square-friendly and cannot be written as the product of nonconstant, square-friendly polynomials. Determine the number of nonconstant, minimally square-friendly polynomials of degree at most $12$.

Consider the set $ S_n$ of all the $ 2^n$ numbers of the type $ 2\pm \sqrt{2 \pm \sqrt {2 \pm ...}},$ where number $ 2$ appears $ n\plus{}1$ times. $ (a)$ Show that all members of $ S_n$ are real. $ (b)$ Find the product $ P_n$ of the elements of $ S_n$.

Solve the system of equations $\begin{cases} \frac{1}{x}+\frac{1}{y}=\frac{1}{6} \\ \frac{3}{x}+\frac{2}{y}=\frac{5}{6} \end{cases}$

For a real number $a,b,c>0$ where $bc-ca-ab=1$ find the maximum value of $$P=\frac{4024}{1+a^2}-\frac{4024}{1+b^2}-\frac{2555}{1+c^2}$$ and find out when that holds . [i](PP-nine)[/i]

Let $f(x) = 1 + \frac{2}{x}$. Put $f_1(x) = f(x)$, $f_2(x) = f(f_1(x))$, $f_3(x) = f(f_2(x))$, $... $. Find the solutions to $x = f_n(x)$ for $n > 0$.

Given that $x$ and $y$ are positive real numbers such that $\frac{5}{x}=\frac{y}{13}=\frac{x}{y}$, find the value of $x^3 + y^3$. Proposed by Ephram Chun

Let $a_1,a_2,..,a_n,b_1,b_2,...,b_n$ be non-negative numbers satisfying the following conditions simultaneously: (1) $\displaystyle\sum_{i=1}^{n} (a_i + b_i) = 1$; (2) $\displaystyle\sum_{i=1}^{n} i(a_i - b_i) = 0$; (3) $\displaystyle\sum_{i=1}^{n} i^2(a_i + b_i) = 10$. Prove that $\text{max}\{a_k,b_k\} \le \dfrac{10}{10+k^2}$ for all $1 \le k \le n$.

Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.

Find all real numbers $a$ and $b$ that satisfy the system of equations: $$\begin{cases} a &= \frac{2}{a+b} \\ \\ b &= \frac{2}{3a-b} \\ \end{cases}$$

Find all functions f : R-->R such that f (f (x) + y^2) = x −1 + (y + 1)f (y) holds for all real numbers x, y

[u]Round 1[/u] [b]p1.[/b] What is the smallest number equal to its cube? [b]p2.[/b] Fhomas has $5$ red spaghetti and $5$ blue spaghetti, where spaghetti are indistinguishable except for color. In how many different ways can Fhomas eat $6$ spaghetti, one after the other? (Two ways are considered the same if the sequence of colors are identical) [b]p3.[/b] Jocelyn labels the three corners of a triangle with three consecutive natural numbers. She then labels each edge with the sum of the two numbers on the vertices it touches, and labels the center with the sum of all three edges. If the total sum of all labels on her triangle is $120$, what is the value of the smallest label? [u]Round 2[/u] [b]p4.[/b] Adam cooks a pie in the shape of a regular hexagon with side length $12$, and wants to cut it into right triangular pieces with angles $30^o$, $60^o$, and $90^o$, each with shortest side $3$. What is the maximum number of such pieces he can make? [b]p5.[/b] If $f(x) =\frac{1}{2-x}$ and $g(x) = 1-\frac{1}{x}$ , what is the value of $f(g(f(g(... f(g(f(2019))) ...))))$, where there are $2019$ functions total, counting both $f$ and $g$? [b]p6.[/b] Fhomas is buying spaghetti again, which is only sold in two types of boxes: a $200$ gram box and a $500$ gram box, each with a fixed price. If Fhomas wants to buy exactly $800$ grams, he must spend $\$8:80$, but if he wants to buy exactly 900 grams, he only needs to spend $\$7:90$! In dollars, how much more does the $500$ gram box cost than the $200$ gram box? [u]Round 3[/u] [b]p7.[/b] Given that $$\begin{cases} a + 5b + 9c = 1 \\ 4a + 2b + 3c = 2 \\ 7a + 8b + 6c = 9\end{cases}$$ what is $741a + 825b + 639c$? [b]p8.[/b] Hexagon $JAMESU$ has line of symmetry $MU$ (i.e., quadrilaterals $JAMU$ and $SEMU$ are reflections of each other), and $JA = AM = ME = ES = 1$. If all angles of $JAMESU$ are $135$ degrees except for right angles at $A$ and $E$, find the length of side $US$. [b]p9.[/b] Max is parked at the $11$ mile mark on a highway, when his pet cheetah, Min, leaps out of the car and starts running up the highway at its maximum speed. At the same time, Max starts his car and starts driving down the highway at $\frac12$ his maximum speed, driving all the way to the $10$ mile mark before realizing that his cheetah is gone! Max then immediately reverses directions and starts driving back up the highway at his maximum speed, nally catching up to Min at the $20$ mile mark. What is the ratio between Max's max speed and Min's max speed? [u]Round 4[/u] [b]p10.[/b] Kevin owns three non-adjacent square plots of land, each with side length an integer number of meters, whose total area is $2019$ m$^2$. What is the minimum sum of the perimeters of his three plots, in meters? [b]p11.[/b] Given a $5\times 5$ array of lattice points, how many squares are there with vertices all lying on these points? [b]p12.[/b] Let right triangle $ABC$ have $\angle A = 90^o$, $AB = 6$, and $AC = 8$. Let points $D,E$ be on side $AC$ such that $AD = EC = 2$, and let points $F,G$ be on side $BC$ such that $BF = FG = 3$. Find the area of quadrilateral $FGED$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2949413p26408203]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given $n\ge 2$ a natural number, $(K,+,\cdot )$ a body with commutative property that $\underbrace{1+...+}_{m}1\ne 0,m=2,...,n,f\in K[X]$ a polynomial of degree $n$ and $G$ a subgroup of the additive group $(K,+,\cdot )$, $G\ne K.$Show that there is $a\in K$ so$f(a)\notin G$.

Let $p, q$ be real numbers. Knowing that there are positive real numbers $a, b, c$, different two by two, such that $$p=\frac{a^2}{(b-c)^2}+\frac{b^2}{(a-c)^2}+\frac{c^2}{(a-b)^2},$$ $$q=\frac{1}{(b-c)^2}+\frac{1}{(a-c)^2}+\frac{1}{(b-a)^2}$$ calculate the value of $$\frac{a}{(b-c)^2}+\frac{b}{(a-c)^2}+\frac{c}{(b-a)^2}$$ in terms of $p, q$.

The square polynomial $x^2+ax+b+1$ has natural roots. Prove that $(a^2+b^2)$ is a composite number.

If $x_0=x_1=1$, and for $n\geq1$ $x_{n+1}=\frac{x_n^2}{x_{n-1}+2x_n}$, find a formula for $x_n$ as a function of $n$.

i) Show there exist (not necessarily distinct) non-negative real numbers $a_1,a_2,\ldots,a_{10}$; $b_1,b_2,\ldots,b_{10}$, with $a_k+b_k \leq 4$ for all $1\leq k \leq 10$, such that $\max\{|a_i-a_j|, |b_i-b_j|\} \geq \dfrac{4}{3} > 1$ for all $1\leq i < j \leq 10$. ii) Prove for any (not necessarily distinct) non-negative real numbers $a_1,a_2,\ldots,a_{11}$; $b_1,b_2,\ldots,b_{11}$, with $a_k+b_k \leq 4$ for all $1\leq k \leq 11$, there exist $1\leq i < j \leq 11$ such that $\max\{|a_i-a_j|, |b_i-b_j|\} \leq 1$. ([i]Dan Schwarz[/i])

Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero. [i]Proposed by Kiran Kedlaya, USA[/i]

Determine the sequence of complex numbers $ \left( x_n\right)_{n\ge 1} $ for which $ 1=x_1, $ and for any natural number $ n, $ the following equality is true: $$ 4\left( x_1x_n+2x_2x_{n-1}+3x_3x_{n-2}+\cdots +nx_nx_1\right) =(1+n)\left( x_1x_2+x_2x_3+\cdots +x_{n-1}x_n +x_nx_{n+1}\right) . $$

Found all functions $f: \mathbb{R} \to \mathbb{R}$, such that for any $x,y \in \mathbb{R}$, \[f(x^2+xy+f(y))=f^2(x)+xf(y)+y.\]

Do there exist two polynomials $P$ and $Q$ with integer coefficient such that i) both $P$ and $Q$ have a coefficient with absolute value bigger than $2021$, ii) all coefficients of $P \cdot Q$ by absolute value are at most $1$.

Let $n$ be a natural number. Find the number of real roots of the following equation: $1+\frac{x}{1}+\frac{x^2}{2}+...+\frac{x^n}{n}=0$.

Let $a, b$ be two reals such that $a+b<2ab$. Prove that $a+b>2$

Let $n$ be a positive integer. Determine the smallest value of the sum $a_1b_1+a_2b_2+...+a_{2n+2}b_{2n+2}$ where $(a_1,a_2,...,a_{2n+2})$ and $(b_1,b_2,...,b_{2n+2})$ are rearrangements of the binomial coefficients $2n+1 \choose 0$, $2n+1 \choose 1$,...,$2n+1 \choose 2n+1$. Justify your answer

Let $a,b,c,d\in\mathbb{Z}_{\ge 0}$, $d\ne 0$ and the function $f:\mathbb{Z}_{\ge 0}\to\mathbb Z_{\ge 0}$ defined by \[f(n)=\left\lfloor \frac{an+b}{cn+d}\right\rfloor\text{ for all } n\in\mathbb{Z}_{\ge 0}.\] Prove that the following are equivalent: [list=1] [*] $f$ is surjective; [*] $c=0$, $b<d$ and $0<a\le d$. [/list] [i]Tiberiu Trif[/i]

Let $B, M$, and $T$ be the three roots of the equation $x^3 + 20x^2 -18x-19 = 0$. What is the value of $|(B + 1)(M + 1)(T + 1)|$?