A flea jumps in a straight numbered line. It jumps first from point $0$ to point $1$. Afterwards, if its last jump was from $A$ to $B$, then the next jump is from $B$ to one of the points $B + (B - A) - 1$, $B + (B - A)$, $B + (B-A) + 1$. Prove that if the flea arrived twice at the point $n$, $n$ positive integer, then it performed at least $\lceil 2\sqrt n\rceil$ jumps.

An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4.$ One vertex of the triangle is $(0,1),$ one altitude is contained in the $y$-axis, and the length of each side is $\sqrt{\frac mn},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Two nonhorizontal, non vertical lines in the $xy$-coordinate plane intersect to form a $45^{\circ}$ angle. One line has slope equal to $6$ times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines? $\textbf{(A)}\ \frac16 \qquad\textbf{(B)}\ \frac23 \qquad\textbf{(C)}\ \frac32 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 6$

[b](a) [/b] Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$. [b](b)[/b] Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$.

Find the number of finite sequences $ \{a_1,a_2,\ldots,a_{2n\plus{}1}\}$, formed with nonnegative integers, for which $ a_1\equal{}a_{2n\plus{}1}\equal{}0$ and $ |a_k \minus{}a_{k\plus{}1}|\equal{}1$, for all $ k\in\{1,2,\ldots,2n\}$.

If $ A$ and $ B$ are fixed points on a given circle and $ XY$ is a variable diameter of the same circle, determine the locus of the point of intersection of lines $ AX$ and $ BY$. You may assume that $ AB$ is not a diameter.

$ 2n\plus{}1$ lines are drawn in the plane, in such a way that every 3 lines define a triangle with no right angles. What is the maximal possible number of acute triangles that can be made in this way?

Let $ABC$ be a triangle with $|AB|=|AC|$. $D$ is the midpoint of $[BC]$. $E$ is the foot of the altitude from $D$ to $AC$. $BE$ cuts the circumcircle of triangle $ABD$ at $B$ and $F$. $DE$ and $AF$ meet at $G$. Prove that $|DG|=|GE|$

For every pair of reals $0 < a < b < 1$, we define sequences $\{x_n\}_{n \ge 0}$ and $\{y_n\}_{n \ge 0}$ by $x_0 = 0$, $y_0 = 1$, and for each integer $n \ge 1$: \begin{align*} x_n & = (1 - a) x_{n - 1} + a y_{n - 1}, \\ y_n & = (1 - b) x_{n - 1} + b y_{n - 1}. \end{align*} The [i]supermean[/i] of $a$ and $b$ is the limit of $\{x_n\}$ as $n$ approaches infinity. Over all pairs of real numbers $(p, q)$ satisfying $\left (p - \textstyle\frac{1}{2} \right)^2 + \left (q - \textstyle\frac{1}{2} \right)^2 \le \left(\textstyle\frac{1}{10}\right)^2$, the minimum possible value of the supermean of $p$ and $q$ can be expressed as $\textstyle\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m + n$. [i]Proposed by Lewis Chen[/i]

What is the minimum value of $ f(x) \equal{} |x \minus{} 1| \plus{} |2x \minus{} 1| \plus{} |3x \minus{} 1| \plus{} \cdots \plus{} |119x \minus{} 1|$? $ \textbf{(A)}\ 49 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 53$

In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0), (0,3), (3,3), (3,1), (5,1)$ and $(5,0)$. The slope of the line through the origin that divides the area of this region exactly in half is [asy] size(200); Label l; l.p=fontsize(6); xaxis("$x$",0,6,Ticks(l,1.0,0.5),EndArrow); yaxis("$y$",0,4,Ticks(l,1.0,0.5),EndArrow); draw((0,3)--(3,3)--(3,1)--(5,1)--(5,0)--(0,0)--cycle,black+linewidth(2));[/asy] $ \textbf{(A)}\ \frac{2}{7} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{2}{3} \qquad\textbf{(D)}\ \frac{3}{4} \qquad\textbf{(E)}\ \frac{7}{9} $

How many hours does it take a train traveling at an average rate of $ 40$ mph between stops to travel $ a$ miles it makes $ n$ stops of $ m$ minutes each? $ \textbf{(A)}\ \frac{3a\plus{}2mn}{120} \qquad \textbf{(B)}\ 3a\plus{}2mn \qquad \textbf{(C)}\ \frac{3a\plus{}2mn}{12} \qquad \textbf{(D)}\ \frac{a\plus{}mn}{40} \qquad \textbf{(E)}\ \frac{a\plus{}40mn}{40}$

Find all the functions $f:[0,1]\rightarrow \mathbb{R}$ for which we have: \[|x-y|^2\le |f(x)-f(y)|\le |x-y|,\] for all $x,y\in [0,1]$.

Let the major axis of an ellipse be $AB$, let $O$ be its center, and let $F$ be one of its foci. $P$ is a point on the ellipse, and $CD$ a chord through $O$, such that $CD$ is parallel to the tangent of the ellipse at $P$. $PF$ and $CD$ intersect at $Q$. Compare the lengths of $PQ$ and $OA$.

A cubic polynomial $f$ with a positive leading coefficient has three different positive zeros. Show that $f'(a)+ f'(b)+ f'(c) > 0$.

Let $P$ be a point in a square whose side are mirror. A ray of light comes from $P$ and with slope $\alpha$. We know that this ray of light never arrives to a vertex. We make an infinite sequence of $0,1$. After each contact of light ray with a horizontal side, we put $0$, and after each contact with a vertical side, we put $1$. For each $n\geq 1$, let $B_{n}$ be set of all blocks of length $n$, in this sequence. a) Prove that $B_{n}$ does not depend on location of $P$. b) Prove that if $\frac{\alpha}{\pi}$ is irrational, then $|B_{n}|=n+1$.

A line through the origin passes through the curve whose equation is $5y=2x^2-9x+10$ at two points whose $x-$coordinates add up to $77.$ Find the slope of the line.

Complex numbers $a$, $b$ and $c$ are the zeros of a polynomial $P(z) = z^3+qz+r$, and $|a|^2+|b|^2+|c|^2=250$. The points corresponding to $a$, $b$, and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h$. Find $h^2$.

The points that satisfy the system $ x \plus{} y \equal{} 1,\, x^2 \plus{} y^2 < 25,$ constitute the following set: $ \textbf{(A)}\ \text{only two points} \qquad \\ \textbf{(B)}\ \text{an arc of a circle}\qquad \\ \textbf{(C)}\ \text{a straight line segment not including the end\minus{}points}\qquad \\ \textbf{(D)}\ \text{a straight line segment including the end\minus{}points}\qquad \\ \textbf{(E)}\ \text{a single point}$

The graph of $ 2x^2 \plus{} xy \plus{} 3y^2 \minus{} 11x \minus{} 20y \plus{} 40 \equal{} 0$ is an ellipse in the first quadrant of the $ xy$-plane. Let $ a$ and $ b$ be the maximum and minimum values of $ \frac {y}{x}$ over all points $ (x, y)$ on the ellipse. What is the value of $ a \plus{} b$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \sqrt {10} \qquad \textbf{(C)}\ \frac72 \qquad \textbf{(D)}\ \frac92 \qquad \textbf{(E)}\ 2\sqrt {14}$

Given points $P(-1,-2)$ and $Q(4,2)$ in the $xy$-plane; point $R(1,m)$ is taken so that $PR+RQ$ is a minimum. Then $m$ equals: $\textbf{(A) }-\tfrac35\qquad \textbf{(B) }-\tfrac25\qquad \textbf{(C) }-\tfrac15\qquad \textbf{(D) }\tfrac15\qquad \textbf{(E) }\text{either }-\tfrac15\text{ or }\tfrac15$

Find the point in the closed unit disc $D=\{ (x,y) | x^2+y^2\le 1 \}$ at which the function $f(x,y)=x+y$ attains its maximum .

Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $P$ and the point $\left(\frac58, \frac38 \right)$ is greater than $\frac12$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

A point $ (x,y)$ in the plane is called a lattice point if both $ x$ and $ y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to $ \textbf{(A)}\ 4.0\qquad \textbf{(B)}\ 4.2\qquad \textbf{(C)}\ 4.5\qquad \textbf{(D)}\ 5.0\qquad \textbf{(E)}\ 5.6$

On the coordinate plane, is there a line family of infinitely many lines $l_1,l_2,\cdots,l_n,\cdots$, satisfying the following? (1) Point$(1,1)\in l_n$ for all $n\in \mathbb{Z}_{+}$. (2) For all $n\in \mathbb{Z}_{+}$,$k_{n+1}=a_n-b_n$, where $k_{n+1}$ is the slope of $l_{n+1}$, $a_n,b_n$ are intercepts of $l_n$ on $x$-axis, $y$-axis. (3) $k_nk_{n+1}\geq0$ for all $n\in \mathbb{Z}_{+}$.