Instead of walking along two adjacent sides of a rectangular field, a boy took a shortcut along the diagonal of the field and saved a distance equal to $ \frac{1}{2}$ the longer side. The ratio of the shorter side of the rectangle to the longer side was: $ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{2}{3} \qquad\textbf{(C)}\ \frac{1}{4} \qquad\textbf{(D)}\ \frac{3}{4} \qquad\textbf{(E)}\ \frac{2}{5}$

A sphere is centered at a point with integer coordinates and passes through the three points $(2,0,0), (0,4,0), (0,0,6),$ but not the origin $(0,0,0)$. If $r$ is the smallest possible radius of the sphere, compute $r^2$.

Justin is being served two different types of chips, A-chips, and B-chips. If there are $3$ B-chips and $5$ A-chips, and if Justin randomly grabs $3$ chips, what is the probability that none of them are A-chips?

Among the $900$ residents of Aimeville, there are $195$ who own a diamond ring, $367$ who own a set of golf clubs, and $562$ who own a garden spade. In addition, each of the $900$ residents owns a bag of candy hearts. There are $437$ residents who own exactly two of these things, and $234$ residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.

Let $ X$ be a set of $ k$ vertexes on a plane such that no three of them are collinear. Let $ P$ be the family of all $ {k \choose 2}$ segments that connect each pair of points. Determine $ \tau(P)$.

Consider the function $f:[0,1)\rightarrow[0,1)$ defined by $f(x)=2x-\lfloor 2x\rfloor$, where $\lfloor 2x\rfloor$ is the greatest integer less than or equal to $2x$. Find the sum of all values of $x$ such that $f^{17}(x)=x.$ [i]Proposed by Matthew Weiss

Given a positive integer $n>1$. Denote $T$ a set that contains all ordered sets $(x;y;z)$ such that $x,y,z$ are all distinct positive integers and $1\leq x,y,z\leq 2n$. Also, a set $A$ containing ordered sets $(u;v)$ is called [i]"connected"[/i] with $T$ if for every $(x;y;z)\in T$ then $\{(x;y),(x;z),(y;z)\} \cap A \neq \varnothing$. a) Find the number of elements of set $T$. b) Prove that there exists a set "connected" with $T$ that has exactly $2n(n-1)$ elements. c) Prove that every set "connected" with $T$ has at least $2n(n-1)$ elements.

Evaluate $ \int_0^{\frac {\pi}{2}} \frac {(\sin x)^{\cos x}}{(\cos x)^{\sin x} \plus{} (\sin x)^{\cos x}} dx$.

There are $n$ lattice points in a general position. (no three points are collinear) A convex polygon $P$ covers the said $n$ points. (the borders are included) Prove that, for large enough $n$ and a positive real $\epsilon$, the perimeter of $P$ is no less than $(\sqrt{2}+\epsilon)n$.

For a positive integer $n$, we consider an $n \times n$ board and tiles with dimensions $1 \times 1, 1 \times 2, ..., 1 \times n$. In how many ways exactly can $\frac12 n (n + 1)$ cells of the board are colored red, so that the red squares can all be covered by placing the $n$ tiles all horizontally, but also by placing all $n$ tiles vertically? Two colorings that are not identical, but by rotation or reflection from the board into each other count as different.

As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE?$ [asy] size(6cm); pair A = (0,10); label("$A$", A, N); pair B = (0,0); label("$B$", B, S); pair C = (10,0); label("$C$", C, S); pair D = (10,10); label("$D$", D, SW); pair EE = (15,11.8); label("$E$", EE, N); pair F = (3,10); label("$F$", F, N); filldraw(D--arc(D,2.5,270,380)--cycle,lightgray); dot(A^^B^^C^^D^^EE^^F); draw(A--B--C--D--cycle); draw(D--EE--F--cycle); label("$110^\circ$", (15,9), SW); [/asy] $\textbf{(A) }160\qquad\textbf{(B) }164\qquad\textbf{(C) }166\qquad\textbf{(D) }170\qquad\textbf{(E) }174$

Let $ABC$ ($BC > AC$) be an acute triangle with circumcircle $k$ centered at $O$. The tangent to $k$ at $C$ intersects the line $AB$ at the point $D$. The circumcircles of triangles $BCD, OCD$ and $AOB$ intersect the ray $CA$ (beyond $A$) at the points $Q, P$ and $K$, respectively, such that $P \in (AK)$ and $K \in (PQ)$. The line $PD$ intersects the circumcircle of triangle $BKQ$ at the point $T$, so that $P$ and $T$ are in different halfplanes with respect to $BQ$. Prove that $TB = TQ$.

There are $n$ holes in a circle. The holes are numbered $1,2,3$ and so on to $n$. In the beginning, there is a peg in every hole except for hole $1$. A peg can jump in either direction over one adjacent peg to an empty hole immediately on the other side. After a peg moves, the peg it jumped over is removed. The puzzle will be solved if all pegs disappear except for one. For example, if $n=4$ the puzzle can be solved in two jumps: peg $3$ jumps peg $4$ to hole $1$, then peg $2$ jumps the peg in $1$ to hole $4$. (See illustration below, in which black circles indicate pegs and white circles are holes.) [center][img]http://i.imgur.com/4ggOa8m.png[/img][/center] [list=a] [*]Can the puzzle be solved for $n=5$? [*]Can the puzzle be solved for $n=2014$? [/list] In each part (a) and (b) either describe a sequence of moves to solve the puzzle or explain why it is impossible to solve the puzzle.

Henry's Hamburger Heaven orders its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two, or three meat patties, and any collection of condiments. How many different kinds of hamburgers can be ordered? $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 256\qquad \textbf{(C)}\ 768\qquad \textbf{(D)}\ 40,\!320\qquad \textbf{(E)}\ 120,\!960$

Let $ABC$ be an isosceles triangle with $AB = AC$. Suppose $P,Q,R$ are points on segments $AC, AB, BC$ respectively such that $AP = QB$, $\angle PBC = 90^\circ - \angle BAC$ and $RP = RQ$. Let $O_1, O_2$ be the circumcenters of $\triangle APQ$ and $\triangle CRP$. Prove that $BR = O_1O_2$. [i]Proposed by Atul Shatavart Nadig[/i]

Given a table in a form of the regular $1000$-gon with sidelength $1$. A Beetle initially is in one of its vertices. All $1000$ vertices are numbered in some order by numbers $1$, $2$, $\ldots$, $1000$ so that initially the Beetle is in the vertex $1$. The Beetle can move only along the edges of $1000$-gon and only clockwise. He starts to move from vertex $1$ and he is moving without stops until he reaches vertex $2$ where he has a stop. Then he continues his journey clockwise from vertex $2$ until he reaches the vertex $3$ where he has a stop, and so on. The Beetle finishes his journey at vertex $1000$. Find the number of ways to enumerate all vertices so that the total length of the Beetle's journey is equal to $2017$.

Points $D$ and $E$ are chosen on the exterior of $\vartriangle ABC$ such that $\angle ADC = \angle BEC = 90^o$. If $\angle ACB = 40^o$, $AD = 7$, $CD = 24$, $CE = 15$, and $BE = 20$, what is the measure of $\angle ABC $ in,degrees?

Call arrangement of $m$ number on the circle [b]$m$-negative[/b], if all numbers are equal to $-1$. On the first step Andrew chooses one number on circle and multiplies it by $-1$. All other steps are similar: instead of the next number(clockwise) he writes its product with the number, written on the previous step. Prove that if $n$-negative arrangement in $k$ steps becomes $n$-negative again, then $(2^n - 1)$-negative after $(2^k - 1)$ steps becomes $(2^n - 1)$-negative again.

A quadruple $(a; b; c; d)$ of positive integers with $a \leq b \leq c \leq d$ is called good if we can colour each integer red, blue, green or purple, in such a way that $i$ of each $a$ consecutive integers at least one is coloured red; $ii$ of each $b$ consecutive integers at least one is coloured blue; $iii$ of each $c$ consecutive integers at least one is coloured green; $iiii$ of each $d$ consecutive integers at least one is coloured purple. Determine all good quadruples with $a = 2.$

The following sequence of letters is written on a board: \[ \text{TNOTNOTNO...TNOTN} \] where the sequence repeats 2024 times. At each step, one of the following operations can be performed: 1. Take two different adjacent letters and replace them with two copies of the missing letter. 2. Take three consecutive identical letters and remove them. After a certain number of steps, only two identical letters remain. Determine which letter it is possible to reach.

$p$ parallel lines are drawn in the plane and $q$ lines perpendicular to them are also drawn. How many rectangles are bounded by the lines?

Three points $K, L$ and $M$ are given in the plane. It is known that they are the midpoints of three successive sides of an erased quadrilateral and that these three sides have the same length. Reconstruct the quadrilateral.

Let $ABC$ be a isosceles triangle, with $AB=AC$. A line $r$ that pass through the incenter $I$ of $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. Let $F$ and $G$ be points on $BC$ such that $BF=CE$ and $CG=BD$. Show that the angle $\angle FIG$ is constant when we vary the line $r$.

Solve the following system of equations over the real numbers: $2x_1 = x_5 ^2 - 23$ $4x_2 = x_1 ^2 + 7$ $6x_3 = x_2 ^2 + 14$ $8x_4 = x_3 ^2 + 23$ $10x_5 = x_4 ^2 + 34$

Find all triples (p,a,m); p is a prime number, $a,m\in \mathbb{N}$, which satisfy: $a\leq 5p^2$ and $(p-1)!+a=p^m$.