Let $r_1, r_2, \cdots, r_7$ be the distinct complex roots of the polynomial $P(x) = x^7 - 7$ Let \[K = \prod_{1 \leq i < j \leq 7} (r_i + r_j)\] that is, the product of all the numbers of the form $r_i + r_j$, where $i$ and $j$ are integers for which $1 \leq i < j \leq 7$. Determine the value of $K^2$.

Let $ ABC $ be an acutangle triangle and $ D $ any point on the $ BC $ side. Let $ E $ be the symmetrical of $ D $ in $ AC $ and $ F $ is the symmetrical $ D $ relative to $ AB $. $ A $ straight $ ED $ intersects straight $ AB $ at $ G $, while straight $ F D $ intersects the line $ AC $ in $ H $. Prove that the points $ A, E, F, G$ and $ H $ are on the same circumference.

$ABCDE$ is a cyclic pentagon, with circumcentre $O$. $AB=AE=CD$. $I$ midpoint of $BC$. $J$ midpoint of $DE$. $F$ is the orthocentre of $\triangle ABE$, and $G$ the centroid of $\triangle AIJ$.$CE$ intersects $BD$ at $H$, $OG$ intersects $FH$ at $M$. Show that $AM\perp CD$.

Find the number of ways to arrange the letters in $LE X I NGTON$ such that the string $LE X$ does not appear.

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of positive numbers such that $$a_n-2a_{n+1}+a_{n+2}\geq 0 \text{ and } \sum_{j=1}^{n} a_j \leq 1, \forall n\in\mathbb{N}.$$ Prove that $0\leq a_{n}-a_{n+1}<\frac{2}{n^2}, \forall n\in\mathbb{N}.$

Find $ \lim_{n\to\infty}(\sum_{i=0}^{n}\frac{1}{n+i})$

The quadrilateral $ABCD$ is an isosceles trapezoid, where $AB\parallel CD$. The trapezoid is inscribed in a circle with radius $R$ and center on side $AB$. Point $E$ lies on the circumscribed circle and is such that $\angle DAE = 90^o$. Given that $\frac{AE}{AB}=\frac34$, calculate the length of the sides of the isosceles trapezoid.

The operation $*$ is defined by $a*b=a+b+ab$, where $a$ and $b$ are real numbers. Find the value of \[\frac{1}{2}*\bigg(\frac{1}{3}*\Big(\cdots*\big(\frac{1}{9}*(\frac{1}{10}*\frac{1}{11})\big)\Big)\bigg).\] [i]2017 CCA Math Bonanza Team Round #3[/i]

Given the areas of the three squares in the figure, what is the area of the interior triangle? [asy] real r=22.61986495; pair A=origin, B=(12,0), C=(12,5); draw(A--B--C--cycle); markscalefactor=0.1; draw(rightanglemark(C, B, A)); draw(scale(12)*shift(0,-1)*unitsquare); draw(scale(5)*shift(12/5,0)*unitsquare); draw(scale(13)*rotate(r)*unitsquare); pair P=shift(0,-1)*(13/sqrt(2) * dir(r+45)), Q=(14.5,1.2), R=(6, -7); label("169", P, N); label("25", Q, N); label("144", R, N); [/asy] $ \textbf{(A)}\ 13\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 60\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 1800$

Let $p$ be a prime number. We know that each natural number can be written in the form \[\sum_{i=0}^{t}a_ip^i (t,a_i \in \mathbb N\cup \{0\},0\le a_i\le p-1)\] Uniquely. Now let $T$ be the set of all the sums of the form \[\sum_{i=0}^{\infty}a_ip^i (0\le a_i \le p-1).\] (This means to allow numbers with an infinite base $p$ representation). So numbers that for some $N\in \mathbb N$ all the coefficients $a_i, i\ge N$ are zero are natural numbers. (In fact we can consider members of $T$ as sequences $(a_0,a_1,a_2,...)$ for which $\forall_{i\in \mathbb N}: 0\le a_i \le p-1$.) Now we generalize addition and multiplication of natural numbers to this set so that it becomes a ring (it's not necessary to prove this fact). For example: $1+(\sum_{i=0}^{\infty} (p-1)p^i)=1+(p-1)+(p-1)p+(p-1)p^2+...$ $=p+(p-1)p+(p-1)p^2+...=p^2+(p-1)p^2+(p-1)p^3+...$ $=p^3+(p-1)p^3+...=...$ So in this sum, coefficients of all the numbers $p^k, k\in \mathbb N$ are zero, so this sum is zero and thus we can conclude that $\sum_{i=0}^{\infty}(p-1)p^i$ is playing the role of $-1$ (the additive inverse of $1$) in this ring. As an example of multiplication consider \[(1+p)(1+p+p^2+p^3+...)=1+2p+2p^2+\cdots\] Suppose $p$ is $1$ modulo $4$. Prove that there exists $x\in T$ such that $x^2+1=0$. [i]Proposed by Masoud Shafaei[/i]

[b](a)[/b] Determine if exist an integer $n$ such that $n^2 -k$ has exactly $10$ positive divisors for each $k = 1, 2, 3.$ [b](b) [/b]Show that the number of positive divisors of $n^2 -4$ is not $10$ for any integer $n.$

The price of a gold ring in a certain universe is proportional to the square of its purity and the cube of its diameter. The purity is inversely proportional to the square of the depth of the gold mine and directly proportional to the square of the price, while the diameter is determined so that it is proportional to the cube root of the price and also directly proportional to the depth of the mine. How does the price vary solely in terms of the depth of the gold mine?

2 squares are painted in blue and 2 squares are painted in red on a $ 3\times 3$ board in such a way that two square with same color is neither at same row nor at same column. In how many different ways can these four squares be painted? $\textbf{(A)}\ 198 \qquad\textbf{(B)}\ 288 \qquad\textbf{(C)}\ 396 \qquad\textbf{(D)}\ 576 \qquad\textbf{(E)}\ 792$

Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ and $Q(x)=b_nx^n+b_{n-1}x^{n-1}+\cdots+b_0$ be two polynomials with integral coefficients such that $a_n-b_n$ is a prime and $a_nb_0-a_0b_n\neq 0,$ and $a_{n-1}=b_{n-1}.$ Suppose that there exists a rational number $r$ such that $P(r)=Q(r)=0.$ Prove that $r\in\mathbb Z.$

What is the smallest integral value of $k$ such that \[ 2x(kx-4)-x^2+6=0 \] has no real roots? $ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

The set of real numbers $ x$ for which \[ \frac{1}{x\minus{}2009}\plus{}\frac{1}{x\minus{}2010}\plus{}\frac{1}{x\minus{}2011}\ge 1\] is the union of intervals of the form $ a<x\le b$. What is the sum of the lengths of these intervals? $ \textbf{(A)}\ \frac{1003}{335} \qquad \textbf{(B)}\ \frac{1004}{335} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac{403}{134} \qquad \textbf{(E)}\ \frac{202}{67}$

Let $I$ be the center of the inscribed circle of $ABC$ and let $I_A$ be the center of the exscribed circle touching the side $BC$. Let $M$ be the midpoint of the side $BC$, and $N$ be the midpoint of the arc $BAC$ of the circumscribed circle of $ABC$ . The point $T$ is symmetric to the point $N$ wrt point $A$. Prove that the points $I_A,M,I,T$ lie on the same circle. (Danilo Hilko)