Find the minimum value of \[\dfrac{(x+1/x)^6-(x^6+1/x^6)-2}{(x+1/x)^3+(x^3+1/x^3)}\] for $x>0$.

Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$ and \[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots \] Determine real number $ a$ such that if $ x_1 > a$, then the sequence is monotonically decreasing, and if $ 0 < x_1 < a$, then the sequence is not monotonic.

Fix $n$. Given $n$ points on Cartesian plane such that no pair of points forms a segment that is parallel to either axes, a pair of points is said to be good if their segment gradient is positive. For which $k$ can there exist a set of $n$ points with exactly $k$ good pairs? [i](Proposed by Ivan Chan Kai Chin)[/i]

Given trapezoid $ABCD$ ($AD\parallel BC$) with $AD \perp AB$ and $T=AC\cap BD$. A circle centered at point $O$ is inscribed in the trapezoid and touches the side $CD$ at point $Q$. Let $P$ be the intersection point (different from $Q$) of the side $CD$ and the circle passing through $T,Q$ and $O$. Prove that $TP \parallel AD$. I. Voronovich

Let $T = \{1,2,...,10\}$. Find the number of bijective functions $f : T\to T$ that satises the following for all $x \in T$: $f(f(x)) = x$ $|f(x) - x| \ge 2$

Find the smallest positive integer that is relatively prime to each of $2, 20, 204,$ and $2048$. [i]Proposed by Yannick Yao[/i]

Let $ w $ be the incircle of triangle $ ABC $. Segments $ BC, CA $ meet with $ w $ at points $ D, E$. A line passing through $ B $ and parallel to $ DE $ meets $ w $ at $ F $ and $ G $. ($ F $ is nearer to $ B $ than $ G $.) Line $ CG $ meets $ w $ at $ H ( \ne G ) $. A line passing through $ G $ and parallel to $ EH $ meets with line $ AC $ at $ I $. Line $ IF $ meets with circle $ w $ at $ J (\ne F ) $. Lines $ CJ $ and $ EG $ meets at $ K $. Let $ l $ be the line passing through $ K $ and parallel to $ JD $. Prove that $ l, IF, ED $ meet at one point.

Suppose that $k \ge 2$ and $n_{1}, n_{2}, \cdots, n_{k}\ge 1$ be natural numbers having the property \[n_{2}\; \vert \; 2^{n_{1}}-1, n_{3}\; \vert \; 2^{n_{2}}-1, \cdots, n_{k}\; \vert \; 2^{n_{k-1}}-1, n_{1}\; \vert \; 2^{n_{k}}-1.\] Show that $n_{1}=n_{2}=\cdots=n_{k}=1$.

A set of $n$ points in space is given, no three of which are collinear and no four of which are co-planar (on a single plane), and each pair of points is connected by a line segment. Initially, all the line segments are colorless. A positive integer $b$ is given and Alice and Bob play the following game. In each turn Alice colors one segment red and then Bob colors up to $b$ segments blue. This is repeated until there are no more colorless segments left. If Alice colors a red triangle, Alice wins. If there are no more colorless segments and Alice hasn’t succeeded in coloring a red triangle, Bob wins. Neither player is allowed to color over an already colored line segment. 1. Prove that if $b < \sqrt{2n - 2} -\frac32$ , then Alice has a winning strategy. 2. Prove that if $b \ge 2\sqrt{n}$, then Bob has a winning strategy.

Given a concyclic quadrilateral $ABCD$ with circumcenter $O$. Let $E$ be the intersection of $AD$ and $BC$, while $F$ be the intersection of $AC$ and $BD$. A circle $w$ are tangent to $BD$ and $AC$ such that $F$ is the orthocenter of $\triangle QEP$ where $PQ$ is a diameter of $w$. Prove that $EO$ passes through the center of $w$.

Determine all integers $ a$ and $ b$, so that $ (a^3\plus{}b)(a\plus{}b^3)\equal{}(a\plus{}b)^4$

Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$. [asy] size(200); defaultpen(fontsize(10)); pair A=origin, B=(14,0), C=(9,12), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), P=centroid(A,B,C); draw(D--A--B--C--A^^B--E^^C--F); dot(A^^B^^C^^P); label("$a$", P--A, dir(-90)*dir(P--A)); label("$b$", P--B, dir(90)*dir(P--B)); label("$c$", P--C, dir(90)*dir(P--C)); label("$d$", P--D, dir(90)*dir(P--D)); label("$d$", P--E, dir(-90)*dir(P--E)); label("$d$", P--F, dir(-90)*dir(P--F)); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, N); label("$P$", P, 1.8*dir(285));[/asy]

Polynomial $p(x)$ with real coefficients, which is different from the constant, has the following property: [i] for any naturals $n$ and $k$ the $\frac{p(n+1)p(n+2)...p(n+k)}{p(1)p(2)...p(k)}$ is an integer.[/i] Prove that this polynomial is divisible by $x$.

A $ 3 \times 3 \times 3$ cube is formed by gluing together 27 standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is 7.) The smallest possible sum of all the numbers showing on the surface of the $ 3 \times 3 \times 3$ cube is $ \text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 84 \qquad \text{(D)}\ 90 \qquad \text{(E)}\ 96$

$512$ persons meet at a meeting[ Under every six of these people there is always at least two who know each other. Prove that there must be six people at this gathering, all mutual know.

Let $ABCD$ be a rectangle with area $2012$. There exist points $E$ on $AB$ and $F$ on $CD$ such that $DE = EF = F B$. Diagonal $AC$ intersects $DE$ at $X$ and $EF$ at $Y$ . Compute the area of triangle $EXY$ .

Homer goes on the $100$-Donut Diet. A $100$-Donut Diet Plan specifies how many of $100$ total donuts Homer will eat each day. The diet requires that the number of donuts he eats does not increase from one day to the next. For example, one $5$-day Donut Diet Plan is $40$, $25$, $25$, $8$, $2$. Are there more $100$-Donut Diet Plans with an odd number of days or plans where Homer eats an odd number of donuts on the first day?

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

Find the sum of all integers $n$ satisfying the following inequality: \[\frac{1}{4}<\sin\frac{\pi}{n}<\frac{1}{3}.\]

A certain town is represented as an infinite plane, which is divided by straight lines into squares. The lines are streets, while the squares are blocks. Along a certain street there stands a policeman on each $100$th intersection . Somewhere in the town there is a bandit , whose position and speed are unknown, but he can move only along the streets. The aim of the police is to see the bandit . Does there exist an algorithm available to the police to enable them to achieve their aim? (A. Andjans, Riga)

At a party, five of Ryan’s friends arrive, each hanging their coats on the coat rack. When they leave, Ryan hands out coats in a random order to his friends. What is the probability that at least half of them receive the right coat? (Half of them is $3$ or more) [i]2015 CCA Math Bonanza Team Round #7[/i]

Suppose $z_0,z_1,\ldots,z_{n-1}$ are complex numbers such that $z_k=e^{2k\pi i/n}$ for $k=0,1,2,\ldots,n-1$. Prove that for any complex number $z$, $\sum_{k=0}^{n-1}|z-z_k|\ge n$.

A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $\overline{AE}$, $\overline{BC}$, and $\overline{CD}$. The plane's intersection with the pyramid has an area that can be expressed as $\sqrt{p}$. Find $p$.

Let $ S$ be the set of all points with coordinates $ (x,y,z)$, where $ x, y,$ and $ z$ are each chosen from the set $ \{ 0, 1, 2\}$. How many equilateral triangles have all their vertices in $ S$? $ \textbf{(A)}\ 72 \qquad \textbf{(B)}\ 76 \qquad \textbf{(C)}\ 80 \qquad \textbf{(D)}\ 84 \qquad \textbf{(E)}\ 88$

The $19$ numbers $472$ , $473$ , $...$ , $490$ are juxtaposed in some order to form a $57$-digit number. Can any of the numbers thus obtained be prime?