A fox stands in the centre of the field which has the form of an equilateral triangle, and a rabbit stands at one of its vertices. The fox can move through the whole field, while the rabbit can move only along the border of the field. The maximal speeds of the fox and rabbit are equal to $u$ and $v$, respectively. Prove that: (a) If $2u>v$, the fox can catch the rabbit, no matter how the rabbit moves. (b) If $2u\le v$, the rabbit can always run away from the fox.

Consider a sequence given by $a_n=a_{n-1}+3a_{n-2}+a_{n-3}$, where $a_0=a_1=a_2=1$. What is the remainder of $a_{2013}$ divided by $7$?

A triangle $ABC$ has sides $BC = a, CA = b, AB = c$ with $a < b < c$ and area $S$. Determine the largest number $u$ and the least number $v$ such that, for every point $P$ inside $\triangle ABC$, the inequality $u \le PD + PE + PF \le v$ holds, where $D,E, F$ are the intersection points of $AP,BP,CP$ with the opposite sides.

Suppose 100 people are gathered around at a park, each with an envelope with their name on it (all their names are distinct). Then, the envelopes are uniformly and randomly permuted between the people. If $N$ is the number of people who end up with their original envelope, find the expected value of $N^5$. [i]Proposed by Michael Duncan[/i]

Let $\mathbb Z$ be the set of all integers. Find all functions $f:\mathbb Z\to\mathbb Z$ satisfying the following conditions: 1. $f(f(x))=xf(x)-x^2+2$ for all $x\in\mathbb Z$; 2. $f$ takes all integer values. [i](I. Voronovich)[/i]

Given in the plane is a lattice and a grid rectangle with sides parallel to the coordinate axes. We divide the rectangle into grid triangles with area $\frac12$. Prove that the number of right angled triangles is at least twice as much as the shorter side of the rectangle. (A grid polygon is a polygon such that both coordinates of each vertex is an integer.)

A natural number $k > 1$ is called [i]good[/i] if there exist natural numbers $$a_1 < a_2 < \cdots < a_k$$ such that $$\dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots + \dfrac{1}{\sqrt{a_k}} = 1$$. Let $f(n)$ be the sum of the first $n$ [i][good[/i] numbers, $n \geq$ 1. Find the sum of all values of $n$ for which $f(n+5)/f(n)$ is an integer.

A polygon can be transformed into a new polygon by making a straight cut, which creates two new pieces each with a new edge. One piece is then turned over and the two new edges are reattached. Can repeated transformations of this type turn a square into a triangle?

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$x^2f(x)+yf(y^2)=f(x+y)f(x^2-xy+y^2)$$ for all $x,y\in\mathbb{R}$.

Let $a_1,a_2 \cdots a_{2n}$ be an arithmetic progression of positive real numbers with common difference $d$. Let $(i)$ $\sum_{i=1}^{n}a_{2i-1}^2 =x$ $(ii)$ $\sum _{i=1}^{n}a_{2i}^2=y$ $(iii)$ $a_n+a_{n+1}=z$ Express $d$ in terms of $x,y,z,n$

A regular triangular pyramid $ABCD$ with the base side $AB=a$ and the lateral edge $AD=b$ is given. Let $M$ and $N$ be the midpoints of $AB$ and $CD$ respectively. A line $\alpha$ through $MN$ intersects the edges $AD$ and $BC$ at $P$ and $Q$, respectively. (a) Prove that $AP/AD=BQ/BC$. (b) Find the ratio $AP/AD$ which minimizes the area of $MQNP$.

In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $ v$, $ w$, $ x$, $ y$, and $ z$. Find $ y \plus{} z$. $ \textbf{(A)}\ 43 \qquad \textbf{(B)}\ 44 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 46 \qquad \textbf{(E)}\ 47$ [asy]unitsize(10mm); defaultpen(linewidth(1pt)); for(int i=0; i<=3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$25$",(0.5,0.5)); label("$z$",(1.5,0.5)); label("$21$",(2.5,0.5)); label("$18$",(0.5,1.5)); label("$x$",(1.5,1.5)); label("$y$",(2.5,1.5)); label("$v$",(0.5,2.5)); label("$24$",(1.5,2.5)); label("$w$",(2.5,2.5));[/asy]

On a examination of $n$ questions a student answers correctly $15$ of the first $20$. Of the remaining questions he answers one third correctly. All the questions have the same credit. If the student's mark is $50\%$, how many different values of $n$ can there be? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ \text{the problem cannot be solved} $

A powderman set a fuse for a blast to take place in $ 30$ seconds. He ran away at a rate of $ 8$ yards per second. Sound travels at the rate of $ 1080$ feet per second. When the powderman heard the blast, he had run approximately: $ \textbf{(A)}\ 200 \text{ yd.} \qquad\textbf{(B)}\ 352 \text{ yd.} \qquad\textbf{(C)}\ 300 \text{ yd.} \qquad\textbf{(D)}\ 245 \text{ yd.} \qquad\textbf{(E)}\ 512 \text{ yd.}$

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following conditions: a. $x+f(y+f(x))=y+f(x+f(y)) \quad \forall x,y \in \mathbb{R}$ b. The set $I=\left\{\frac{f(x)-f(y)}{x-y}\mid x,y\in \mathbb{R},x\neq y \right\}$ is an interval. [i]Proposed by Navid Safaei[/i]

The diagram below shows a shaded region bounded by two concentric circles where the outer circle has twice the radius of the inner circle. The total boundary of the shaded region has length $36\pi$. Find $n$ such that the area of the shaded region is $n\pi$. [img]https://cdn.artofproblemsolving.com/attachments/4/5/c9ffdc41c633cc61127ef585a45ee5e6c0f88d.png[/img]

Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}$, $BC = CD = 43$, and $\overline{AD} \perp \overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. GIven that $OP = 11$, the length $AD$ can be written in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m + n$? $\textbf{(A)}\: 65\qquad\textbf{(B)}\: 132\qquad\textbf{(C)}\: 157\qquad\textbf{(D)}\: 194\qquad\textbf{(E)}\: 215$

It is easy to prove that in a right triangle the sum of the radii of the incircle and three excircles is equal to the perimeter. Prove that the opposite statement is also true. [i]Proposed by I. Weinstein[/i]

Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point inside $ABC$, so let the points $D, E, F$ be on $BC, AC, AB$ respectively so that the Miquel point of $DEF$ with respect to $ABC$ is $P$. Let the reflections of $D, E, F$ over the midpoints of the sides that they lie on be $R, S, T$. Let the Miquel point of $RST$ with respect to the triangle $ABC$ be $Q$. Show that $OP = OQ$. [i]Proposed by Yang Liu[/i]

Let $ABCD$ be a trapezoid with $\angle A = \angle B = 90^o$ and a point $E$ lies on the segment $CD$. Denote $(\omega)$ as incircle of triangle $ABE$ and it is tangent to $AB,AE,BE$ respectively at $P, F,K$. Suppose that $KF$ cuts $BC,AD$ at $M,N$ and $PM,PN$ cut $(\omega)$ at $H, T$. Prove that $PH = PT$.

At the faculty of mathematics study $40$ boys and $10$ girls. Every girl acquaintance with all boys, who older than her, or tall(higher) than her. Prove that there exist two boys such that the sets of acquainted-girls of the boys are same.

Find the highest power of 2 that is a factor of the number $$ L_n = (n+1)(n+2)... 2n,$$ where $n$is a natural number.

Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to him in the circle. Then each person computes and announces the average of the numbers of his two neighbors. The figure shows the average announced by each person ([u]not[/u] the original number the person picked). The number picked by the person who announced the average $6$ was [asy] label("(1)", (0,.9)); label("(2)", (.4,.65)); label("(3)", (.8,.25)); label("(4)", (.8,-.2)); label("(5)", (.4,-.65)); label("(6)", (0,-.9)); label("(7)", (-.4,-.65)); label("(8)", (-.8,-.2)); label("(9)", (-.8,.25)); label("(10)", (-.4,.65)); [/asy] $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ \text{not uniquely determined from the given information} $

Let $a,b,c$ be positive real numbers such that $ab+bc+ca=1$. Prove that \[\sqrt[4]{\frac{\sqrt{3}}{a}+6\sqrt{3}b}+\sqrt[4]{\frac{\sqrt{3}}{b}+6\sqrt{3}c}+\sqrt[4]{\frac{\sqrt{3}}{c}+6\sqrt{3}a}\le\frac{1}{abc}\] When does inequality hold?

Triangle $ ABC$ is inscribed in circle $ \omega$. The tangent lines to $ \omega$ at $ B$ and $ C$ meet at $ T$. Point $ S$ lies on ray $ BC$ such that $ AS \perp AT$. Points $ B_1$ and $ C_1$ lie on ray $ ST$ (with $ C_1$ in between $ B_1$ and $ S$) such that $ B_1T \equal{} BT \equal{} C_1T$. Prove that triangles $ ABC$ and $ AB_1C_1$ are similar to each other.