Determine the numbers $x,y$, with $x$ integer and $y$ rational, for which equality holds: $$5(x^2+xy+y^2) = 7(x+2y)$$

A sequence starts with $2017$ as its first term and each subsequent term is the sum of cubes of the digits in the previous number. What is the $2017$th term of this sequence? [i]2017 CCA Math Bonanza Individual Round #3[/i]

Find all polynomials $f, g, h$ with real coefficients, such that $f(x)^2+(x+1)g(x)^2=(x^3+x)h(x)^2$

[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].

Given four numbers $x, y, z, t$, let $(a, b, c, d)$ be a permutation of $(x, y, z, t)$ and set $x_1 =|a- b|$, $y_1 = |b-c|$, $z_1 = |c-d|$, and $t_1 = |d -a|$. From $x_1, y_1, z_1, t_1$, form in the same fashion the numbers $x_2, y_2, z_2, t_2$, and so on. It is known that $x_n = x, y_n = y, z_n = z, t_n = t$ for some $n$. Find all possible values of $(x, y, z, t)$.

Solve in reals: \begin{eqnarray*}a^2=b^3+c^3 \\ b^2=c^3+d^3 \\ c^2=d^3+e^3 \\ d^2=e^3+a^3 \\ e^2=a^3+b^3 \end{eqnarray*}

Let $a, b, c, d$ be distinct non-zero real numbers satisfying the following two conditions: $ac = bd$ and $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 4$. Determine the largest possible value of the expression $\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}$.

Let $a,b,c$ be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: $ax^2+bx+c, bx^2+cx+a,$ and $cx^2+ax+b $.

A sequence of numbers $a_n, n = 1,2, \ldots,$ is defined as follows: $a_1 = \frac{1}{2}$ and for each $n \geq 2$ \[ a_n = \frac{2 n - 3}{2 n} a_{n-1}. \] Prove that $\sum^n_{k=1} a_k < 1$ for all $n \geq 1.$

Let $x_1$, $x_2$,$x_3$, $y_1$, $y_2$, and $y_3 $ be positive real numbers, such that $x_1 + y_2 = x_2 + y_3 = x_3 + y_1 =1$. Prove that $$ x_1y_1 + x_2y_2 + x_3y_3 <1$$

For an integer $n > 0$, denote by $\mathcal F(n)$ the set of integers $m > 0$ for which the polynomial $p(x) = x^2 + mx + n$ has an integer root. [list=a] [*] Let $S$ denote the set of integers $n > 0$ for which $\mathcal F(n)$ contains two consecutive integers. Show that $S$ is infinite but \[ \sum_{n \in S} \frac 1n \le 1. \] [*] Prove that there are infinitely many positive integers $n$ such that $\mathcal F(n)$ contains three consecutive integers. [/list] [i]Ivan Borsenco[/i]

Find all $x$ and $y$ which are rational multiples of $\pi$ with $0<x<y<\frac{\pi}{2}$ and $\tan x+\tan y =2$.

Given three distinct real numbers $a$, $b$, and $c$, show that at least two of the three following equations \[(x-a)(x-b)=x-c\] \[(x-c)(x-b)=x-a\] \[(x-c)(x-a)=x-b\] have real solutions.

$a,b,c,d\in\mathbb{R}$, solve the system of equations: \[ \begin{cases} a^3+b=c \\ b^3+c=d \\ c^3+d=a \\ d^3+a=b \end{cases} \]

Prove that if $a,b,c \ge 1$ and $a+b+c=9$, then $\sqrt{ab+bc+ca} \le \sqrt{a} +\sqrt{b} + \sqrt{c}$

For each positive integer $n$, show that the polynomial: $$P_n(x)=\sum _{k=0}^n2^k\binom{2n}{2k}x^k(x-1)^{n-k}$$ has $n$ real roots.

Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions: (i) $f(x)$ is an integer if and only if $x$ is an integer; (ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.

Let $h \ge 3$ be an integer and $X$ the set of all positive integers that are greater than or equal to $2h$. Let $S$ be a nonempty subset of $X$ such that the following two conditions hold: [list] [*]if $a + b \in S$ with $a \ge h, b \ge h$, then $ab \in S$; [*]if $ab \in S$ with $a \ge h, b \ge h$, then $a + b \in S$.[/list] Prove that $S = X$.

Prove that there exists a real $c<\frac{3}{4}$, such that for each sequence $x_1, x_2, \ldots$ satisfying $0 \leq x_i \leq 1$ for all $i$, there exist infinitely many $(m, n)$ with $m>n$, such that $$|x_m-x_n|\leq \frac{c} {m}.$$

Find four distinct positive integers, $a$, $b$, $c$, and $d$, such that each of the four sums $a+b+c$, $a+b+d$,$a+c+d$, and $b+c+d$ is the square of an integer. Show that infinitely many quadruples $(a,b,c,d)$ with this property can be created.

Let $A$ be a subset of the irrational numbers such that the sum of any two distinct elements of it be a rational number. Prove that $A$ has two elements at most.

Let $ a_1, $ $ a_2, $ $ \ldots, $ $ a_ {99} $ be positive real numbers such that $ ia_j + ja_i \ge i + j $ for all $ 1 \le i <j \le 99. $ Prove , that $ (a_1 + 1) (a_2 + 2) \ldots (a_ {99} +99) \ge 100!$ .

Let $a,b$ be positive real numbers. Define $m(a,b)$ as the minimum of $\[ a,\frac{1}{b} \text{ and } \frac{1}{a}+b.\]$ Find the maximum of $m(a,b).$

There are $2012$ distinct points in the plane, each of which is to be coloured using one of $n$ colours, so that the numbers of points of each colour are distinct. A set of $n$ points is said to be [i]multi-coloured [/i]if their colours are distinct. Determine $n$ that maximizes the number of multi-coloured sets.

Let $f(t)=t^3+t$. Decide if there exist rational numbers $x, y$ and positive integers $m, n$ such that $xy=3$ and: \begin{align*} \underbrace{f(f(\ldots f(f}_{m \ times}(x))\ldots)) = \underbrace{f(f(\ldots f(f}_{n \ times}(y))\ldots)). \end{align*}