During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^3,z^5,z^7,\ldots,z^{2013}$ in that order; on Sunday, he begins at $1$ and delivers milk to houses located at $z^2,z^4,z^6,\ldots,z^{2012}$ in that order. Eli always walks directly (in a straight line) between two houses. If the distance he must travel from his starting point to the last house is $\sqrt{2012}$ on both days, find the real part of $z^2$.

Prove that the rectangle $m\times n$ table can be filled with exact squares so, that the sums in the rows and the sums in the columns will be exact squares also.

Let $m \leq n$ be natural numbers. Starting with the product $t=m\cdot (m+1) \cdot (m+2) \cdot \cdots \cdot n$, let $T_{m, n}$ be the sum of products that can be obtained from deleting from $t$ pairs of consecutive integers (this includes $t$ itself). In the case where all the numbers are deleted, we assume the number $1$. For example, $T_{2, 7} = 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 + 2 \cdot 3 \cdot 4 \cdot 5 + 2 \cdot 3 \cdot 4 \cdot 7 + 2 \cdot 3 \cdot 6 \cdot 7 + 2 \cdot 5 \cdot 6 \cdot 7 + 4 \cdot 5 \cdot 6 \cdot 7 + 2 \cdot 3 + 2 \cdot 5 + 2 \cdot 7 + 4 \cdot 7 + 6 \cdot 7 + 1 = 5040 + 120 + 168 + 252 + 420 + 840 + 6 + 10 + 14 + 20 + 28 + 42 + 1 = 6961$. Taking $T_{n+1, n} = 1$. Show that $T_{m, n+1}=T_{m, k-1} \cdot T_{k+2, n+1} + T_{m, k} \cdot T_{k+1, n+1}$ for all $1 \leq m \leq k \leq n$.

A polynomial $P(x)$ is called [i]nice[/i] if $P(0) = 1$ and the nonzero coefficients of $P(x)$ alternate between $1$ and $-1$ when written in order. Suppose that $P(x)$ is nice, and let $m$ and $n$ be two relatively prime positive integers. Show that \[Q(x) = P(x^n) \cdot \frac{(x^{mn} - 1)(x-1)}{(x^m-1)(x^n-1)}\] is nice as well.

Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]

Prove that no pair of different positive integers $(m, n)$ exist, such that $\frac{4m^{2}n^{2}-1}{(m^{2}-n^2)^{2}}$ is an integer.

On an infinite (in both directions) strip of squares, indexed by the integers, are placed several stones (more than one may be placed on a single square). We perform a sequence of moves of one of the following types: (a) Remove one stone from each of the squares $n - 1$ and $n$ and place one stone on square $n + 1$. (b) Remove two stones from square $n$ and place one stone on each of the squares $n + 1$, $n - 2$. Prove that any sequence of such moves will lead to a position in which no further moves can be made, and moreover that this position is independent of the sequence of moves. [i]D. Fon-der-Flaas[/i]

Let $x$ and $y$ be real numbers satisfying the conditions $x + y = 4$ and $xy = 3$. Compute the value of $(x - y)^2$. A. $0$ B. $1$ C. $4$ D. $9$ E.$ -1$

Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.

If $x > 0$ and $x^2 +\frac{1}{x^2}= 14$, find $x^5 +\frac{1}{x^5}$.

Let $ x_1$, $ x_2$, $ \ldots$, $ x_n$ be real numbers satisfying the conditions: \[ \left\{\begin{array}{cccc} |x_1 \plus{} x_2 \plus{} \cdots \plus{} x_n | & \equal{} & 1 & \ \\ |x_i| & \leq & \displaystyle \frac {n \plus{} 1}{2} & \ \textrm{ for }i \equal{} 1, 2, \ldots , n. \end{array} \right. \] Show that there exists a permutation $ y_1$, $ y_2$, $ \ldots$, $ y_n$ of $ x_1$, $ x_2$, $ \ldots$, $ x_n$ such that \[ | y_1 \plus{} 2 y_2 \plus{} \cdots \plus{} n y_n | \leq \frac {n \plus{} 1}{2}. \]

Let $a_1,a_2,a_3,...$ be a sequence of positive real numbers such that: (i) For all positive integers $m,n$, we have $a_{mn}=a_ma_n$ (ii) There exists a positive real number $B$ such that for all positive integers $m,n$ with $m<n$, we have $a_m < Ba_n$ Find all possible values of $\log_{2015}(a_{2015}) - \log_{2014}(a_{2014})$

Find a function $f$ such that $f(t^2 +t +1) = t$ for all real $t \ge 0$

Let $s \geq 2$ and $n \geq k \geq 2$ be integes, and let $A$ be a subset of $\{1, 2, . . . , n\}^k$ of size at least $2sk^2n^{k-2}$ such that any two members of $A$ share some entry. Prove that there are an integer $p \leq k$ and $s+2$ members $A_1, A_2, . . . , A_{s+2}$ of $A$ such that $A_i$ and $A_j$ share the $p$-th entry alone, whenever $i$ and $j$ are distinct. [i]Miroslav Marinov, Bulgaria[/i]

Determine all functions $ f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfy: $ f(x\plus{}f(y))\equal{}f(x)\plus{}y$ for all $ x,y \in \mathbb{N}$.

In triangle $ABC$, $K$ and $L$ are points on the side $BC$ ($K$ being closer to $B$ than $L$) such that $BC \cdot KL = BK \cdot CL$ and $AL$ bisects $\angle KAC$. Show that $AL \perp AB.$

Let \( n > 2 \) be an integer, \( k > 1 \) a real number, and \( x_1, x_2, \ldots, x_n \) be positive real numbers such that \( x_1 \cdot x_2 \cdots x_n = 1 \). Prove that: \[ \frac{1 + x_1^k}{1 + x_2} + \frac{1 + x_2^k}{1 + x_3} + \cdots + \frac{1 + x_n^k}{1 + x_1} \geq n. \] When does equality hold?

Triangle $ABC$ has incenter $I$. Let $D$ be the foot of the perpendicular from $A$ to side $BC$. Let $X$ be a point such that segment $AX$ is a diameter of the circumcircle of triangle $ABC$. Given that $ID = 2$, $IA = 3$, and $IX = 4$, compute the inradius of triangle $ABC$.

A semicircle is inscribed in a semicircle of radius $2$ as shown. Find the radius of the smaller semicircle. [img]https://cdn.artofproblemsolving.com/attachments/c/1/c60cd40eaecfe417aca46ce4fd386fe22af85b.png[/img]

ABC is a triangle with A=90 and C=30.Let M be the midpoint of BC. Let W be a circle passing through A tangent in M to BC. Let P be the circumcircle of ABC. W is intersecting AC in N and P in M. prove that MN is perpendicular to BC.

Let $a$ be a positive integer, $a<100$, and $a^3+23$ is a multiple of $24$. Then, the number of such $a$ is $\text{(A)}4\qquad\text{(B)}5\qquad\text{(C)}9\qquad\text{(D)}10$

[color=darkred]The sides $AD$ and $CD$ of a tangent quadrilateral $ABCD$ touch the incircle $\varphi$ at $P$ and $Q,$ respectively. If $M$ is the midpoint of the chord $XY$ determined by $\varphi$ on the diagonal $BD,$ prove that $\angle AMP = \angle CMQ.$[/color]

$A$ and $B$ are two cylindrical containers that contain water. The height of the water at$ A$ is $1000$ cm and at $B$, $350$ cm. Using a pump, water is transferred from $A$ to $B$. It is noted that, in container $A$, the height of the water decreases $4$ cm per minute and in $B$ it increases $9$ cm per minute. After how much time, since the pump was started, will the heights at $A$ and $B$ be the same?

Alice, Bob, and Carol are playing a game. Each turn, one of them says one of the $3$ players' names, chosen from {Alice, Bob, Carol} uniformly at random. Alice goes first, Bob goes second, Carol goes third, and they repeat in that order. Let $E$ be the expected number of names that are have been said when, for the first time, all $3$ names have been said twice. If $E = \tfrac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$. (Include the last name to be said twice in your count.)

Jake, Jonathan, and Joe are playing a dice game involving polyhedron dice. The dice are as follows: 4 sides, 6 sides, 12 sides, and 20 sides. An n-sided dice has the numbers 1 through n labeled on the sides. Jake starts by selecting a 4-sided die and a 20-sided die. The amount of points that a player gets is the sum of the numbers on the rolled dice. Jonathan then selects a 12-sided die and an 20-sided die. Finally, Joe selects a 20-sided die and a 6-sided die. [list=a] [*]What is the probability that Joe places last? [*]What is the probability that Joe places second? [*]What is the probability that Joe places first? [*]What is the probability that there is a three-way tie? [/list] [i]Proposed by beanielove2[/i]