Crystal has a running course marked out for her daily run. She starts this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles is this last portion of her run? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ \sqrt2 \qquad \textbf{(C)}\ \sqrt3 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 2\sqrt2$

Let $AD, BE$ and $CF$ be the medians of triangle $ABC$. The points $X$ and $Y$ are the reflections of $F$ about $AD$ and $BE$, respectively. Prove that the circumcircles of triangles $BEX$ and $ADY$ are concentric.

Which terms must be removed from the sum \[ \frac12 \plus{} \frac14 \plus{} \frac16 \plus{} \frac18 \plus{} \frac1{10} \plus{} \frac1{12} \]if the sum of the remaining terms is equal to $ 1$? $ \textbf{(A)}\ \frac14\text{ and }\frac18 \qquad \textbf{(B)}\ \frac14\text{ and }\frac1{12} \qquad \textbf{(C)}\ \frac18\text{ and }\frac1{12} \qquad \textbf{(D)}\ \frac16\text{ and }\frac1{10} \qquad \textbf{(E)}\ \frac18\text{ and }\frac1{10}$

[b]Q11.[/b] Let be given a sequense $a_1=5, \; a_2=8$ and $a_{n+1}=a_n+3a_{n-1}, \qquad n=1,2,3,...$ Calculate the greatest common divisor of $a_{2011}$ and $a_{2012}$.

Find all integer solutions to $2 x^4 + 1 = y^2.$

Let $B_1$ be the foot of the altitude from the vertex $B$ in the acute-angled $\triangle ABC$. Let $D$ be the midpoint of side $AB$, and $O$ be the circumcentre of $\triangle ABC$. Line $B_1D$ meets line $CO$ at $E$. Prove that the points $B, C, B_1$, and $E$ lie on a circle.

Suppose we have a hexagonal grid in the shape of a hexagon of side length $4$ as shown at left. Define a “chunk” to be four tiles, two of which are adjacent to the other three, and the other two of which are adjacent to just two of the others. The three possible rotations of these are shown at right. [img]https://cdn.artofproblemsolving.com/attachments/a/7/147d8aa2c149918ab855db1e945d389433446a.png[/img] In how many ways can we choose a chunk from the grid?

Let $ABC$ be a triangle, let $M$ be the midpoint of $AB$, and let $N$ be the midpoint of $CM$. Let $X$ be a point satisfying both $\angle XMC = \angle MBC$ and $\angle XCM = \angle MCB$ such that $X$ and $B$ lie on opposite sides of $CM$. Let $\omega$ be the circumcircle of triangle $AMX$. $(a)$ Show that $CM$ is tangent to $\omega$. $(b)$ Show that the lines $NX$ and $AC$ intersect on $\omega$

In a quadrilateral $ABCD$ points $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$, respectively. The line through $M$ and $N$ meets $AB$ and $CD$ at $M'$ and $N'$, respectively. Prove that if $MM' = NN'$, then $AD // BC$.

Let $f$ be an increasing mapping from the family of subsets of a given finite set $H$ into itself, i.e. such that for every $X \subseteq Y\subseteq H$ we have $f (X )\subseteq f (Y )\subseteq H .$ Prove that there exists a subset $H_{0}$ of $H$ such that $f (H_{0}) = H_{0}.$

In triangle $\, ABC, \,$ angle $\,A\,$ is twice angle $\,B,\,$ angle $\,C\,$ is obtuse, and the three side lengths $\,a,b,c\,$ are integers. Determine, with proof, the minimum possible perimeter.

Prove that for any sequence $a_1, a_2, \ldots , a_{2002}$ of non-negative integers written in the usual decimal notation with $a_1 > 0$ there exists an integer $n$ such that $n^2$ starts with digits $a_1, a_2, \ldots , a_{2002}$ (in this order).

Determine all integers $s \ge 4$ for which there exist positive integers $a$, $b$, $c$, $d$ such that $s = a+b+c+d$ and $s$ divides $abc+abd+acd+bcd$. [i]Proposed by Ankan Bhattacharya and Michael Ren[/i]

Danielle writes a sign '+' or '-' in each of the next $64$ spaces: $$\_\_1 \_\_2 \_\_3 \_\_4 \text{ }.... \text{ }\_\_63 \_\_64=2024$$ such that the equality holds. Find the largest number of negative signs Danielle can use.

Older television screens have an aspect ratio of $ 4: 3$. That is, the ratio of the width to the height is $ 4: 3$. The aspect ratio of many movies is not $ 4: 3$, so they are sometimes shown on a television screen by 'letterboxing' - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of $ 2: 1$ and is shown on an older television screen with a $ 27$-inch diagonal. What is the height, in inches, of each darkened strip? [asy]unitsize(1mm); defaultpen(linewidth(.8pt)); filldraw((0,0)--(21.6,0)--(21.6,2.7)--(0,2.7)--cycle,grey,black); filldraw((0,13.5)--(21.6,13.5)--(21.6,16.2)--(0,16.2)--cycle,grey,black); draw((0,2.7)--(0,13.5)); draw((21.6,2.7)--(21.6,13.5));[/asy]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2.25 \qquad \textbf{(C)}\ 2.5 \qquad \textbf{(D)}\ 2.7 \qquad \textbf{(E)}\ 3$

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(2m+2n)=f(m)f(n)$ for all natural numbers $m,n$.

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(f(x)^2+f(y^2))=(x-y)f(x-f(y))$

Construct on the real projective plane a continuous curve, consisting of simple points, which is not a straight line and is intersected in a single point by every tangent and every secant of a given conic. [i]F. Karteszi[/i]

Given a segment $AB$ in the plane, choose on it a point $M$ different from $A$ and $B$. Two equilateral triangles $\triangle AMC$ and $\triangle BMD$ in the plane are constructed on the same side of segment $AB$. The circumcircles of the two triangles intersect in point $M$ and another point $N$. (The [b]circumcircle[/b] of a triangle is the circle that passes through all three of its vertices.) (a) Prove that lines $AD$ and $BC$ pass through point $N$. (b) Prove that no matter where one chooses the point $M$ along segment $AB$, all lines $MN$ will pass through some fixed point $K$ in the plane.

Find all prime numbers $ p,q$ less than 2005 and such that $ q|p^2 \plus{} 4$, $ p|q^2 \plus{} 4$.

Show that there is some rational number in the interval $(0,1)$ that can be expressed as a sum of $2021$ reciprocals of positive integers, but cannot be expressed as a sum of $2020$ reciprocals of positive integers.

Find all primes $p \ge 3$ such that $p- \lfloor p/q \rfloor q$ is a square-free integer for any prime $q<p$.

If permutations of the numbers $2, 3,4,..., 102$ are denoted by $a_i,a_2, a_3,...,a_{101}$, find all such permutations in which $a_k$ is divisible by $k$ for all $k$.

A function $ f: R^3\rightarrow R$ for all reals $ a,b,c,d,e$ satisfies a condition: \[ f(a,b,c)\plus{}f(b,c,d)\plus{}f(c,d,e)\plus{}f(d,e,a)\plus{}f(e,a,b)\equal{}a\plus{}b\plus{}c\plus{}d\plus{}e\] Show that for all reals $ x_1,x_2,\ldots,x_n$ ($ n\geq 5$) equality holds: \[ f(x_1,x_2,x_3)\plus{}f(x_2,x_3,x_4)\plus{}\ldots \plus{}f(x_{n\minus{}1},x_n,x_1)\plus{}f(x_n,x_1,x_2)\equal{}x_1\plus{}x_2\plus{}\ldots\plus{}x_n\]

For integer $n\geq4$, find the smallest integer $f(n)$, such that for any positive integer $m$, in any subset with $f(n)$ elements of the set $\{m, m+1, \cdots, m+n-1\}$ there are at least three elements that are relatively prime .